&Digraphs\ ;

\n[BM06] Boyer, J. M.\ and Myrvold, W. J.,\n On the Cutting Edge: Simplified\ O(n) Planarity by Edge\n Addition,\n in Gr\ aph Algorithms and Applications 5,\n {WORLD} {S\ CIENTIFIC}\n (2006),\n 241--273.\n

\n[CK86] Calder\ bank, R. and Kantor, W. M.,\n The geometry of two-weig\ ht codes,\n Bull. London Math. Soc.,\n 18 (2)\n (1986),\n 97--122.\n

\n[Gab00] Gabow, H. N.,\n Path-based depth-first search for strong and biconnected\n \ components,\n Information Processing Letters,\n 74 (34)\n (\ 2000),\n 107 - 114.\ \n

\n[JK07] Junttila, T. and\ Kaski, P. (Applegate, D., Brodal, G. S., Panario,\ D. and Sedgewick, R., Eds.),\n Engineering an effi\ cient canonical labeling tool for large and\n sparse graphs,\n \ in Proceedings of the Ninth Workshop on Algorithm Eng\ ineering and\n Experiments and the Fourth Workshop on Analytic Algorithms and\ \n Combinatorics,\n SIAM\n\ (2007),\n 135--149\ .\n

\n[MP14] McKay, B. D. an\ d Piperno, A.,\n Practical graph isomorphism, II,\ \n Journal of Symbolic Computation,\n 60 (0)\n (2014),\n 94 - 112.\n

\n[vLS81\ ] van Lint, J. H. and Schrijver, A.,\n <\ i class='BibTitle'>Construction of strongly regular graphs, two-weight codes a\ nd\n partial geometries by finite fields,\n Combinatorica,\n 1 (1)\n (1981),\n 63--73.\n

1 \n The Digraphs\ package\n \n

1.1 \ Introduction\n\n

\n
1.1-1 Def\ initions\n\n

2 Installing Digraphs\n
2.1 For those in a hurry\n\n
\n
2.2 Optional package\ dependencies\n\n
\n
  2.2-1 The Grape package\n\n
\n
2.\ 3 Compiling the kernel module\n\n\n
&nb\ sp;2.4 Rebuilding the documentation\n\n
\n
  2.4-1 DigraphsMakeDoc\n
\n
2.5 Testing your installation\n\n
\n
  \ 2.5-1 DigraphsTestInstall\n
  2.5-2 DigraphsTestStandard\ \n
  2.5-3 DigraphsTestExtreme\n
\n
\n", 3 := "
3 Creating digraphs\n
3.1 Creating digraphs\n\n
\n
<\ span class=\"nocss\">  3.1-1 IsDigraph\n
  3.1-2 Is\ MutableDigraph\n
\   3.1-3 IsImmutab\ leDigraph\n
\ ; 3.1-4 IsCayleyDigrap\ h\n
  <\ /span>3.1-5 IsDigraphWithAdjacency\ Function\n
\ 3.1-6 DigraphByOutNei\ ghboursType\n
&nb\ sp; 3.1-7 Digraph<\ /span>\n
  \ 3.1-8 DigraphByAdjacencyMatrix\ \n
  3.1-9 DigraphByEdges\n\
  3.1-10 EdgeOrbitsDigraph\n
  3.1-11 DigraphByInNeighbours\n
  3.1-12 CayleyDigraph\n
\n<\ div class=\"ContSect\"> 3.2 Ch\ anging representations\n\n
\n
  3.2-1 AsBinaryRelation\n
  3.2-2 AsDigraph\n  3.2-3 Graph\n
  3.2-4 As\ Graph\n
&nb\ sp;3.2-5 AsTransformation\n
\n
3.3 \ New digraphs from old\n\n
\n
&nb\ sp;3.3-1 DigraphImmutableCo\ py\n
  \ 3.3-2 DigraphImmutableCopyI\ fImmutable\n
&nbs\ p; 3.3-3 InducedSubdig\ raph\n
&nbs\ p;3.3-4 ReducedDigraph<\ /span>\n
  \ 3.3-5 MaximalSymmetricSubdigraph\n
  3.3-6 MaximalAntiSymmetricSubdi\ graph\n
&nb\ sp;3.3-7 UndirectedSpanning\ Forest\n
&n\ bsp;3.3-8 QuotientDigraph\n
  3.3-9 DigraphReverse\ \n
  3.3-10 DigraphDual\n
  3.3-11 DigraphSymmetricClosure\n
  3.3-12 DigraphTransitiveClosure\n
  3.3-13 DigraphTransitiveReduction\n
  3.3-14 DigraphAddVertex\n
  3.3-15 DigraphAddVertices\n
  3.3-16 DigraphAddEdge\n
  3.3-17 DigraphAddEdgeOrbit\n\
  3.3-18 DigraphAddEdges\n
<\ span class=\"nocss\">  3.3-19 DigraphRemoveVertex\n
  3.3-20 DigraphRemoveVertices\n
  3.3-21 DigraphRemoveEdge\n
  3.\ 3-22 DigraphRemoveEdgeOrbit\n
  3.\ 3-23 DigraphRemoveEdges\n
  3.3-2\ 4 DigraphRemoveLoops\n
  3.3-25 Di\ graphRemoveAllMultipleEdges\n
  3.\ 3-26 DigraphReverseEdges\n
  3.3-2\ 7 DigraphDisjointUnion\n
  3.3-28 \ DigraphEdgeUnion\n
  3.3-29 Digra\ phJoin\n
&n\ bsp;3.3-30 DigraphCartesian\ Product\n
&\ nbsp;3.3-31 DigraphDirectPr\ oduct\n
&nb\ sp;3.3-32 DigraphCartesianP\ roductProjections\n
  3.3-33 Digra\ phDirectProductProjections\n
  3.3\ -34 LineDigraph\n
  3.3-35 LineUnd\ irectedDigraph\n
\   3.3-36 DoubleDi\ graph\n
&nb\ sp;3.3-37 BipartiteDoubleDi\ graph\n
&nb\ sp;3.3-38 DigraphAddAllLoop\ s\n
  <\ /span>3.3-39 DistanceDigraph\n
  <\ a href=\"chap3_mj.html#X86F9CCEA839ABC48\">3.3-40 DigraphClosure\n<\ span class=\"ContSS\">
  3.3-41 DigraphMycielskian\n
\n
3.4 Random digraphs\n\n
\n
  3.4-1 RandomDigraph\n
  3.4-2 RandomMultiDigraph\n
  3.4-3 RandomTournament\n<\ br />  3.4-4 RandomLattice\n
\n
3.5 Standard examples\n\n
\n
  3.5-1 ChainDigraph\n
  3.5-2\ CompleteDigraph\n
  3.5-3 Comple\ teBipartiteDigraph\n
  3.5-4 Compl\ eteMultipartiteDigraph\n
  3.5-5 C\ ycleDigraph\n
&nb\ sp; 3.5-6 EmptyDigraph\ \n
  3.5-7 JohnsonDigraph\n
  3.5-8 PetersenGraph\n
  3.5-9 GeneralisedPetersenGraph\n
\n
\n", 4 := "
4 Operators\n
4.1 Operators for digraphs\n\n
\n
  4.1-1 IsSubdigraph\n
  4.1-2 Is\ UndirectedSpanningTree\n
\n
\n", 5 := "
5 Attributes and operations\n
5.1 Vertices and\ edges\n\n
\n
  5.1-1 DigraphVertices\n
  5.1-2 DigraphNrVertices\n
  5.1-3 DigraphEdges\n
  5.1-4 Di\ graphNrEdges\n
&n\ bsp; 5.1-5 DigraphSink\ s\n
  <\ /span>5.1-6 DigraphSources\n
  5.1-7 DigraphTopologicalSort\n
  5.1-8 DigraphVertexLabel\n<\ span class=\"ContSS\">
  5.1-9 DigraphVertexLabels\n
  5.1-10 DigraphEdgeLabel\n
  5.1-11 DigraphEdgeLabels\n
  5.1-12 DigraphInEdges\n\
  5.1-13 DigraphOutEdges\n
<\ span class=\"nocss\">  5.1-14 IsDigraphEdge\n
  5.1\ -15 IsMatching\n
\   5.1-16 DigraphM\ aximalMatching\n
\   5.1-17 DigraphM\ aximumMatching\n
\n
5.2 Neighbours and degree\n\n
\n
  5.2-1 A\ djacencyMatrix\n
\   5.2-2 Character\ isticPolynomial\n
  5.2-3 BooleanA\ djacencyMatrix\n
\   5.2-4 DigraphAd\ jacencyFunction\n
  5.2-5 DigraphR\ ange\n
&nbs\ p;5.2-6 OutNeighbours\n
  <\ a href=\"chap5_mj.html#X85C7AA5A81DA6E11\">5.2-7 InNeighbours\n
  5.2-8 OutDegrees\n
  5.2-9 InDegrees\n
  5.2-10 OutDegreeOfVertex\n
  \ 5.2-11 OutNeighboursOfVertex\n
  5\ .2-12 InDegreeOfVertex\n
  5.2-13 \ InNeighboursOfVertex\n
  5.2-14 Di\ graphLoops\n
&nbs\ p; 5.2-15 PartialOrder\ DigraphMeetOfVertices\n
  5.2-16 D\ egreeMatrix\n
&nb\ sp; 5.2-17 LaplacianMa\ trix\n
\n
5.3 Reachability and connectivity\n\n
\n
  5.3-1 Dig\ raphDiameter\n
&n\ bsp; 5.3-2 DigraphShor\ testDistance\n
&n\ bsp; 5.3-3 DigraphShor\ testDistances\n
&\ nbsp; 5.3-4 DigraphLon\ gestDistanceFromVertex\n
  5.3-5 D\ igraphDistanceSet\n
  5.3-6 Digrap\ hGirth\n
&n\ bsp;5.3-7 DigraphOddGirth\n
  5.3-8 DigraphUndirectedGirth\n
  5.3-9 DigraphConnectedComponents\ \n
  5.3-10 DigraphConnectedCompon\ ent\n
  \ ;5.3-11 DigraphStronglyConn\ ectedComponents\n
  5.3-12 Digraph\ StronglyConnectedComponent\n
  5.3\ -13 DigraphBicomponents\n
  5.3-1\ 4 ArticulationPoints\n
  5.3-15 Br\ idges\n
&nb\ sp;5.3-16 StrongOrientation\ \n
  5.3-17 DigraphPeriod\n
  5.3-18 DigraphFloydWarshall\ \n
  5.3-19 IsReachable\n
  5.3-20 DigraphPath\n
  5.3-21 DigraphShortestPath\n\
  5.3-22 IteratorOfPaths\n
<\ span class=\"nocss\">  5.3-23 DigraphAllSimpleCircuits\n
  5.3-24 DigraphLongestSimpleCircuit\n\
  5.3-25 DigraphLayers\n
  5.3-26 DigraphDegeneracy\n
  5\ .3-27 DigraphDegeneracyOrdering\n
  5.3-28 HamiltonianPath\n
  5.3-\ 29 NrSpanningTrees\n
  5.3-30 Digr\ aphDijkstra\n
\n
5.4 Cayley graphs of groups\n\n
\n
  5.4-1 Gro\ upOfCayleyDigraph\n
  5.4-2 Genera\ torsOfCayleyDigraph\n
\n
5.5 Associated semigroups\ \n\n
\n
  5.\ 5-1 AsSemigroup\n
  5.5-2 AsSemigr\ oup\n
\n
\ 5.6 Planarity\n\n
\n
  5.6-1 KuratowskiPlanarSubdigrap\ h\n
  <\ /span>5.6-2 KuratowskiOuterPlanarS\ ubdigraph\n
\ ; 5.6-3 PlanarEmbeddin\ g\n
  <\ /span>5.6-4 OuterPlanarEmbedding\n
  5.6-5 SubdigraphHomeomorphicToK\ 23\n
\n
\n", 6 := "
6 Properties of digraphs\n
6.1 Edge propertie\ s\n\n
\n  6.1-1 DigraphHasLoops\n
  6.1-2 IsAntiSymmetricDigraph\n
  6.1-3 IsBipartiteDigraph\n
  6.\ 1-4 IsCompleteBipartiteDigraph\n
  6.1-5 IsCompleteDigraph\n
  6.1-6\ IsCompleteMultipartiteDigraph\n
  6.1-7 IsEmptyDigraph\n
  6.1-8 Is\ EquivalenceDigraph\n
  6.1-9 IsFun\ ctionalDigraph\n
\   6.1-10 IsMultiD\ igraph\n
&n\ bsp;6.1-11 IsReflexiveDigra\ ph\n
  \ 6.1-12 IsSymmetricDigraph\n
  6.1-13 IsTournament\ \n
  6.1-14 IsTransitiveDigraph\n<\ span class=\"ContSS\">
  6.1-15 IsPreorderDigraph\n
  6.1-16 IsPartialOrderDigraph\n
  6.1-17 IsMeetSemilatticeDigraph\n
\n
6.2 Regularity\n\n
\n
  6.2-1 IsInRegularDigraph\n
  6.2-2 IsOutRegularDigraph\n
  6.2-3 IsRegularDigraph\n\
  6.2-4 IsDistanceRegularDigraph\n
\n
6.3 Connectiv\ ity and cycles\n\n
\n
  6.3-1 IsAcyclicDigraph\n
  6.3-2 IsChainDigraph\n
  6.3-3 IsConnectedDigraph\n
  6.3-4 IsBiconnectedDigraph\n
  \ 6.3-5 IsBridgelessDigraph\n
  6.3-\ 6 IsStronglyConnectedDigraph\n
  6\ .3-7 IsAperiodicDigraph\n
  6.3-8\ IsDirectedTree\n
  6.3-9 IsUndire\ ctedTree\n
\ 6.3-10 IsEulerianDigr\ aph\n
  \ ;6.3-11 IsHamiltonianDigrap\ h\n
  <\ /span>6.3-12 IsCycleDigraph\n
\n
6.4 Planarity\n\n
\ \n
  6.4-1 IsPlanarDigraph\n
  6.4-2 IsOuterPlanarDigraph\n
\n
&nb\ sp;6.5 Homomorphisms and transformations\n\n
\n
  6.5-1 IsDigraphCore\n\
  6.5-2 IsEdgeTransitive\n
  6.5-3 IsVertexTransitive\n
\ \n
\n", 7 := "
7 Homomorphisms\n
7.1 Acting on digraphs\n\n
\n
  7.1-1 OnDigraphs\n
  7.1-2 OnMu\ ltiDigraphs\n
\n
7.2 Isomorphisms and canonical labellings\n\n
\n
  7.2-1 DigraphsUseNauty\n
  7.\ 2-2 AutomorphismGroup\n
  7.2-3 Bl\ issAutomorphismGroup\n
  7.2-4 Nau\ tyAutomorphismGroup\n
  7.2-5 Auto\ morphismGroup\n
&\ nbsp; 7.2-6 Automorphi\ smGroup\n
&\ nbsp;7.2-7 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-8 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-9 BlissCanonicalDi\ graph\n
&nb\ sp;7.2-10 DigraphGroup<\ /span>\n
  \ 7.2-11 DigraphOrbits\n<\ span class=\"ContSS\">
  7.2-12 DigraphOrbitReps\n
  7.2-13 DigraphSchreierVector\n
  7.2-14 DigraphStabilizer\n
  7.2-15 IsIsomorphicDigraph\n
  7.2-16 IsIsomorphicDigraph\n
  7.2-17 IsomorphismDigraphs\n
  7.2-18 IsomorphismDigraphs\n\
  7.2-19 RepresentativeOutNeighbours\n
  7.2-20 IsDigraphIsomorphism\n
  7.2-21 IsDigraphColouring\n
\n
7.3 Homomorphis\ ms of digraphs\n\n
\n
  7.3-1 HomomorphismDigraphsFinder\n
  7.3-2 DigraphHomomorphism\n
  7.3-3 HomomorphismsDigraphs\n
  7.3-4 DigraphMonomorphism\n
  7.3-5 MonomorphismsDigraphs\n
  7.3-6 DigraphEpimorphism\n  7.3-7 EpimorphismsDigraphs\n
  7.3-8 DigraphEmbedding\n
  7.3-9 EmbeddingsDigraphs\n
  7.3-\ 10 IsDigraphHomomorphism\n
  7.3-1\ 1 IsDigraphEmbedding\n
  7.3-12 Di\ graphsRespectsColouring\n
  7.3-1\ 3 GeneratorsOfEndomorphismMonoid\n
  7.3-14 DigraphColouring\n
  7.3\ -15 DigraphGreedyColouring\n
  7.3\ -16 DigraphWelshPowellOrder\n
  7.\ 3-17 ChromaticNumber\n
  7.3-18 Di\ graphCore\n
\n
\n", 8 := "
8 Finding cliques and independent sets\n\
8.1 F\ inding cliques\n\n
\n
  8.1-1 IsClique\n
  8.1-2 CliquesFinder\n
  8\ .1-3 DigraphClique\n
  8.1-4 Digra\ phMaximalCliques\n
  8.1-5 Clique\ Number\n
\n
8.2 Finding independent sets\n\n\
\n
  8.2-1 IsInde\ pendentSet\n
&nbs\ p; 8.2-2 DigraphIndepe\ ndentSet\n
\ 8.2-3 DigraphMaximalI\ ndependentSets\n
\n
\n", 9 := "
9 Visualising and IO\n
9.1 Visualising a digra\ ph\n\n
\n  9.1-1 Splash\n
  9.1-2\ DotDigraph\n
&nb\ sp; 9.1-3 DotSymmetric\ Digraph\n
&\ nbsp;9.1-4 DotPartialOrderD\ igraph\n
&n\ bsp;9.1-5 DotPreorderDigrap\ h\n
  <\ /span>9.1-6 DotHighlightedDigraph<\ /a>\n
\n
9.2\ Reading and writing digraphs to a file\n\n
\n
  9.2-1\ String\n
&\ nbsp;9.2-2 DigraphFromGraph\ 6String\n
&\ nbsp;9.2-3 Graph6String\ \n
  9.2-4 DigraphFile\n
  9.2-5 ReadDigraphs\n
  9.2-6 WriteDigraphs\n<\ br />  9.2-7 IteratorFromDigraphFile\n<\ br />  9.2-8 DigraphPlainTextLineEncoder\n
  9.2-9 TournamentLineDecoder\n
  9.2-10 AdjacencyMatrixUpperTriangleLineDecoder\n
  9.2-11 TCodeDecoder\n
  9.2-12 PlainTextString\n
  9.2-13 WritePlainTextDigraph\n\
  9.2-14 WriteDIMACSDigraph\n
\n
\n", A := "
A \n Grape to Digraphs Command Map\n \n
\ ;A.1 \n Functions to construct and modify graphs\n \n\ \n
\n
A.2 \n Functions to inspect graphs, vertices and edges\n \ \n\n
\n
A.3 \n Functions to determine regularity properties of \ graphs\n \n\n
\n
A.4 \n Some special vertex subsets o\ f a graph\n \n\n
\n
A.5 \n Functions to construct new\ graphs from old\n \n\n
\n
A.6 \n Vertex-Colouring an\ d Complete Subgraphs\n \n\n
\n
A.7 \n Automorphism g\ roups and isomorphism testing for graphs\n \n\n
\n\n" ), chapsectlinkstext := rec( 1 := "
1 \n The Digraphs\ package\n \n
1.1 \ Introduction\n\n
\n
  1.1-1 Def\ initions\n\n
\n
\n", 2 := "
2 Installing Digraphs\n
2.1 For those in a hurry\n\n
\n
2.2 Optional package\ dependencies\n\n
\n
  2.2-1 The Grape package\n\n
\n
2.\ 3 Compiling the kernel module\n\n\n
&nb\ sp;2.4 Rebuilding the documentation\n\n
\n
  2.4-1 DigraphsMakeDoc\n
\n
2.5 Testing your installation\n\n
\n
  \ 2.5-1 DigraphsTestInstall\n
  2.5-2 DigraphsTestStandard\ \n
  2.5-3 DigraphsTestExtreme\n
\n
\n", 3 := "
3 Creating digraphs\n
3.1 Creating digraphs\n\n
\n
<\ span class=\"nocss\">  3.1-1 IsDigraph\n
  3.1-2 Is\ MutableDigraph\n
\   3.1-3 IsImmutab\ leDigraph\n
\ ; 3.1-4 IsCayleyDigrap\ h\n
  <\ /span>3.1-5 IsDigraphWithAdjacency\ Function\n
\ 3.1-6 DigraphByOutNei\ ghboursType\n
&nb\ sp; 3.1-7 Digraph<\ /span>\n
  \ 3.1-8 DigraphByAdjacencyMatrix\ \n
  3.1-9 DigraphByEdges\n\
  3.1-10 EdgeOrbitsDigraph\n
  3.1-11 DigraphByInNeighbours\n
  3.1-12 CayleyDigraph\n
\n<\ div class=\"ContSect\"> 3.2 Ch\ anging representations\n\n
\n
  3.2-1 AsBinaryRelation\n
  3.2-2 AsDigraph\n  3.2-3 Graph\n
  3.2-4 As\ Graph\n
&nb\ sp;3.2-5 AsTransformation\n
\n
3.3 \ New digraphs from old\n\n
\n
&nb\ sp;3.3-1 DigraphImmutableCo\ py\n
  \ 3.3-2 DigraphImmutableCopyI\ fImmutable\n
&nbs\ p; 3.3-3 InducedSubdig\ raph\n
&nbs\ p;3.3-4 ReducedDigraph<\ /span>\n
  \ 3.3-5 MaximalSymmetricSubdigraph\n
  3.3-6 MaximalAntiSymmetricSubdi\ graph\n
&nb\ sp;3.3-7 UndirectedSpanning\ Forest\n
&n\ bsp;3.3-8 QuotientDigraph\n
  3.3-9 DigraphReverse\ \n
  3.3-10 DigraphDual\n
  3.3-11 DigraphSymmetricClosure\n
  3.3-12 DigraphTransitiveClosure\n
  3.3-13 DigraphTransitiveReduction\n
  3.3-14 DigraphAddVertex\n
  3.3-15 DigraphAddVertices\n
  3.3-16 DigraphAddEdge\n
  3.3-17 DigraphAddEdgeOrbit\n\
  3.3-18 DigraphAddEdges\n
<\ span class=\"nocss\">  3.3-19 DigraphRemoveVertex\n
  3.3-20 DigraphRemoveVertices\n
  3.3-21 DigraphRemoveEdge\n
  3.\ 3-22 DigraphRemoveEdgeOrbit\n
  3.\ 3-23 DigraphRemoveEdges\n
  3.3-2\ 4 DigraphRemoveLoops\n
  3.3-25 Di\ graphRemoveAllMultipleEdges\n
  3.\ 3-26 DigraphReverseEdges\n
  3.3-2\ 7 DigraphDisjointUnion\n
  3.3-28 \ DigraphEdgeUnion\n
  3.3-29 Digra\ phJoin\n
&n\ bsp;3.3-30 DigraphCartesian\ Product\n
&\ nbsp;3.3-31 DigraphDirectPr\ oduct\n
&nb\ sp;3.3-32 DigraphCartesianP\ roductProjections\n
  3.3-33 Digra\ phDirectProductProjections\n
  3.3\ -34 LineDigraph\n
  3.3-35 LineUnd\ irectedDigraph\n
\   3.3-36 DoubleDi\ graph\n
&nb\ sp;3.3-37 BipartiteDoubleDi\ graph\n
&nb\ sp;3.3-38 DigraphAddAllLoop\ s\n
  <\ /span>3.3-39 DistanceDigraph\n
  <\ a href=\"chap3_mj.html#X86F9CCEA839ABC48\">3.3-40 DigraphClosure\n<\ span class=\"ContSS\">
  3.3-41 DigraphMycielskian\n
\n
3.4 Random digraphs\n\n
\n
  3.4-1 RandomDigraph\n
  3.4-2 RandomMultiDigraph\n
  3.4-3 RandomTournament\n<\ br />  3.4-4 RandomLattice\n
\n
3.5 Standard examples\n\n
\n
  3.5-1 ChainDigraph\n
  3.5-2\ CompleteDigraph\n
  3.5-3 Comple\ teBipartiteDigraph\n
  3.5-4 Compl\ eteMultipartiteDigraph\n
  3.5-5 C\ ycleDigraph\n
&nb\ sp; 3.5-6 EmptyDigraph\ \n
  3.5-7 JohnsonDigraph\n
  3.5-8 PetersenGraph\n
  3.5-9 GeneralisedPetersenGraph\n
\n
\n", 4 := "
4 Operators\n
4.1 Operators for digraphs\n\n
\n
  4.1-1 IsSubdigraph\n
  4.1-2 Is\ UndirectedSpanningTree\n
\n
\n", 5 := "
5 Attributes and operations\n
5.1 Vertices and\ edges\n\n
\n
  5.1-1 DigraphVertices\n
  5.1-2 DigraphNrVertices\n
  5.1-3 DigraphEdges\n
  5.1-4 Di\ graphNrEdges\n
&n\ bsp; 5.1-5 DigraphSink\ s\n
  <\ /span>5.1-6 DigraphSources\n
  5.1-7 DigraphTopologicalSort\n
  5.1-8 DigraphVertexLabel\n<\ span class=\"ContSS\">
  5.1-9 DigraphVertexLabels\n
  5.1-10 DigraphEdgeLabel\n
  5.1-11 DigraphEdgeLabels\n
  5.1-12 DigraphInEdges\n\
  5.1-13 DigraphOutEdges\n
<\ span class=\"nocss\">  5.1-14 IsDigraphEdge\n
  5.1\ -15 IsMatching\n
\   5.1-16 DigraphM\ aximalMatching\n
\   5.1-17 DigraphM\ aximumMatching\n
\n
5.2 Neighbours and degree\n\n
\n
  5.2-1 A\ djacencyMatrix\n
\   5.2-2 Character\ isticPolynomial\n
  5.2-3 BooleanA\ djacencyMatrix\n
\   5.2-4 DigraphAd\ jacencyFunction\n
  5.2-5 DigraphR\ ange\n
&nbs\ p;5.2-6 OutNeighbours\n
  <\ a href=\"chap5_mj.html#X85C7AA5A81DA6E11\">5.2-7 InNeighbours\n
  5.2-8 OutDegrees\n
  5.2-9 InDegrees\n
  5.2-10 OutDegreeOfVertex\n
  \ 5.2-11 OutNeighboursOfVertex\n
  5\ .2-12 InDegreeOfVertex\n
  5.2-13 \ InNeighboursOfVertex\n
  5.2-14 Di\ graphLoops\n
&nbs\ p; 5.2-15 PartialOrder\ DigraphMeetOfVertices\n
  5.2-16 D\ egreeMatrix\n
&nb\ sp; 5.2-17 LaplacianMa\ trix\n
\n
5.3 Reachability and connectivity\n\n
\n
  5.3-1 Dig\ raphDiameter\n
&n\ bsp; 5.3-2 DigraphShor\ testDistance\n
&n\ bsp; 5.3-3 DigraphShor\ testDistances\n
&\ nbsp; 5.3-4 DigraphLon\ gestDistanceFromVertex\n
  5.3-5 D\ igraphDistanceSet\n
  5.3-6 Digrap\ hGirth\n
&n\ bsp;5.3-7 DigraphOddGirth\n
  5.3-8 DigraphUndirectedGirth\n
  5.3-9 DigraphConnectedComponents\ \n
  5.3-10 DigraphConnectedCompon\ ent\n
  \ ;5.3-11 DigraphStronglyConn\ ectedComponents\n
  5.3-12 Digraph\ StronglyConnectedComponent\n
  5.3\ -13 DigraphBicomponents\n
  5.3-1\ 4 ArticulationPoints\n
  5.3-15 Br\ idges\n
&nb\ sp;5.3-16 StrongOrientation\ \n
  5.3-17 DigraphPeriod\n
  5.3-18 DigraphFloydWarshall\ \n
  5.3-19 IsReachable\n
  5.3-20 DigraphPath\n
  5.3-21 DigraphShortestPath\n\
  5.3-22 IteratorOfPaths\n
<\ span class=\"nocss\">  5.3-23 DigraphAllSimpleCircuits\n
  5.3-24 DigraphLongestSimpleCircuit\n\
  5.3-25 DigraphLayers\n
  5.3-26 DigraphDegeneracy\n
  5\ .3-27 DigraphDegeneracyOrdering\n
  5.3-28 HamiltonianPath\n
  5.3-\ 29 NrSpanningTrees\n
  5.3-30 Digr\ aphDijkstra\n
\n
5.4 Cayley graphs of groups\n\n
\n
  5.4-1 Gro\ upOfCayleyDigraph\n
  5.4-2 Genera\ torsOfCayleyDigraph\n
\n
5.5 Associated semigroups\ \n\n
\n
  5.\ 5-1 AsSemigroup\n
  5.5-2 AsSemigr\ oup\n
\n
\ 5.6 Planarity\n\n
\n
  5.6-1 KuratowskiPlanarSubdigrap\ h\n
  <\ /span>5.6-2 KuratowskiOuterPlanarS\ ubdigraph\n
\ ; 5.6-3 PlanarEmbeddin\ g\n
  <\ /span>5.6-4 OuterPlanarEmbedding\n
  5.6-5 SubdigraphHomeomorphicToK\ 23\n
\n
\n", 6 := "
6 Properties of digraphs\n
6.1 Edge propertie\ s\n\n
\n  6.1-1 DigraphHasLoops\n
  6.1-2 IsAntiSymmetricDigraph\n
  6.1-3 IsBipartiteDigraph\n
  6.\ 1-4 IsCompleteBipartiteDigraph\n
  6.1-5 IsCompleteDigraph\n
  6.1-6\ IsCompleteMultipartiteDigraph\n
  6.1-7 IsEmptyDigraph\n
  6.1-8 Is\ EquivalenceDigraph\n
  6.1-9 IsFun\ ctionalDigraph\n
\   6.1-10 IsMultiD\ igraph\n
&n\ bsp;6.1-11 IsReflexiveDigra\ ph\n
  \ 6.1-12 IsSymmetricDigraph\n
  6.1-13 IsTournament\ \n
  6.1-14 IsTransitiveDigraph\n<\ span class=\"ContSS\">
  6.1-15 IsPreorderDigraph\n
  6.1-16 IsPartialOrderDigraph\n
  6.1-17 IsMeetSemilatticeDigraph\n
\n
6.2 Regularity\n\n
\n
  6.2-1 IsInRegularDigraph\n
  6.2-2 IsOutRegularDigraph\n
  6.2-3 IsRegularDigraph\n\
  6.2-4 IsDistanceRegularDigraph\n
\n
6.3 Connectiv\ ity and cycles\n\n
\n
  6.3-1 IsAcyclicDigraph\n
  6.3-2 IsChainDigraph\n
  6.3-3 IsConnectedDigraph\n
  6.3-4 IsBiconnectedDigraph\n
  \ 6.3-5 IsBridgelessDigraph\n
  6.3-\ 6 IsStronglyConnectedDigraph\n
  6\ .3-7 IsAperiodicDigraph\n
  6.3-8\ IsDirectedTree\n
  6.3-9 IsUndire\ ctedTree\n
\ 6.3-10 IsEulerianDigr\ aph\n
  \ ;6.3-11 IsHamiltonianDigrap\ h\n
  <\ /span>6.3-12 IsCycleDigraph\n
\n
6.4 Planarity\n\n
\ \n
  6.4-1 IsPlanarDigraph\n
  6.4-2 IsOuterPlanarDigraph\n
\n
&nb\ sp;6.5 Homomorphisms and transformations\n\n
\n
  6.5-1 IsDigraphCore\n\
  6.5-2 IsEdgeTransitive\n
  6.5-3 IsVertexTransitive\n
\ \n
\n", 7 := "
7 Homomorphisms\n
7.1 Acting on digraphs\n\n
\n
  7.1-1 OnDigraphs\n
  7.1-2 OnMu\ ltiDigraphs\n
\n
7.2 Isomorphisms and canonical labellings\n\n
\n
  7.2-1 DigraphsUseNauty\n
  7.\ 2-2 AutomorphismGroup\n
  7.2-3 Bl\ issAutomorphismGroup\n
  7.2-4 Nau\ tyAutomorphismGroup\n
  7.2-5 Auto\ morphismGroup\n
&\ nbsp; 7.2-6 Automorphi\ smGroup\n
&\ nbsp;7.2-7 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-8 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-9 BlissCanonicalDi\ graph\n
&nb\ sp;7.2-10 DigraphGroup<\ /span>\n
  \ 7.2-11 DigraphOrbits\n<\ span class=\"ContSS\">
  7.2-12 DigraphOrbitReps\n
  7.2-13 DigraphSchreierVector\n
  7.2-14 DigraphStabilizer\n
  7.2-15 IsIsomorphicDigraph\n
  7.2-16 IsIsomorphicDigraph\n
  7.2-17 IsomorphismDigraphs\n
  7.2-18 IsomorphismDigraphs\n\
  7.2-19 RepresentativeOutNeighbours\n
  7.2-20 IsDigraphIsomorphism\n
  7.2-21 IsDigraphColouring\n
\n
7.3 Homomorphis\ ms of digraphs\n\n
\n
  7.3-1 HomomorphismDigraphsFinder\n
  7.3-2 DigraphHomomorphism\n
  7.3-3 HomomorphismsDigraphs\n
  7.3-4 DigraphMonomorphism\n
  7.3-5 MonomorphismsDigraphs\n
  7.3-6 DigraphEpimorphism\n  7.3-7 EpimorphismsDigraphs\n
  7.3-8 DigraphEmbedding\n
  7.3-9 EmbeddingsDigraphs\n
  7.3-\ 10 IsDigraphHomomorphism\n
  7.3-1\ 1 IsDigraphEmbedding\n
  7.3-12 Di\ graphsRespectsColouring\n
  7.3-1\ 3 GeneratorsOfEndomorphismMonoid\n
  7.3-14 DigraphColouring\n
  7.3\ -15 DigraphGreedyColouring\n
  7.3\ -16 DigraphWelshPowellOrder\n
  7.\ 3-17 ChromaticNumber\n
  7.3-18 Di\ graphCore\n
\n
\n", 8 := "
8 Finding cliques and independent sets\n\
8.1 F\ inding cliques\n\n
\n
  8.1-1 IsClique\n
  8.1-2 CliquesFinder\n
  8\ .1-3 DigraphClique\n
  8.1-4 Digra\ phMaximalCliques\n
  8.1-5 Clique\ Number\n
\n
8.2 Finding independent sets\n\n\
\n
  8.2-1 IsInde\ pendentSet\n
&nbs\ p; 8.2-2 DigraphIndepe\ ndentSet\n
\ 8.2-3 DigraphMaximalI\ ndependentSets\n
\n
\n", 9 := "
9 Visualising and IO\n
9.1 Visualising a digra\ ph\n\n
\n  9.1-1 Splash\n
  9.1-2\ DotDigraph\n
&nb\ sp; 9.1-3 DotSymmetric\ Digraph\n
&\ nbsp;9.1-4 DotPartialOrderD\ igraph\n
&n\ bsp;9.1-5 DotPreorderDigrap\ h\n
  <\ /span>9.1-6 DotHighlightedDigraph<\ /a>\n
\n
9.2\ Reading and writing digraphs to a file\n\n
\n
  9.2-1\ String\n
&\ nbsp;9.2-2 DigraphFromGraph\ 6String\n
&\ nbsp;9.2-3 Graph6String\ \n
  9.2-4 DigraphFile\n
  9.2-5 ReadDigraphs\n
  9.2-6 WriteDigraphs\n<\ br />  9.2-7 IteratorFromDigraphFile\n<\ br />  9.2-8 DigraphPlainTextLineEncoder\n
  9.2-9 TournamentLineDecoder\n
  9.2-10 AdjacencyMatrixUpperTriangleLineDecoder\n
  9.2-11 TCodeDecoder\n
  9.2-12 PlainTextString\n
  9.2-13 WritePlainTextDigraph\n\
  9.2-14 WriteDIMACSDigraph\n
\n
\n", A := "
A \n Grape to Digraphs Command Map\n \n
\ ;A.1 \n Functions to construct and modify graphs\n \n\ \n
\n
A.2 \n Functions to inspect graphs, vertices and edges\n \ \n\n
\n
A.3 \n Functions to determine regularity properties of \ graphs\n \n\n
\n
A.4 \n Some special vertex subsets o\ f a graph\n \n\n
\n
A.5 \n Functions to construct new\ graphs from old\n \n\n
\n
A.6 \n Vertex-Colouring an\ d Complete Subgraphs\n \n\n
\n
A.7 \n Automorphism g\ roups and isomorphism testing for graphs\n \n\n
\n\n" ), chaptitle := rec( 1 := "\n The Digraphs package\n ", 2 := "Installing Digraphs<\ /strong>", 3 := "Creating digraphs", 4 := "Operators", 5 := "Attributes and operations", 6 := "Properties of digraphs", 7 := "Homomorphisms", 8 := "Finding cliques and independent sets", 9 := "Visualising and IO", A := "\n Grape to Digraphs Command Map\n " ), content := [ rec( content := "Book SYSTEM \"gapdoc.dtd\"\n [ Digraphs\">\n http://www.maths.qmul.ac.uk/~leonard/grape/Grape\">\n http://www.tcs.tkk.fi/Software/bliss/bliss\">\n https://github.com/graph-algorithms/edge-addition-planarity-suiteedge-addition-planarity-suite\">\n ht\ tp://pallini.di.uniroma1.it/nauty\">\n https://github.com/gap-packages/NautyTracesInter\ faceNautyTracesInterface\">\n \n http://gap-p\ ackages.github.io/io/\nIO\">\n \n http://http:/\ /gap-packages.github.io/orb/\n Orb\">\n https://github.com/gap-packages/datastructuresdatastructures\">\n \n\n\n\n\n\n\n\n\n\n ]", count := [ 0, 0, 0, 1 ], name := "XMLDOCTYPE", root := ~ ), rec( attributes := rec( Name := "digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ) ], count := [ 0, 0, 0, 1 ], name := "Title", next := 2023, root := ~, start := 1998, stop := 2022 ), rec( attributes := rec( ), content := [ rec( content := "Version 1.3.0", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Version", next := 2062, root := ~, start := 2026, stop := 2061 ), rec( attributes := rec( ), content := [ rec( content := "Jan De Beule\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "jdebeule@cage.ugent.be", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2128, root := ~, start := 2091, stop := 2127 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://homepages.vub.ac.b\ e/~jdbeule", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2189, root := ~, start := 2133, stop := 2188 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2201, root := ~, start := 2066, stop := 2200 ), rec( attributes := rec( ), content := [ rec( content := "Julius Jonušas\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "julius.jonusas@tuwien.ac.\ at", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2279, root := ~, start := 2237, stop := 2278 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://julius.jonusas.wor\ k", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2331, root := ~, start := 2284, stop := 2330 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2343, root := ~, start := 2205, stop := 2342 ), rec( attributes := rec( ), content := [ rec( content := "James D. Mitchell\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "jdm3@st-and.ac.uk", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2409, root := ~, start := 2377, stop := 2408 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "https://jdbm.me", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2450, root := ~, start := 2414, stop := 2449 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2462, root := ~, start := 2347, stop := 2461 ), rec( attributes := rec( ), content := [ rec( content := "Michael Torpey\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "mct25@st-andrews.ac.uk", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2530, root := ~, start := 2493, stop := 2529 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "https://mtorpey.github.io\ ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2581, root := ~, start := 2535, stop := 2580 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2593, root := ~, start := 2466, stop := 2592 ), rec( attributes := rec( ), content := [ rec( content := "Wilf A. Wilson\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "gap@wilf-wilson.net", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2658, root := ~, start := 2624, stop := 2657 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://wilf.me", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2698, root := ~, start := 2663, stop := 2697 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2710, root := ~, start := 2597, stop := 2709 ), rec( attributes := rec( ), content := [ rec( content := "Stuart Burrell", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2745, root := ~, start := 2714, stop := 2744 ), rec( attributes := rec( ), content := [ rec( content := "Reinis Cirpons", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2780, root := ~, start := 2749, stop := 2779 ), rec( attributes := rec( ), content := [ rec( content := "Luke Elliott", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2813, root := ~, start := 2784, stop := 2812 ), rec( attributes := rec( ), content := [ rec( content := "Max Horn", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2842, root := ~, start := 2817, stop := 2841 ), rec( attributes := rec( ), content := [ rec( content := "Christopher Jefferson", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2884, root := ~, start := 2846, stop := 2883 ), rec( attributes := rec( ), content := [ rec( content := "Markus Pfeiffer", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2920, root := ~, start := 2888, stop := 2919 ), rec( attributes := rec( ), content := [ rec( content := "Chris Russell", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2954, root := ~, start := 2924, stop := 2953 ), rec( attributes := rec( ), content := [ rec( content := "Finn Smith", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2985, root := ~, start := 2958, stop := 2984 ), rec( attributes := rec( ), content := [ rec( content := "Murray White", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 3018, root := ~, start := 2989, stop := 3017 ), rec( attributes := rec( ), content := [ rec( content := "© 2014-20 by\n Jan De Beule, \ Julius Jonušas, James D. Mitchell, Michael\n Torpey, Wilf A. Wilson et al.\ ", count := [ 0, 0, 1, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "P", next := 3168, root := ~, start := 3164, stop := 3167 ), rec( content := "\n\n ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is free software; you can redis\ tribute it and/or modify it\n under the terms of the ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "GNU General Public License" ), content := [ rec( content := "\n http://www.fsf.org/\ licenses/gpl.html", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "URL", next := 3357, root := ~, start := 3271, stop := 3356 ), rec( content := " as published by the Free Softwa\ re\n Foundation; either version 3 of the License, or (at your option) any l\ ater\n version.\n ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 1, 1 ], name := "Copyright", next := 3498, root := ~, start := 3022, stop := 3497 ), rec( attributes := rec( ), content := [ rec( content := "\n The ", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package is a ", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " package containing methods for \ graphs,\n digraphs, and multidigraphs.\n ", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 2, 1 ], name := "Abstract", next := 3636, root := ~, start := 3502, stop := 3635 ), rec( attributes := rec( ), content := [ rec( content := "\n We would like to thank Chri\ stopher Jefferson for his help in including \n ", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ) , content := [ rec( content := "http://www.tcs.tk\ k.fi/Software/bliss/", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 0, 0, 3, 1 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 0, 0, 3, 1 ], name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " in ", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".\n\n This package's methods \ for computing digraph homomorphisms are based on work\n by Max Neunhöffer,\ and independently Artur Schäfer.\n ", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], name := "Acknowledgements", next := 3931, root := ~, start := 3640, stop := 3930 ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "TitlePage", next := 3945, root := ~, start := 1984, stop := 3944 ), rec( attributes := rec( ), content := 0, count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "TableOfContents", next := 3966, root := ~, start := 3948, stop := 3965 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "The Digraphs package" ), content := [ rec( content := "\n ", count := [ 1, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The ", count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\n ", count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 4067, root := ~, start := 4018, stop := 4066 ), rec( content := "\n ", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package overview", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "Index", next := 4127, root := ~, start := 4070, stop := 4126 ), rec( content := "\n\n", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Introduction", count := [ 1, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 4169, root := ~, start := 4138, stop := 4168 ), rec( content := "\n\n This is the manual\ for version 1.3.0 of the ", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package.\n This packag\ e was developed at the University of St Andrews by:\n ", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Jan De Be\ ule,", count := [ 1, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 5 ], name := "Item", next := 4345, root := ~, start := 4319, stop := 4344 ), rec( attributes := rec( ), content := [ rec( content := "Julius Jo\ nušas,", count := [ 1, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 7 ], name := "Item", next := 4383, root := ~, start := 4350, stop := 4382 ), rec( attributes := rec( ), content := [ rec( content := "James D. \ Mitchell,", count := [ 1, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 9 ], name := "Item", next := 4419, root := ~, start := 4388, stop := 4418 ), rec( attributes := rec( ), content := [ rec( content := "Michael C\ . Torpey, and", count := [ 1, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 11 ], name := "Item", next := 4459, root := ~, start := 4424, stop := 4458 ), rec( attributes := rec( ), content := [ rec( content := "Wilf A. W\ ilson.", count := [ 1, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 13 ], name := "Item", next := 4492, root := ~, start := 4464, stop := 4491 ) ], count := [ 1, 1, 0, 4 ], mathmode := "Text", name := "List", next := 4502, root := ~, start := 4308, stop := 4501 ), rec( content := "\n Additional contribut\ ions were made by:\n ", count := [ 1, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Stuart Bu\ rrell,", count := [ 1, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 17 ], name := "Item", next := 4585, root := ~, start := 4557, stop := 4584 ), rec( attributes := rec( ), content := [ rec( content := "Reinis Ci\ rpons,", count := [ 1, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 19 ], name := "Item", next := 4618, root := ~, start := 4590, stop := 4617 ), rec( attributes := rec( ), content := [ rec( content := "Luke Elli\ ott,", count := [ 1, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 21 ], name := "Item", next := 4649, root := ~, start := 4623, stop := 4648 ), rec( attributes := rec( ), content := [ rec( content := "Max Horn,\ ", count := [ 1, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 23 ], name := "Item", next := 4676, root := ~, start := 4654, stop := 4675 ), rec( attributes := rec( ), content := [ rec( content := "Christoph\ er Jefferson,", count := [ 1, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 25 ], name := "Item", next := 4716, root := ~, start := 4681, stop := 4715 ), rec( attributes := rec( ), content := [ rec( content := "Markus Pf\ eiffer,", count := [ 1, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 27 ], name := "Item", next := 4750, root := ~, start := 4721, stop := 4749 ), rec( attributes := rec( ), content := [ rec( content := "Chris Rus\ sell,", count := [ 1, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 29 ], name := "Item", next := 4782, root := ~, start := 4755, stop := 4781 ), rec( attributes := rec( ), content := [ rec( content := "Finn Smit\ h, and", count := [ 1, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 31 ], name := "Item", next := 4815, root := ~, start := 4787, stop := 4814 ), rec( attributes := rec( ), content := [ rec( content := "Maria Tsa\ lakou,", count := [ 1, 1, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 33 ], name := "Item", next := 4848, root := ~, start := 4820, stop := 4847 ), rec( attributes := rec( ), content := [ rec( content := "Murray Wh\ yte.", count := [ 1, 1, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 35 ], name := "Item", next := 4879, root := ~, start := 4853, stop := 4878 ) ], count := [ 1, 1, 0, 16 ], mathmode := "Text", name := "List", next := 4889, root := ~, start := 4546, stop := 4888 ), rec( content := "\n\n The ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package contains a vari\ ety of methods for efficiently creating\n and storing mutable and immutable d\ igraphs and computing information about\n them. Full explanations of all t\ he functions contained in the package are\n provided below. ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "P", next := 5151, root := ~, start := 5147, stop := 5150 ), rec( content := "\n\n If the ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package is available, i\ t will be loaded automatically.\n Digraphs created with the ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package can be converte\ d to ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := "\n graphs with ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Graph" ) , content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 5333, root := ~, start := 5314, stop := 5332 ), rec( content := ", and conversely ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " graphs can be\n conver\ ted to ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " objects with ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph" ), content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 5432, root := ~, start := 5411, stop := 5431 ), rec( content := ". ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " is not\n required for \ ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to run. ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "P", next := 5488, root := ~, start := 5484, stop := 5487 ), rec( content := "\n\n The ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " tool ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "JK07" ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Cite", next := 5527, root := ~, start := 5509, stop := 5526 ), rec( content := " is included in this pac\ kage. It\n is an open-source tool for computing automorphism groups and cano\ nical forms\n of graphs, written by Tommi Junttila and Petteri Kaski. Several\ of the methods\n in the ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package rely on ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := ". If the ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "NautyTracesInte\ rface" ), content := [ rec( content := "https://github.com/gap-packages/NautyT\ racesInterface", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ) , rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "NautyTrac\ esInterface", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := "\n package for GAP is a\ vailable then it is also possible to use ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "http://pa\ llini.di.uniroma1.it/", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 124, root := ~, start := 100, stop := 123 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "MP14" ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Cite", next := 5889, root := ~, start := 5867, stop := 5888 ), rec( content := " for computing automorph\ ism groups and canonical forms\n in ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ". See Section \n ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Isomorphisms and canoni\ cal labellings" ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Ref", next := 6027, root := ~, start := 5976, stop := 6026 ), rec( content := " for more details. \n \ ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "P", next := 6054, root := ~, start := 6050, stop := 6053 ), rec( content := "\n\n From version 1.0.0\ of this package, digraphs can be either mutable or immutable.\n Mutable digr\ aphs can be changed in-place by many of the methods in the\n package, which a\ voids unnecessary copying. Immutable digraphs cannot be\n changed in-place, b\ ut their advantage is that the value of an attribute of\n an immutable digrap\ h is only ever computed once. Mutable digraphs can be\n converted into immuta\ ble digraphs in-place using ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref", Func := "MakeImmutable" ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Ref", next := 6535, root := ~, start := 6489, stop := 6534 ), rec( content := ". One of the motivation\ s for introducing mutable\n digraphs in version 1.0.0 was that in practice th\ e authors often wanted\n to create a digraph and immediately modify it (remov\ ing certain edges,\n loops, and so on). Before version 1.0.0, this involved c\ opying the digraph\n several times, with each copy being discarded almost imm\ ediately.\n From version 1.0.0, this unnecessary copying can be eliminated by\ \n first creating a mutable digraph, then changing it in-place, and finally\ \n converting the mutable digraph to an immutable one in-place\n (if desirab\ le).\n\n ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Definitions" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Definitio\ ns", count := [ 1, 1, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 2 ], mathmode := "Text", name := "Heading", next := 7166, root := ~, start := 7136, stop := 7165 ), rec( content := "\n\n For the \ purposes of this package and its documentation, the following\n definitions\ apply: ", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "P", next := 7269, root := ~, start := 7265, stop := 7268 ), rec( content := "\n\n A ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7291, root := ~, start := 7277, stop := 7290 ), rec( content := " ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E=(E^0,E^\ 1,r,s)", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7314, root := ~, start := 7292, stop := 7313 ), rec( content := ", also known as \ a ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed\\ n graph", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7359, root := ~, start := 7332, stop := 7358 ), rec( content := ", consists of a \ set of vertices ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E^0", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7401, root := ~, start := 7391, stop := 7400 ), rec( content := " and a set of ed\ ges\n ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E^1", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7435, root := ~, start := 7425, stop := 7434 ), rec( content := " together with f\ unctions ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s, r: E^1\ \\to E^0", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7484, root := ~, start := 7460, stop := 7483 ), rec( content := ", called the\n \ ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "source", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7514, root := ~, start := 7501, stop := 7513 ), rec( content := " and ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "range", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7531, root := ~, start := 7519, stop := 7530 ), rec( content := ", respectively.\\ n\n The source and range of an edge is respectively the values of ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s, r", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7625, root := ~, start := 7614, stop := 7624 ), rec( content := " at\n that ed\ ge.\n\n An edge is called a ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "loop", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7680, root := ~, start := 7669, stop := 7679 ), rec( content := " if its source a\ nd range are the same.\n\n A digraph is called a ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "multidigr\ aph", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7765, root := ~, start := 7746, stop := 7764 ), rec( content := " if there exist \ two or more edges\n with the same source and the same range. ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "P", next := 7848, root := ~, start := 7844, stop := 7847 ), rec( content := "\n\n A ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed \ walk", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 7876, root := ~, start := 7856, stop := 7875 ), rec( content := " on a digraph is\ a sequence of alternating vertices\n and edges ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v_1, e_1\ , v_2, e_2, ..., e_{n-1}, v_n)", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 7988, root := ~, start := 7942, stop := 7987 ), rec( content := " such that each \ edge\n ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e_i", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8023, root := ~, start := 8013, stop := 8022 ), rec( content := " has source ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_i", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8045, root := ~, start := 8035, stop := 8044 ), rec( content := " and range ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_{i+1}", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8070, root := ~, start := 8056, stop := 8069 ), rec( content := ". A ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed\\ n path", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8100, root := ~, start := 8074, stop := 8099 ), rec( content := " is a directed w\ alk where no vertex (and hence no edge) is\n repeated. A ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed \ circuit", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8199, root := ~, start := 8176, stop := 8198 ), rec( content := " is a directed w\ alk where ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_1 =\n \ v_n", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8247, root := ~, start := 8225, stop := 8246 ), rec( content := ", and a ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed \ cycle", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8276, root := ~, start := 8255, stop := 8275 ), rec( content := " is a directed c\ ircuit where where no\n vertex is repeated, except for ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_1 = v_n\ ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8365, root := ~, start := 8349, stop := 8364 ), rec( content := ". ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "P", next := 8372, root := ~, start := 8368, stop := 8371 ), rec( content := "\n\n The ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "length", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8395, root := ~, start := 8382, stop := 8394 ), rec( content := " of a directed w\ alk ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v_1, e_1\ , v_2, e_2, ..., e_{n-1},\n v_n)", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8467, root := ~, start := 8415, stop := 8466 ), rec( content := " is equal to ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n-1", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8490, root := ~, start := 8480, stop := 8489 ), rec( content := ", the number of \ edges it contains. A\n directed walk (or path) ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v_1, e_1\ , v_2, e_2, ..., e_{n-1}, v_n)", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8602, root := ~, start := 8556, stop := 8601 ), rec( content := " is\n sometim\ es called a directed walk (or path) ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "from vert\ ex ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ _1", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8678, root := ~, start := 8668, stop := 8677 ), rec( content := " to\n \ vertex ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ _n", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8705, root := ~, start := 8695, stop := 8704 ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8709, root := ~, start := 8653, stop := 8708 ), rec( content := ". A directed wa\ lk of zero length, i.e. a sequence\n ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v)", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8774, root := ~, start := 8764, stop := 8773 ), rec( content := " for some vertex\ ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8799, root := ~, start := 8791, stop := 8798 ), rec( content := ", is called ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "trivial", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8825, root := ~, start := 8811, stop := 8824 ), rec( content := ". A trivial\n \ directed walk is considered to be both a circuit and a cycle, as is the\n \ empty directed walk ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "()", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8946, root := ~, start := 8937, stop := 8945 ), rec( content := ". A ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "simple ci\ rcuit", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8971, root := ~, start := 8950, stop := 8970 ), rec( content := " is another name\ for a\n non-trivial and non-empty directed cycle.", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "P", next := 9043, root := ~, start := 9039, stop := 9042 ), rec( content := "\n\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 1 ], mathmode := "Text", name := "Subsection", next := 9060, root := ~, start := 7104, stop := 9059 ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 9073, root := ~, start := 4129, stop := 9072 ), rec( content := "\n", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 9084, root := ~, start := 3977, stop := 9083 ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " intro ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Installing Digraphs" ) , content := [ rec( content := "\n ", count := [ 2, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Installing ", count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ) ], count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 9183, root := ~, start := 9143, stop := 9182 ), rec( content := "\n\n ", count := [ 2, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "For those in a hurry" ), content := [ rec( content := "\n ", count := [ 2, 1, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "For those in a hu\ rry", count := [ 2, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 9269, root := ~, start := 9230, stop := 9268 ), rec( content := "\n In this section we\ give a brief description of how to start using\n ", count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".", count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "P", next := 9359, root := ~, start := 9355, stop := 9358 ), rec( content := "\n\n It is assumed th\ at you have a working copy of ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " with version number\n \ 4.9.0 or higher. The most up-to-date version of ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " and instructions\n o\ n how to install it can be obtained from the main ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " webpage\n ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://www.gap-sy\ stem.org", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "URL", next := 9625, root := ~, start := 9589, stop := 9624 ), rec( content := ".", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "P", next := 9630, root := ~, start := 9626, stop := 9629 ), rec( content := "\n\n The following is\ a summary of the steps that should lead to a\n successful installation of \ ", count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ":\n\n ", count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ensure that the ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 1, 0, 7 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 2, 1, 0, 7 ], name := \ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " package\ version 4.5.1 or higher is available.\n ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 1, 0, 7 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 2, 1, 0, 7 ], name := \ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " must be\ compiled before ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be \ loaded.\n ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], name := "Item", next := 9921, root := ~, start := 9760, stop := 9920 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ensure that the ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Orb" ), content := [ rec( content := "\n http://http://gap-\ packages.github.io/orb/", count := [ 2, 1, 0, 9 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "URL\ ", next := 86, root := ~, start := 17, stop := 85 ), rec( content := "\n " , count := [ 2, 1, 0, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Alt", next := 98, root := ~, start := 1, stop := 97 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Orb", count := [ 2, 1, 0, 9 ], name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Package", next := 137, root := ~, start := 115 , stop := 136 ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 98, stop := 142 ), rec( content := " package\ version 4.8.2 or higher is available.\n ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Orb" ), content := [ rec( content := "\n http://http://gap-\ packages.github.io/orb/", count := [ 2, 1, 0, 9 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "URL\ ", next := 86, root := ~, start := 17, stop := 85 ), rec( content := "\n " , count := [ 2, 1, 0, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Alt", next := 98, root := ~, start := 1, stop := 97 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Orb", count := [ 2, 1, 0, 9 ], name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Package", next := 137, root := ~, start := 115 , stop := 136 ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 98, stop := 142 ), rec( content := " has bet\ ter performance when compiled, but although compilation is\n recommende\ d, it is not required to be compiled for ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to be\n\ loaded.\n ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Item", next := 10201, root := ~, start := 9929, stop := 10200 ), rec( attributes := rec( ), content := [ rec( content := " ensure t\ hat the ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "datastructures" ), content := [ rec( content := "https://github\ .com/gap-packages/datastructures", count := [ 2, 1, 0, 11 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], mathmode := "Tex\ t", name := "URL", next := 103, root := ~, start := 17, stop := 102 ) ], count := [ 2, 1, 0, 11 ], name := "Alt", next := 109, root := ~, start := 1, stop := 108 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "datastructures", count := [ 2, 1, 0, 1\ 1 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Packa\ ge", next := 162, root := ~, start := 129, stop := 161 ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "Alt", next := 168, root := ~, start := 109, stop := 167 ), rec( content := " package\ version\n 0.2.5 or higher is available. \n ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Item", next := 10330, root := ~, start := 10209, stop := 10329 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ his step is optional:", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "B", next := 10382, root := ~, start := 10353, stop := 10381 ), rec( content := " certain\ functions in ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " require\ \n the ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 1, 0, 13 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ to be available; see Section\n ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Subsect := "The \ Grape package" ), content := 0, count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Ref", next := 10522, root := ~, start := 10488, stop := 10521 ), rec( content := " for ful\ l details. To use these\n functions make sure\n that the ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 1, 0, 13 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ version 4.8.1 or higher is available.\n If ", count := [ 2, 1, 0, 13 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 1, 0, 13 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " is not \ available, then ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be \ used as normal with\n the exception that the functions listed in Subsec\ tion\n ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Subsect := "The \ Grape package" ), content := 0, count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Ref", next := 10844, root := ~, start := 10810, stop := 10843 ), rec( content := " will no\ t work.\n ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Item", next := 10873, root := ~, start := 10338, stop := 10872 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ his step is optional:", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "B", next := 10931, root := ~, start := 10902, stop := 10930 ), rec( content := " certain\ functions in ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " require\ \n the ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 2, 1, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 2, 1, 0, 15 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 2, 1\ , 0, 15 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " package\ to be available.\n ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " TODO \n\ see Section\n for full d\ etails. ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n \ If you want to make use of these functions, please ensure that the\n \ ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 2, 1, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 2, 1, 0, 15 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 2, 1\ , 0, 15 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " package\ version 0.2 or\n higher is available. If ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 2, 1, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 2, 1, 0, 15 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 2, 1\ , 0, 15 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is not \ available,\n then ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be \ used as normal with the exception that \n functions whose names conta\ in ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ auty", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Q", next := 11490, root := ~, start := 11478, stop := 11489 ), rec( content := " will no\ t work.\n ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Item", next := 11519, root := ~, start := 10887, stop := 11518 ), rec( attributes := rec( ), content := [ rec( content := "\n \ download the package archive ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.3.0.tar.gz", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "F", next := 11598, root := ~, start := 11571, stop := 11597 ), rec( content := " from\n \ ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "the Dig\ raph package webpage" ), content := [ rec( content := "http://gap-packages.git\ hub.io/Digraphs/", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "URL", next := 11697, root := ~, start := 11612, stop := 11696 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], name := "Item", next := 11712, root := ~, start := 11527, stop := 11711 ), rec( attributes := rec( ), content := [ rec( content := "\n \ unzip and untar the file, this should create a directory called\n ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.3.0", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "F", next := 11827, root := ~, start := 11807, stop := 11826 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 19 ], name := "Item", next := 11842, root := ~, start := 11720, stop := 11841 ), rec( attributes := rec( ), content := [ rec( content := "\n \ locate the ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 11886, root := ~, start := 11876, stop := 11885 ), rec( content := " directo\ ry of your ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " directo\ ry, which contains\n the directories ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ib", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 11971, root := ~, start := 11961, stop := 11970 ), rec( content := ", ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ oc", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 11983, root := ~, start := 11973, stop := 11982 ), rec( content := " and so \ on. Move the directory\n ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.3.0", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12042, root := ~, start := 12022, stop := 12041 ), rec( content := " into th\ e ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12062, root := ~, start := 12052, stop := 12061 ), rec( content := " directo\ ry.\n ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], name := "Item", next := 12087, root := ~, start := 11850, stop := 12086 ), rec( attributes := rec( ), content := [ rec( content := "\n \ it is necessary to compile the ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ .\n Inside the ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg/digraphs-1.3.0", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "F", next := 12204, root := ~, start := 12180, stop := 12203 ), rec( content := " directo\ ry, type\n ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ n./configure\nmake", count := [ 2, 1, 0, 24 ], name := "PCDATA", root := ~ ) ] , count := [ 2, 1, 0, 24 ], mathmode := "Text", name := "Listing", next := 12277, root := ~, start := 12229, stop := 12276 ), rec( content := "\n \ Further information about this step can be found in Section\n ", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Compili\ ng the kernel module" ), content := 0, count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "Ref", next := 12395, root := ~, start := 12354, stop := 12394 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], name := "Item", next := 12410, root := ~, start := 12095, stop := 12409 ), rec( attributes := rec( ), content := [ rec( content := "\n \ start ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " in the \ usual way (i.e. type ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "C", next := 12483, root := ~, start := 12473, stop := 12482 ), rec( content := " at the \ command line).\n ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 27 ], name := "Item", next := 12519, root := ~, start := 12418, stop := 12518 ), rec( attributes := rec( ), content := [ rec( content := "\n \ type ", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ oadPackage(\"digraphs\");", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "C", next := 12578, root := ~, start := 12547, stop := 12577 ), rec( content := "\n \ ", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 29 ], name := "Item", next := 12592, root := ~, start := 12527, stop := 12591 ) ], count := [ 2, 1, 0, 6 ], mathmode := "Text", name := "List", next := 12604, root := ~, start := 9746, stop := 12603 ), rec( content := "\n\n If you want to c\ heck that the package is working correctly, you\n should run some of the te\ sts described in Section\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Testing your installati\ on" ), content := 0, count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "Ref", next := 12771, root := ~, start := 12732, stop := 12770 ), rec( content := ".\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 12785, root := ~, start := 9187, stop := 12784 ), rec( content := "\n\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Optional package dependencies"\ ), content := [ rec( content := "\n ", count := [ 2, 2, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Optional package \ dependencies", count := [ 2, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 13050, root := ~, start := 13002, stop := 13049 ), rec( content := "\n\n ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "************************\ ********************************************", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n The ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package is written in " , count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " and C code and requires\ the\n ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ rec( content := "\n ht\ tp://gap-packages.github.io/io/\n", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 2, 0, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "IO", count := [ 2, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], name := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " package. The ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ rec( content := "\n ht\ tp://gap-packages.github.io/io/\n", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 2, 0, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "IO", count := [ 2, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], name := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " package is used to read\ and write transformations,\n partial permutations, and bipartitions to a f\ ile. ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "P", next := 13345, root := ~, start := 13341, stop := 13344 ), rec( content := "\n\n ", count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "************************\ ********************************************", count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "The Grape package" ), content := [ rec( content := "\n ", count := [ 2, 2, 1, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "The Grape\ package", count := [ 2, 2, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 2 ], mathmode := "Text", name := "Heading", next := 13513, root := ~, start := 13477, stop := 13512 ), rec( content := "\n\n The ", count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ 2, 2, 1, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 2, 1, 3 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ) , content := [ rec( content := "G\ rape", count := [ 2, 2, 1, 3 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 3 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package must be\ available for the following operations to be\n available:\n\n ", count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( attributes :=\ rec( Oper := "Graph" ), content := 0, count := [ 2, 2, 1, 5 ], mathmode := "T\ ext", name := "Ref", next := 13658, root := ~, start := 13639, stop := 13657 ) , rec( content := "\ with a digraph argument", count := [ 2, 2, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 2, 1, 5 ], name := "Item", next := 13689, root := ~, start := 13633, stop := 13688 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Attr := "AsGraph" ), content := 0, count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "Ref", next := 13725, root := ~, start := 13704, stop := 13724\ ), rec( content := " with a digraph argument", count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 7 ], name := "Item", next := 13756, root := ~, start := 13698, stop := 13755 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Oper := "Digraph" ), content := 0, count := [ 2, 2, 1, 9 ], mathmode := \ "Text", name := "Ref", next := 13792, root := ~, start := 13771, stop := 13791\ ), rec( content := " with a ", count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( content := "http://www.math\ s.qmul.ac.uk/~leonard/grape/", count := [ 2, 2, 1, 9 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 9 ], mathmode := "Text", name\ := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 2, 1, 9 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "Grape", count := [ 2, 2, 1, 9 ], name := "PCDATA", root := ~ ) ], count := [ \ 2, 2, 1, 9 ], name := "Package", next := 137, root := ~, start := 113, stop :=\ 136 ) ], count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := "\ graph argument", count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 2, 2, 1, 9 ], name := "Item", next := 13829, root := ~, start := 13765, stop := 13828 ) ], count := [ 2, 2, 1, 4 ], mathmode := "Text", name := "List", next := 13843, root := ~, start := 13618, stop := 13842 ), rec( content := "\n\n If ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ 2, 2, 1, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 2, 1, 11 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ) , content := [ rec( content := "G\ rape", count := [ 2, 2, 1, 11 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " is not availabl\ e, then ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs" , count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be used as \ normal with\n the exception that the functions above will not work.\n \ ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 1 ], mathmode := "Text", name := "Subsection", next := 14000, root := ~, start := 13432, stop := 13999 ), rec( content := "\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 14013, root := ~, start := 12950, stop := 14012 ), rec( content := "\n\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Compiling the kernel module" ) , content := [ rec( content := "\n ", count := [ 2, 3, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Compiling the ker\ nel module", count := [ 2, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 14274, root := ~, start := 14228, stop := 14273 ), rec( content := "\n The ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package has a ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " kernel component in\n \ C which should be compiled. This component contains certain low-level\n f\ unctions required by ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".\n ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "P", next := 14454, root := ~, start := 14450, stop := 14453 ), rec( content := "\n\n It is not possib\ le to use the ", count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package without compili\ ng it.", count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "P", next := 14534, root := ~, start := 14530, stop := 14533 ), rec( content := "\n\n To compile the k\ ernel component inside the ", count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "pkg/digraphs-1.3.\ 0", count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "F", next := 14607, root := ~, start := 14583, stop := 14606 ), rec( content := "\n directory, type\n" , count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n\n./configure\n\ make", count := [ 2, 3, 0, 6 ], name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 6 ], mathmode := "Text", name := "Listing", next := 14677, root := ~, start := 14628, stop := 14676 ), rec( content := "\n ", count := [ 2, 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "P", next := 14686, root := ~, start := 14682, stop := 14685 ), rec( content := "\n\n If you installed\ the package in another 'pkg' directory than the standard 'pkg'\n directory\ in your ", count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " installation, then you \ have to do two things. Firstly\n during compilation you have to use the opt\ ion '--with-gaproot=PATH' of the\n 'configure' script where 'PATH' is a pat\ h to the main GAP root directory (if\n not given the default '../..' is ass\ umed).", count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "P", next := 15064, root := ~, start := 15060, stop := 15063 ), rec( content := "\n\n If you installed\ ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " on several architecture\ s, you must execute the\n configure/make step for each of the architectures\ . You can either do this\n immediately after configuring and compiling GAP \ itself on this architecture, or\n alternatively set the environment\n va\ riable 'CONFIGNAME' to the name of the configuration you used when compiling\n\ GAP before running './configure'. Note however that your compiler choice \ and\n flags (environment variables 'CC' and 'CFLAGS') need to be chosen to \ match the\n setup of the original GAP compilation. For example you have to \ specify 32-bit\n or 64-bit mode correctly!\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 15712, root := ~, start := 14178, stop := 15711 ), rec( content := "\n\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Rebuilding the documentation" ), content := [ rec( content := "\n ", count := [ 2, 4, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Rebuilding the do\ cumentation", count := [ 2, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 15975, root := ~, start := 15928, stop := 15974 ), rec( content := "\n The ", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package comes complete \ with pdf, html, and text versions of\n the documentation. However, you migh\ t find it necessary, at some point, to\n rebuild the documentation. To rebu\ ild the documentation, please use the\n function ", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphsMakeDoc" ), content := 0, count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "Ref", next := 16252, root := ~, start := 16223, stop := 16251 ), rec( content := ".\n\n ", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsMakeDoc\ " ), content := 0, count := [ 2, 4, 1, 2 ], mathmode := "Text", name := "Func", next := 16311, root := ~, start := 16274, stop := 16310 ), rec( attributes := rec( ), content := [ rec( content := "Nothing", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 16340, root := ~, start := 16314, stop := 16339 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function should be called with no argument to compile the ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := "\n do\ cumentation.\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 16470, root := ~, start := 16343, stop := 16469 ) ], count := [ 2, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 16484, root := ~, start := 16259, stop := 16483 ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 16498, root := ~, start := 15877, stop := 16497 ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Testing your installation" ), content := [ rec( content := "\n ", count := [ 2, 5, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Testing your inst\ allation", count := [ 2, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 16755, root := ~, start := 16711, stop := 16754 ), rec( content := "\n\n In this section \ we describe how to test that ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is working as\n inte\ nded. To test that ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is installed correctly \ use\n ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphsTestInstall" ), content := 0, count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Ref", next := 16934, root := ~, start := 16901, stop := 16933 ), rec( content := " or for more extensive t\ ests use\n ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphsTestStandard" ) , content := 0, count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Ref", next := 17005, root := ~, start := 16971, stop := 17004 ), rec( content := ". ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "P", next := 17011, root := ~, start := 17007, stop := 17010 ), rec( content := "\n\n If something goe\ s wrong, then please review the instructions in Section ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "For those in a hurry" ) , content := 0, count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "Ref", next := 17129, root := ~, start := 17089, stop := 17128 ), rec( content := " and ensure that ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " has been\n properly \ installed. If you continue having problems, please use the ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "issue tracker" ), content := [ rec( content := "https://github.co\ m/gap-packages/Digraphs/issues", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "URL", next := 17324, root := ~, start := 17239, stop := 17323 ), rec( content := "\n to report the issu\ es you are having.\n\n ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsTestIns\ tall" ), content := 0, count := [ 2, 5, 1, 2 ], mathmode := "Text", name := "Func", next := 17427, root := ~, start := 17386, stop := 17426 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "K", next := 17450, root := ~, start := 17439, stop := 17449 ), rec( content := " or ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "K", next := 17466, root := ~, start := 17454, stop := 17465 ), rec( content := ".", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 17477, root := ~, start := 17430, stop := 17476 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function can be called without arguments to test your installation of\n " , count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is work\ ing correctly. These tests should take no more than a\n few seconds to comp\ lete. To test more comprehensively that\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is work\ ing correctly, use ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sTestStandard" ), content := 0, count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Ref", next := 17788, root := ~, start := 17754, stop := 17787 ), rec( content := ".\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 17806, root := ~, start := 17480, stop := 17805 ) ], count := [ 2, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 17820, root := ~, start := 17371, stop := 17819 ), rec( content := "\n\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsTestSta\ ndard" ), content := 0, count := [ 2, 5, 2, 2 ], mathmode := "Text", name := "Func", next := 17883, root := ~, start := 17841, stop := 17882 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "K", next := 17906, root := ~, start := 17895, stop := 17905 ), rec( content := " or ", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "K", next := 17922, root := ~, start := 17910, stop := 17921 ), rec( content := ".", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 17933, root := ~, start := 17886, stop := 17932 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function can be called without arguments to test all of the methods\n inc\ luded in ", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".\n T\ hese tests should take less than a minute to complete. ", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "P", next := 18119, root := ~, start := 18115, stop := 18118 ), rec( content := "\n\n \ To quickly test that ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is inst\ alled correctly use ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sTestInstall" ), content := 0, count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 18221, root := ~, start := 18184, stop := 18220 ), rec( content := ". For a\ more thorough test, use ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sTestExtreme" ), content := 0, count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 18291, root := ~, start := 18254, stop := 18290 ), rec( content := ".\n ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 18309, root := ~, start := 17936, stop := 18308 ) ], count := [ 2, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 18323, root := ~, start := 17826, stop := 18322 ), rec( content := "\n\n ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsTestExt\ reme" ), content := 0, count := [ 2, 5, 3, 2 ], mathmode := "Text", name := "Func", next := 18385, root := ~, start := 18344, stop := 18384 ), rec( attributes := rec( ), content := [ rec( content := "Nothing." , count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 18415, root := ~, start := 18388, stop := 18414 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function should be called with no argument. It executes a series of\n ver\ y demanding tests, which measure the performance of a variety of\n function\ s on large examples. These tests take a long time to\n complete, at least \ several minutes. ", count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "P", next := 18691, root := ~, start := 18687, stop := 18690 ), rec( content := "\n\n \ For these tests to complete, the digraphs library ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-lib", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "C", next := 18766, root := ~, start := 18747, stop := 18765 ), rec( content := " must\n \ be downloaded and placed in the ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "C", next := 18823, root := ~, start := 18808, stop := 18822 ), rec( content := " directo\ ry in a subfolder\n named ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-lib", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "F", next := 18878, root := ~, start := 18859, stop := 18877 ), rec( content := ".\n T\ his library can be found on the ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " website\ .\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 18953, root := ~, start := 18418, stop := 18952 ) ], count := [ 2, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 18967, root := ~, start := 18329, stop := 18966 ), rec( content := "\n\n\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 18982, root := ~, start := 16663, stop := 18981 ), rec( content := "\n\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n\n", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 19156, root := ~, start := 9103, stop := 19155 ), rec( content := "\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " installing ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Creating digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Creating digraphs", count := [ 3, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 19251, root := ~, start := 19215, stop := 19250 ), rec( content := "\n\n In this chapter we describ\ e how to create digraphs.", count := [ 3, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 0, 0, 4 ], mathmode := "Text", name := "P", next := 19310, root := ~, start := 19306, stop := 19309 ), rec( content := "\n\n ", count := [ 3, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Creating digraphs\ ", count := [ 3, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 19359, root := ~, start := 19323, stop := 19358 ), rec( content := "\n ", count := [ 3, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsDigraph", Type := "Category" ), content := 0, count := [ 3, 1, 1, 2 ], mathmode := "Text", name := "Filt", next := 19419, root := ~, start := 19379, stop := 19418 ), rec( attributes := rec( ), content := [ rec( content := "\n Eve\ ry digraph in ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " belong\ s to the category\n ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraph", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "C", next := 19518, root := ~, start := 19502, stop := 19517 ), rec( content := ". Some \ basic attributes and operations for digraphs\n are ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Ref", next := 19608, root := ~, start := 19579, stop := 19607 ), rec( content := ", ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Edges" ), content := 0, count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Ref", next := 19636, root := ~, start := 19610, stop := 19635 ), rec( content := ", and\n \ ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Ref", next := 19673, root := ~, start := 19646, stop := 19672 ), rec( content := ".\n " , count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 19693, root := ~, start := 19422, stop := 19692 ) ], count := [ 3, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 19707, root := ~, start := 19364, stop := 19706 ), rec( content := "\n\n ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsMutableDigrap\ h", Type := "Category" ), content := 0, count := [ 3, 1, 2, 2 ], mathmode := "Text", name := "Filt", next := 19775, root := ~, start := 19728, stop := 19774 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMutableDigraph", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "C", next := 19819, root := ~, start := 19796, stop := 19818 ), rec( content := " is a sy\ nonym for ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsDigra\ ph" ), content := 0, count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Ref", next := 19861, root := ~, start := 19837, stop := 19860 ), rec( content := " and ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsMutable" ), content := 0, count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Ref", next := 19909, root := ~, start := 19866, stop := 19908 ), rec( content := ". A mut\ able digraph may be changed\n in-place by methods in the ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ , and is not attribute-storing\n – see ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsAttributeStoringRep" ), content := 0, count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Ref", next := 20088, root := ~, start := 20037, stop := 20087 ), rec( content := ".\n " , count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "P", next := 20098, root := ~, start := 20094, stop := 20097 ), rec( content := "\n\n \ A mutable digraph may be converted into an immutable attribute-storing\n di\ graph by calling ", count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "MakeImmutable" ), content := 0, count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "Ref", next := 20240, root := ~, start := 20198, stop := 20239 ), rec( content := " on the\\ n digraph.\n ", count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 20279, root := ~, start := 19778, stop := 20278 ) ], count := [ 3, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 20293, root := ~, start := 19713, stop := 20292 ), rec( content := "\n\n ", count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsImmutableDigr\ aph", Type := "Category" ), content := 0, count := [ 3, 1, 3, 2 ], mathmode := "Text", name := "Filt", next := 20363, root := ~, start := 20314, stop := 20362 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sImmutableDigraph", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "C", next := 20409, root := ~, start := 20384, stop := 20408 ), rec( content := " is a su\ bcategory of ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsDigra\ ph" ), content := 0, count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 20454, root := ~, start := 20430, stop := 20453 ), rec( content := ".\n D\ igraphs that lie in ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sImmutableDigraph", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "C", next := 20506, root := ~, start := 20481, stop := 20505 ), rec( content := " are imm\ utable and\n attribute-storing. In particular, they lie in\n ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsAttributeStoringRep" ), content := 0, count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 20630, root := ~, start := 20579, stop := 20629 ), rec( content := ". ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "P", next := 20636, root := ~, start := 20632, stop := 20635 ), rec( content := "\n\n \ A mutable digraph may be converted to an immutable\n digraph that lies in t\ he category ", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sImmutableDigraph", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "C", next := 20756, root := ~, start := 20731, stop := 20755 ), rec( content := " by call\ ing\n ", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "MakeImmutable" ), content := 0, count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "Ref", next := 20814, root := ~, start := 20772, stop := 20813 ), rec( content := " on the \ digraph.", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "P", next := 20834, root := ~, start := 20830, stop := 20833 ), rec( content := "\n\n \ The operation ", count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ MutableCopy" ), content := 0, count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "Ref", next := 20886, root := ~, start := 20854, stop := 20885 ), rec( content := " can be \ used\n to construct a mutable copy of an immutable digraph. It is however n\ ot\n possible to convert an immutable digraph into a mutable digraph in-pla\ ce.\n ", count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 21068, root := ~, start := 20366, stop := 21067 ) ], count := [ 3, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 21082, root := ~, start := 20299, stop := 21081 ), rec( content := "\n\n ", count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsCayleyDigraph\ ", Type := "Category" ), content := 0, count := [ 3, 1, 4, 2 ], mathmode := "Text", name := "Filt", next := 21149, root := ~, start := 21103, stop := 21148 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCayleyDigraph", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "C", next := 21192, root := ~, start := 21170, stop := 21191 ), rec( content := " is a su\ bcategory of ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraph", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "C", next := 21229, root := ~, start := 21213, stop := 21228 ), rec( content := ". Digrap\ hs that\n are Cayley digraphs of a group and that are constructed by the op\ eration\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CayleyD\ igraph" ), content := 0, count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "Ref", next := 21353, root := ~, start := 21326, stop := 21352 ), rec( content := " are con\ structed in this category, and are\n always immutable.\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 21436, root := ~, start := 21152, stop := 21435 ) ], count := [ 3, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 21450, root := ~, start := 21088, stop := 21449 ), rec( content := "\n\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsDigraphWithAd\ jacencyFunction", Type := "Category" ), content := 0, count := [ 3, 1, 5, 2 ], mathmode := "Text", name := "Filt", next := 21532, root := ~, start := 21471, stop := 21531 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphWithAdjacencyFunction", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "C", next := 21590, root := ~, start := 21553, stop := 21589 ), rec( content := " is a su\ bcategory of ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraph", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "C", next := 21627, root := ~, start := 21611, stop := 21626 ), rec( content := ".\n D\ igraphs that are ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ reated", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "E", next := 21665, root := ~, start := 21651, stop := 21664 ), rec( content := " using a\ n adjacency function are constructed\n in this category.\n ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 21748, root := ~, start := 21535, stop := 21747 ) ], count := [ 3, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 21762, root := ~, start := 21456, stop := 21761 ), rec( content := "\n\n ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "DigraphByOutNei\ ghboursType" ), content := 0, count := [ 3, 1, 6, 2 ], mathmode := "Text", name := "Var", next := 21823, root := ~, start := 21783, stop := 21822 ), rec( attributes := rec( Name := "DigraphFamily" ), content := 0, count := [ 3, 1, 6, 4 ], mathmode := "Text", name := "Fam", next := 21853, root := ~, start := 21826, stop := 21852 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ type of all digraphs is ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphByOutNeighboursType", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "C", next := 21935, root := ~, start := 21902, stop := 21934 ), rec( content := ".\n T\ he family of all digraphs is ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFamily", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "C", next := 21991, root := ~, start := 21971, stop := 21990 ), rec( content := ".\n ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "Description", next := 22009, root := ~, start := 21856, stop := 22008 ) ], count := [ 3, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 22023, root := ~, start := 21768, stop := 22022 ), rec( content := "\n\n ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt,] obj[, so\ urce, range]", Name := "Digraph" ), content := 0, count := [ 3, 1, 7, 2 ], mathmode := "Text", name := "Oper", next := 22105, root := ~, start := 22044, stop := 22104 ), rec( attributes := rec( Arg := "[filt,] list, fu\ nc", Label := "for a list and function", Name := "Digraph" ), content := 0, count := [ 3, 1, 7, 4 ], mathmode := "Text", name := "Oper", next := 22198, root := ~, start := 22108, stop := 22197 ), rec( attributes := rec( Arg := "[filt,] G, list,\ act, adj", Label := "for a group, list, function, and function", Name := "Digraph" ), content := 0, count := [ 3, 1, 7, 6 ], mathmode := "Text", name := "Oper", next := 22315, root := ~, start := 22201, stop := 22314 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Returns", next := 22347, root := ~, start := 22318, stop := 22346 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 22410, root := ~, start := 22399, stop := 22409 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 22556, root := ~, start := 22545, stop := 22555 ), rec( content := " is ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Ref", next := 22590, root := ~, start := 22560, stop := 22589 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 22658, root := ~, start := 22647, stop := 22657 ), rec( content := " is ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Ref", next := 22698, root := ~, start := 22662, stop := 22697 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 22783, root := ~, start := 22772, stop := 22782 ), rec( content := " is not \ present, then ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Ref", next := 22841, root := ~, start := 22805, stop := 22840 ), rec( content := " is used\ by default.\n ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "for a list (i.e. an adjacency list)", coun\ t := [ 3, 1, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 8 ], name := "Mar\ k", next := 22927, root := ~, start := 22879, stop := 22926 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 9\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "obj", count := [ 3, 1, 7, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := \ "Text", name := "A", next := 22962, root := ~, start := 22952, stop := 22961 ) , rec( cont\ ent := " is a list of lists of positive integers in\n the range from ", coun\ t := [ 3, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "1", count := [ 3, 1, 7, 9 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], math\ mode := "Text", name := "C", next := 23037, root := ~, start := 23029, stop :=\ 23036 ), rec( content := " to ", count := [ 3, 1, 7, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "Length(", count := [ 3, 1, 7, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ), rec( attributes := rec( ), content := [ rec( content := "obj",\ count := [ 3, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ],\ count := [ 3, 1, 7, 9 ], mathmode := "Text", name := "A", next := 23061, root\ := ~, start := 23051, stop := 23060 ), rec( content := ")", count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "C", next := 23066, root := ~, start := 23\ 041, stop := 23065 ), rec( content := ", then this function\n returns t\ he digraph with vertices\n ", count := [ 3, 1, 7, 9 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "E ^ 0 = ", count := [ 3, 1, 7, 9 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := "Text", name :=\ "M", next := 23152, root := ~, start := 23137, stop := 23151 ), rec( attribut\ es := rec( ), content := [ rec( content := "[1 .. Length(", count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := \ rec( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 9 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode\ := "Text", name := "A", next := 23178, root := ~, start := 23168, stop := 231\ 77 ), rec( content := ")]", count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "C", next := 23184, root := ~, start := 23152, stop := 23183 ), rec( conten\ t := ",\n and edges corresponding to the entries of ", count := [ 3, 1,\ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 9 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmod\ e := "Text", name := "A", next := 23246, root := ~, start := 23236, stop := 23\ 245 ), rec( content := ". ", count := [ 3, 1, 7, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 3, 1, 7, 10 ], mathmode := "Text", name := "P", next := 23252, root := ~, star\ t := 23248, stop := 23251 ), rec( content := "\n\n More precisely, ther\ e is an edge from vertex ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "C", next :=\ 23315, root := ~, start := 23307, stop := 23314 ), rec( content := " to ", coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "j", count := [ 3, 1, 7, 10 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "C", next := 23327, root := ~, start := 23319, stop :=\ 23326 ), rec( content := " if and\n only if ", count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "j", count := [ 3, 1, 7, 10 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 23359, root := ~, start := 23351, stop := 23358 ), rec( conte\ nt := " is in ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := re\ c( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode\ := "Text", name := "A", next := 23379, root := ~, start := 23369, stop := 233\ 78 ), rec( content := "[i]", count := [ 3, 1, 7, 10 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 23386, root := ~, start := 23366, stop := 23385 ), rec( conte\ nt := "; the source of this edge is\n ", count := [ 3, 1, 7, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 3, 1, 7, 10 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 23431, root := ~, start := 23423, stop := 23430 ), rec( conte\ nt := " and the range is ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "j", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "C", next :=\ 23457, root := ~, start := 23449, stop := 23456 ), rec( content := ". If ", coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "j", count := [ 3, 1, 7, 10 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "C", next := 23470, root := ~, start := 23462, stop :=\ 23469 ), rec( content := " occurs in\n ", count := [ 3, 1, 7, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( attributes := rec( ), content := [ rec( content := "obj", count\ := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "A", next := 23502, root := \ ~, start := 23492, stop := 23501 ), rec( content := "[i]", count := [ 3, 1, 7,\ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7\ , 10 ], mathmode := "Text", name := "C", next := 23509, root := ~, start := 23\ 489, stop := 23508 ), rec( content := " with multiplicity ", count := [ 3, 1, \ 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "k", count := [ 3, 1, 7, 10 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmod\ e := "Text", name := "C", next := 23536, root := ~, start := 23528, stop := 23\ 535 ), rec( content := ", then there are ", count := [ 3, 1, 7, 10 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "k", count := [ 3, 1, 7, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 23561, root := ~, start := 23553, stop := 23560 ), rec( conte\ nt := "\n edges from ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "C", next :=\ 23589, root := ~, start := 23581, stop := 23588 ), rec( content := " to ", coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "j", count := [ 3, 1, 7, 10 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "C", next := 23601, root := ~, start := 23593, stop :=\ 23600 ), rec( content := ".\n ", count := [ 3, 1, 7, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], name := "Item\ ", next := 23616, root := ~, start := 22934, stop := 23615 ), rec( attributes :\ = rec( ), content := [ rec( content := "for three lists", count := [ 3, 1, 7,\ 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 11 ], name := "Mar\ k", next := 23652, root := ~, start := 23624, stop := 23651 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 1\ 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 12 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode \ := "Text", name := "A", next := 23687, root := ~, start := 23677, stop := 2368\ 6 ), rec( content := " is a duplicate-free list, and ", count := [ 3, 1, 7, 12\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "source", count := [ 3, 1, 7, 12 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmod\ e := "Text", name := "A", next := 23731, root := ~, start := 23718, stop := 23\ 730 ), rec( content := " and\n ", count := [ 3, 1, 7, 12 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "range", count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "A", next := 23756, root := ~, start := 23744, stop := 23755 ), rec( conte\ nt := " are lists of equal length consisting of positive integers\n in \ the list ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[1 .. Lengt\ h(", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "obj", count := [\ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := \ [ 3, 1, 7, 12 ], mathmode := "Text", name := "A", next := 23861, root := ~, st\ art := 23851, stop := 23860 ), rec( content := ")]", count := [ 3, 1, 7, 12 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ] , math\ mode := "Text", name := "C", next := 23867, root := ~, start := 23835, stop :=\ 23866 ), rec( content := ", then this function\n returns a digraph wit\ h vertices\n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "E ^ 0 = ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "M", next \ := 23951, root := ~, start := 23936, stop := 23950 ), rec( attributes := rec( \ ), content := [ rec( content := "[1 .. Length(", count := [ 3, 1, 7, 12 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), c\ ontent := [ rec( content := "obj", count := [ 3, 1, 7, 12 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Tex\ t", name := "A", next := 23977, root := ~, start := 23967, stop := 23976 ), re\ c( content := ")]", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C" , next\ := 23983, root := ~, start := 23951, stop := 23982 ), rec( content := ", and\ \n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Length(", c\ ount := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), re\ c( attributes := rec( ), content := [ rec( content := "source", count := [ 3,\ 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3\ , 1, 7, 12 ], mathmode := "Text", name := "A", next := 24020, root := ~, start\ := 24007, stop := 24019 ), rec( content := ")", count := [ 3, 1, 7, 12 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], math\ mode := "Text", name := "C", next := 24025, root := ~, start := 23997, stop :=\ 24024 ), rec( content := " edges. For each ", count := [ 3, 1, 7, 12 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 3, 1, 7, 12 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "C", next := 24051, root := ~, start := 24043, stop := 24050 ), rec( conte\ nt := " in\n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "[1 .. Length(", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "sou\ rce", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "A", next := 2409\ 2, root := ~, start := 24079, stop := 24091 ), rec( content := ")]", count := \ [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next := 24098, root := ~, star\ t := 24063, stop := 24097 ), rec( content := " there exists an edge with sourc\ e\n vertex ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\ source[i]", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next :\ = 24163, root := ~, start := 24147, stop := 24162 ), rec( content := " and ran\ ge vertex ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "range[i]", \ count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next := 24196, root := \ ~, start := 24181, stop := 24195 ), rec( content := ".\n See ", count :\ = [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( Attr := "DigraphSource" ), content := 0, count := [ 3, 1, 7, 12\ ], mathmode := "Text", name := "Ref", next := 24237, root := ~, start := 2421\ 0, stop := 24236 ), rec( content := " and ", count := [ 3, 1, 7, 12 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "D\ igraphRange" ), content := 0, count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "Ref", next := 24268, root := ~, start := 24242, stop := 24267 ), rec( cont\ ent := ".\n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1\ , 7, 13 ], mathmode := "Text", name := "P", next := 24282, root := ~, start :=\ 24278, stop := 24281 ), rec( content := "\n\n The vertices of the digr\ aph will be labelled by the elements of\n ", count := [ 3, 1, 7, 13 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "obj", count := [ 3, 1, 7, 13 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 13 ], mathmode := "Text", name\ := "A", next := 24374, root := ~, start := 24364, stop := 24373 ), rec( conte\ nt := ".\n ", count := [ 3, 1, 7, 13 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 7, 12 ], name := "Item", next := 24389, root := ~, start := 236\ 59, stop := 24388 ), rec( attributes := rec( ), content := [ rec( content := \ "for an integer, and two lists", count := [ 3, 1, 7, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 14 ], name := "Mark", next := 2443\ 9, root := ~, start := 24397, stop := 24438 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n if ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "obj", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", next := 2\ 4474, root := ~, start := 24464, stop := 24473 ), rec( content := " is an inte\ ger, and ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "source", co\ unt := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", next := 24507, root := \ ~, start := 24494, stop := 24506 ), rec( content := " and\n ", count :=\ [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "range", count := [ 3, 1, 7, 1\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, \ 15 ], mathmode := "Text", name := "A", next := 24532, root := ~, start := 2452\ 0, stop := 24531 ), rec( content := " are lists of equal length consisting of \ positive integers\n in the list ", count := [ 3, 1, 7, 15 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "[1 .. ", count := [ 3, 1, 7, 15 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( con\ tent := "obj", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", nex\ t := 24630, root := ~, start := 24620, stop := 24629 ), rec( content := "]", c\ ount := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next := 24635, root := \ ~, start := 24611, stop := 24634 ), rec( content := ", then this function\n \ returns a digraph with vertices\n ", count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "E ^ 0 = ", count := [ 3, 1, 7, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "T\ ext", name := "M", next := 24719, root := ~, start := 24704, stop := 24718 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[1 .. ", count := [ 3, 1, 7, \ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 15 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode\ := "Text", name := "A", next := 24738, root := ~, start := 24728, stop := 247\ 37 ), rec( content := "]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name\ := "C", next := 24743, root := ~, start := 24719, stop := 24742 ), rec( conte\ nt := ", and\n ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "Length(", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", \ root := ~ ), rec( attributes := rec( ), content := [ rec( content := "source"\ , count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) \ ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", next := 24780, r\ oot := ~, start := 24767, stop := 24779 ), rec( content := ")", count := [ 3, \ 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3,\ 1, 7, 15 ], mathmode := "Text", name := "C", next := 24785, root := ~, start \ := 24757, stop := 24784 ), rec( content := " edges. For each ", count := [ 3,\ 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "i", count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "C", next := 24811, root := ~, start := 24803, stop :=\ 24810 ), rec( content := " in\n ", count := [ 3, 1, 7, 15 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "[1 .. Length(", count := [ 3, 1, 7, 15 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "source", count := [ 3, 1, 7, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name \ := "A", next := 24852, root := ~, start := 24839, stop := 24851 ), rec( conten\ t := ")]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next :=\ 24858, root := ~, start := 24823, stop := 24857 ), rec( content := " there ex\ ists an edge with source\n vertex ", count := [ 3, 1, 7, 15 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "source[i]", count := [ 3, 1, 7, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Tex\ t", name := "C", next := 24923, root := ~, start := 24907, stop := 24922 ), rec( cont\ ent := " and range vertex ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "range[i]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next\ := 24956, root := ~, start := 24941, stop := 24955 ), rec( content := ".\n \ See ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( Attr := "DigraphSource" ), content := 0, coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "Ref", next := 24997, root :\ = ~, start := 24970, stop := 24996 ), rec( content := " and ", count := [ 3, 1\ , 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( Attr := "DigraphRange" ), content := 0, count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "Ref", next := 25028, root := ~, start := 25002, stop \ := 25027 ), rec( content := ".\n ", count := [ 3, 1, 7, 15 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], name := "Ite\ m", next := 25043, root := ~, start := 24446, stop := 25042 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a list and a function", count := [ \ 3, 1, 7, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 16 ], name := "Mar\ k", next := 25089, root := ~, start := 25051, stop := 25088 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 1\ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "list", count := [ 3, 1, 7, 17 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode\ := "Text", name := "A", next := 25125, root := ~, start := 25114, stop := 251\ 24 ), rec( content := " is a list and ", count := [ 3, 1, 7, 17 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "func", count := [ 3, 1, 7, 17 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name\ := "A", next := 25151, root := ~, start := 25140, stop := 25150 ), rec( conte\ nt := " is a function taking 2\n arguments that are elements of ", coun\ t := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "list", count := [ 3, 1, 7, 17\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 1\ 7 ], mathmode := "Text", name := "A", next := 25225, root := ~, start := 25214 , stop\ := 25224 ), rec( content := ", and ", count := [ 3, 1, 7, 17 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "func", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name :=\ "A", next := 25242, root := ~, start := 25231, stop := 25241 ), rec( content \ := " returns\n ", count := [ 3, 1, 7, 17 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "true", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "K", next :=\ 25270, root := ~, start := 25259, stop := 25269 ), rec( content := " or ", coun\ t := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "false", count := [ 3, 1, 7, 1\ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, \ 17 ], mathmode := "Text", name := "K", next := 25286, root := ~, start := 2527\ 4, stop := 25285 ), rec( content := ", then this operation creates a digraph w\ ith\n vertices ", count := [ 3, 1, 7, 17 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "[1 .. Length(", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "A", next := 2537\ 5, root := ~, start := 25364, stop := 25374 ), rec( content := ")]", count := \ [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 25381, root := ~, star\ t := 25348, stop := 25380 ), rec( content := " and an edge from vertex\n \ ", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "i", count := [ 3\ , 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 25422, root := ~, star\ t := 25414, stop := 25421 ), rec( content := " to vertex ", count := [ 3, 1, 7\ , 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "j", count := [ 3, 1, 7, 17 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode :\ = "Text", name := "C", next := 25441, root := ~, start := 25433, stop := 25440\ ), rec( content := " if and only if\n ", count := [ 3, 1, 7, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( attributes := rec( ), content := [ rec( content := "func", coun\ t := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], cou\ nt := [ 3, 1, 7, 17 ], mathmode := "Text", name := "A", next := 25479, root :=\ ~, start := 25468, stop := 25478 ), rec( content := "(", count := [ 3, 1, 7, \ 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "list", count := [ 3, 1, 7, 17 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmod\ e := "Text", name := "A", next := 25491, root := ~, start := 25480, stop := 25\ 490 ), rec( content := "[i], ", count := [ 3, 1, 7, 17 ], mathmode := "Text", \ name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( c\ ontent := "list", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "A", \ next := 25507, root := ~, start := 25496, stop := 25506 ), rec( content := "[j\ ])", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 25515 , root\ := ~, start := 25465, stop := 25514 ), rec( content := " returns ", count := [ \ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "true", count := [ 3, 1, 7, 17 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ] , math\ mode := "Text", name := "K", next := 25535, root := ~, start := 25524, stop :=\ 25534 ), rec( content := ".\n ", count := [ 3, 1, 7, 17 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], name := "Ite\ m", next := 25550, root := ~, start := 25096, stop := 25549 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a group, a list, and two function\ s", count := [ 3, 1, 7, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 18 ], name := "Mar\ k", next := 25609, root := ~, start := 25558, stop := 25608 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n The arguments will be ", coun\ t := [ 3, 1, 7, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G, list, act, adj", count := \ [ 3, 1, 7, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 19 ], mathmode := "Text", name := "A", next := 25677, root := ~, star\ t := 25653, stop := 25676 ), rec( content := ". ", count := [ 3, 1, 7, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := 0, count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "P", next := 2\ 5683, root := ~, start := 25679, stop := 25682 ), rec( content := "\n\n \ Let ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count := \ [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 25705, root := ~, star\ t := 25697, stop := 25704 ), rec( content := " be a group acting on the object\ s in ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "list", count \ := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 25753, root := ~ , star\ t := 25742, stop := 25752 ), rec( content := " via\n the action ", coun\ t := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "act", count := [ 3, 1, 7, 20 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20\ ], mathmode := "Text", name := "A", next := 25787, root := ~, start := 25777, stop\ := 25786 ), rec( content := ", and let ", count := [ 3, 1, 7, 20 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "adj", count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "A", next := 25807, root := ~, start := 25797, stop := 25806 ), rec( conte\ nt := " be a function taking\n two objects from ", count := [ 3, 1, 7, \ 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "list", count := [ 3, 1, 7, 20 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmod\ e := "Text", name := "A", next := 25865, root := ~, start := 25854, stop := 25\ 864 ), rec( content := " as arguments and returning ", count := [ 3, 1, 7, 20 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "true", count := [ 3, 1, 7, 20 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text" , name\ := "C", next := 25904, root := ~, start := 25893, stop := 25903 ), rec( conte\ nt := " or\n ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "false", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "C", next := \ 25928, root := ~, start := 25916, stop := 25927 ), rec( content := ". The func\ tion ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "adj", count :\ = [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 25953, root := ~, star\ t := 25943, stop := 25952 ), rec( content := " will describe the adjacency\n \ between objects from ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "list", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next\ := 26022, root := ~, start := 26011, stop := 26021 ), rec( content := ", whic\ h is invariant under the action of\n ", count := [ 3, 1, 7, 20 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "G", count := [ 3, 1, 7, 20 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "A", next := 26079, root := ~, start := 26071, stop := 26078 ), rec( conte\ nt := ". This variant of the constructor returns a digraph with\n verti\ ces the objects of ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "list", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 26\ 179, root := ~, start := 26168, stop := 26178 ), rec( content := " and directe\ d edges ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[x, y]", co\ unt := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 20 ], mathmode := "Text", name := "C", next := 26212, root := \ ~, start := 26199, stop := 26211 ), rec( content := "\n when ", count :\ = [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "f(x, y)", count := [ 3, 1, 7,\ 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7\ , 20 ], mathmode := "Text", name := "C", next := 26240, root := ~, start := 26\ 226, stop := 26239 ), rec( content := " is ", count := [ 3, 1, 7, 20 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "true", count := [ 3, 1, 7, 20 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "C", next := 26255, root := ~, start := 26244, stop := 26254 ), rec( conte\ nt := ". ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "P", next := 26261, root := ~, start := 26257, stop :=\ 26260 ), rec( content := "\n\n The action of the group ", count := [ 3\ , 1, 7, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "G", count := [ 3, 1, 7, 21 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "A", next := 26303, root := ~, start := 26295, stop :=\ 26302 ), rec( content := " on the objects in ", count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "list", count := [ 3, 1, 7, 21 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 21 ], mathmode := "Text" , name\ := "A", next := 26333, root := ~, start := 26322, stop := 26332 ), rec( conte\ nt := " is stored\n in the attribute ", count := [ 3, 1, 7, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := \ "DigraphGroup" ), content := 0, count := [ 3, 1, 7, 21 ], mathmode := "Text", name\ := "Ref", next := 26395, root := ~, start := 26369, stop := 26394 ), rec( cont\ ent := ", and is used to speed up\n operations like ", count := [ 3, 1,\ 7, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Attr := "DigraphDiameter" ), content := 0, count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "Ref", next := 26474, root := ~, start := 26445, stop \ := 26473 ), rec( content := ".\n ", count := [ 3, 1, 7, 21 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 19 ], name := "Ite\ m", next := 26489, root := ~, start := 25616, stop := 26488 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a Grape package graph", count := [ \ 3, 1, 7, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 22 ], name := "Mar\ k", next := 26535, root := ~, start := 26497, stop := 26534 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 2\ 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 23 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], mathmode \ := "Text", name := "A", next := 26570, root := ~, start := 26560, stop := 2656\ 9 ), rec( content := " is a ", count := [ 3, 1, 7, 23 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content :=\ [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "ht\ tp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ 3, 1, 7, 23 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], mathmod\ e := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ] , coun\ t := [ 3, 1, 7, 23 ], name := "Alt", next := 93, root := ~, start := 1, stop :\ = 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes\ := rec( ), content := [ rec( content := "Grape", count := [ 3, 1, 7, 23 ], n\ ame := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], name := "Package", n\ ext := 137, root := ~, start := 113, stop := 136 ) ], count := [ 3, 1, 7, 23 ] , math\ mode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 14\ 2 ), rec( content := " package graph (i.e. a record for which the\n fun\ ction ", count := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := [ rec( content := "IsGraph", co\ unt := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 23 ], mathmode := "Text", name := "C", next := 26658, root := \ ~, start := 26644, stop := 26657 ), rec( content := " returns ", count := [ 3,\ 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "true", count := [ 3, 1, 7, 23 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], math\ mode := "Text", name := "K", next := 26678, root := ~, start := 26667, stop :=\ 26677 ), rec( content := "), then this function returns\n a digraph is\ omorphic to ", count := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "obj", count\ := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 23 ], mathmode := "Text", name := "A", next := 26750, root := \ ~, start := 26740, stop := 26749 ), rec( content := ".\n ", count := [ 3,\ 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], name := "Ite\ m", next := 26765, root := ~, start := 26542, stop := 26764 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a binary relation", count := [ 3,\ 1, 7, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 24 ], name := "Mar\ k", next := 26807, root := ~, start := 26773, stop := 26806 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 2\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 25 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode \ := "Text", name := "A", next := 26842, root := ~, start := 26832, stop := 2684\ 1 ), rec( content := " is a binary relation on the points ", count := [ 3, 1, \ 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "[1 .. n]", count := [ 3, 1, 7, 25 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], math\ mode := "Text", name := "C", next := 26893, root := ~, start := 26878, stop :=\ 26892 ), rec( content := " for\n some posititve integer ", count := [ \ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "n", count := [ 3, 1, 7, 25 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], math\ mode := "Text", name := "M", next := 26937, root := ~, start := 26929, stop :=\ 26936 ), rec( content := ", then this function returns the digraph\n d\ efined by ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "obj", count\ := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "A", next := 27007, root := \ ~, start := 26997, stop := 27006 ), rec( content := ". Specifically, this func\ tion returns a digraph\n which has ", count := [ 3, 1, 7, 25 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "n", count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "M", next := 27081, root := ~, start := 27073, stop := 27080 ), rec( conte\ nt := " vertices, and which has an edge with source ", count := [ 3, 1, 7, 25 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 3, 1, 7, 25 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "C", next := 27134, root := ~, start := 27126, stop := 27133 ), rec( conte\ nt := "\n and range ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "j", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "C", next :=\ 27161, root := ~, start := 27153, stop := 27160 ), rec( content := " if and o\ nly if ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[i,j]", cou\ nt := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "C", next := 27189, root := \ ~, start := 27177, stop := 27188 ), rec( content := " is a pair in\n th\ e binary relation ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\ obj", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "A", next := 2724\ 1, root := ~, start := 27231, stop := 27240 ), rec( content := ".\n ", coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], name := "Ite\ m", next := 27256, root := ~, start := 26814, stop := 27255 ) ], count := [ 3, 1, 7, 8 ], mathmode := "Text", name := "List", next := 27268, root := ~, start := 22866, stop := 27267 ), rec( content := "\n\n \ ", count := [ 3, 1, 7, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [2, 5, 8, 10], [2, 3, 4, 2, 5, 6, 8, 9, 10], [1],\n> \ [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],\n> [1, 5, 9], [1, 2, 7,\ 8], [3, 5]]);\n\ngap> gr :\ = Digraph([\"a\", \"b\", \"c\"], [\"a\"], [\"b\"]);\n\ngap> gr := Digraph(5, [1, 2, 2, 4, 1, 1], [2, 3, 5, 5, 1, \ 1]);\n\ngap> Petersen := Grap\ h(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y) return Intersection(x\ , y) = []; end);;\ngap> Digraph(Petersen);\n\ngap> gr := Digraph([1 .. 10], ReturnTrue);\n", count := [ 3, 1, 7, 28 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 28 ], mathmode := "Text", name := "Example", next := 28010, root := ~, start := 27274, stop := 28009 ), rec( content := "\n\n \ The next example illustrates the uses of the fourth and fifth variants of\n \ this constructor. The resulting digraph is a strongly regular graph, and\ \n it is actually the point graph of the van Lint-Schrijver partial\n \ geometry, ", count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "vLS81" ) , content := 0, count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "Cite", next := 28277, root := ~, start := 28258, stop := 28276 ), rec( content := ". The a\ lgebraic description\n is taken from the seminal paper of Calderbank and \ Kantor ", count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "CK86" ), content := 0, count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "Cite", next := 28395, root := ~, start := 28369, stop := 28394 ), rec( content := ".\n", count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> f := GF(3 ^ 4);\nGF(3^4)\ngap> gamma := First(f, x -> Order(x) = 5);\nZ(\ 3^4)^64\ngap> L := Union([Zero(f)], List(Group(gamma)));\n[ 0*Z(3), Z(3)^0, Z(\ 3^4)^16, Z(3^4)^32, Z(3^4)^48, Z(3^4)^64 ]\ngap> omega := Union(List(L, x -> L\ ist(Difference(L, [x]), y -> x - y)));\n[ Z(3)^0, Z(3), Z(3^4)^5, Z(3^4)^7, Z(\ 3^4)^8, Z(3^4)^13, Z(3^4)^15, \n Z(3^4)^16, Z(3^4)^21, Z(3^4)^23, Z(3^4)^24, \ Z(3^4)^29, Z(3^4)^31, \n Z(3^4)^32, Z(3^4)^37, Z(3^4)^39, Z(3^4)^45, Z(3^4)^4\ 7, Z(3^4)^48, \n Z(3^4)^53, Z(3^4)^55, Z(3^4)^56, Z(3^4)^61, Z(3^4)^63, Z(3^4\ )^64, \n Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^77, Z(3^4)^79 ]\ngap> adj :=\ function(x, y)\n> return x - y in omega;\n> end;\nfunction( x, y ) ... end\ \ngap> digraph := Digraph(AsList(f), adj);\n\ngap> group := Group(Z(3));;\ngap> act := \\*;\n\ngap> digraph := Digraph(group, List(f), act, adj);\n\n", count := [ 3, 1, 7, 30 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 30 ], mathmode := "Text", name := "Example", next := 29365, root := ~, start := 28397, stop := 29364 ), rec( content := "\n ", count := [ 3, 1, 7, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Description", next := 29382, root := ~, start := 22350, stop := 29381 ) ], count := [ 3, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 29396, root := ~, start := 22029, stop := 29395 ), rec( content := "\n\n ", count := [ 3, 1, 7, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByAdjace\ ncyMatrix" ), content := 0, count := [ 3, 1, 8, 2 ], mathmode := "Text", name := "Oper", next := 29475, root := ~, start := 29417, stop := 29474 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 29507, root := ~, start := 29478, stop := 29506 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 29570, root := ~, start := 29559, stop := 29569 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 29716, root := ~, start := 29705, stop := 29715 ), rec( content := " is ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 29750, root := ~, start := 29720, stop := 29749 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 29818, root := ~, start := 29807, stop := 29817 ), rec( content := " is ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 29858, root := ~, start := 29822, stop := 29857 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 29943, root := ~, start := 29932, stop := 29942 ), rec( content := " is not \ present, then ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 30001, root := ~, start := 29965, stop := 30000 ), rec( content := " is used\ by default.", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "P", next := 30025, root := ~, start := 30021, stop := 30024 ), rec( content := "\n\n \ If ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "A", next := 30045, root := ~, start := 30034, stop := 30044 ), rec( content := " is the \ adjacency matrix of a digraph in the sense of\n ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "Ref", next := 30132, root := ~, start := 30103, stop := 30131 ), rec( content := ", then t\ his operation returns the digraph\n which is defined by ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "A", next := 30209, root := ~, start := 30198, stop := 30208 ), rec( content := ". ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "P", next := 30215, root := ~, start := 30211, stop := 30214 ), rec( content := "\n\n \ Alternatively, if ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "A", next := 30250, root := ~, start := 30239, stop := 30249 ), rec( content := " is a sq\ uare boolean matrix, then this\n operation returns the digraph with ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ength(", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 30342, root := ~, start := 30328, stop := 30341 ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "A", next := 30353, root := ~, start := 30342, stop := 30352 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 30361, root := ~, start := 30353, stop := 30360 ), rec( content := "\n ve\ rtices which has the edge ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 30406, root := ~, start := 30394, stop := 30405 ), rec( content := " if and \ only if\n ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "A", next := 30437, root := ~, start := 30426, stop := 30436 ), rec( attributes := rec( ), content := [ rec( content := "[\ i][j]", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 30450, root := ~, start := 30437, stop := 30449 ), rec( content := " is ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "K", next := 30465, root := ~, start := 30454, stop := 30464 ), rec( content := ". ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 8, 6 ], mathmode := "Text", name := "P", next := 30471, root := ~, start := 30467, stop := 30470 ), rec( content := "\n\n \ ", count := [ 3, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphByAdjacencyMatrix([\n> [0, 1, 0, 2, 0],\n> [1, 1, 1, 0, 1],\n> [0\ , 3, 2, 1, 1],\n> [0, 0, 1, 0, 1],\n> [2, 0, 0, 0, 0]]);\n\ngap> D := DigraphByAdjacencyMatrix([\n> [true,\ false, true],\n> [false, false, true],\n> [false, true, false]]);\n\ngap> OutNeighbours(D);\n[ [ 1, 3 ], [ 3 ],\ [ 2 ] ]\ngap> D := DigraphByAdjacencyMatrix(IsMutableDigraph, \n> [[true, fal\ se, true],\n> [false, false, true],\n> [false, true, false]]);\n\n", count := [ 3, 1, 8, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 7 ], mathmode := "Text", name := "Example", next := 31067, root := ~, start := 30477, stop := 31066 ), rec( content := "\n ", count := [ 3, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 31084, root := ~, start := 29510, stop := 31083 ) ], count := [ 3, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 31098, root := ~, start := 29402, stop := 31097 ), rec( content := "\n\n ", count := [ 3, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]list[, n\ ]", Name := "DigraphByEdges" ), content := 0, count := [ 3, 1, 9, 2 ], mathmode := "Text", name := "Oper", next := 31172, root := ~, start := 31119, stop := 31171 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Returns", next := 31204, root := ~, start := 31175, stop := 31203 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 31267, root := ~, start := 31256, stop := 31266 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 31413, root := ~, start := 31402, stop := 31412 ), rec( content := " is ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Ref", next := 31447, root := ~, start := 31417, stop := 31446 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 31515, root := ~, start := 31504, stop := 31514 ), rec( content := " is ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Ref", next := 31555, root := ~, start := 31519, stop := 31554 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 31640, root := ~, start := 31629, stop := 31639 ), rec( content := " is not \ present, then ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Ref", next := 31698, root := ~, start := 31662, stop := 31697 ), rec( content := " is used\ by default.", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "P", next := 31722, root := ~, start := 31718, stop := 31721 ), rec( content := "\n\n \ If ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "A", next := 31742, root := ~, start := 31731, stop := 31741 ), rec( content := " is list\ of pairs of positive integers, then this function\n returns the digraph wi\ th the minimum number of vertices ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "C", next := 31869, root := ~, start := 31861, stop := 31868 ), rec( content := " such th\ at\n its list equal ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "A", next := 31910, root := ~, start := 31899, stop := 31909 ), rec( content := ".", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "P", next := 31915, root := ~, start := 31911, stop := 31914 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "A", next := 31961, root := ~, start := 31953, stop := 31960 ), rec( content := " is a po\ sitive integer with\n ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "n", count := [ 3, 1, 9, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := \ "Text", name := "A", next := 32004, root := ~, start := 31996, stop := 32003 ) , rec( content := "\ >= m", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "C", next := 32013, root := ~, start := 31993, stop := 32012 ), rec( content := " (with " , count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "C", next := 32028, root := ~, start := 32020, stop := 32027 ), rec( content := " defined\ as above),\n then this function returns the digraph with ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "A", next := 32104, root := ~, start := 32096, stop := 32103 ), rec( content := " vertice\ s and list\n ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "A", next := 32138, root := ~, start := 32127, stop := 32137 ), rec( content := ". ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "P", next := 32144, root := ~, start := 32140, stop := 32143 ), rec( content := "\n\n \ See ", count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Edges" ), content := 0, count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "Ref", next := 32180, root := ~, start := 32154, stop := 32179 ), rec( content := ".\n\n \ ", count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphByEdges(\n> [[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],\n> [4, 6], \ [5, 2], [5, 4], [5, 6], [6, 6]]);\n\ngap> DigraphByEdges(\n> [[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],\n> [4, 6\ ], [5, 2], [5, 4], [5, 6], [6, 6]], 12);\n\ngap> DigraphByEdges(IsMutableDigraph, \n> [[1, 3], [2, 1], [2, 3],\ [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);\n\n", count := [ 3, 1, 9, 7 ], name := "PCDAT\ A", root := ~ ) ], count := [ 3, 1, 9, 7 ], mathmode := "Text", name := "Example", next := 32710, root := ~, start := 32187, stop := 32709 ), rec( content := "\n ", count := [ 3, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 32727, root := ~, start := 31207, stop := 32726 ) ], count := [ 3, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 32741, root := ~, start := 31104, stop := 32740 ), rec( content := "\n\n ", count := [ 3, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "G, edges[, n]", Name := "EdgeOrbitsDigra\ ph" ), content := 0, count := [ 3, 1, 10, 2 ], mathmode := "Text", name := "Oper", next := 32814, root := ~, start := 32762, stop := 32813 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable digraph.\n ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "Returns", next := 32865, root := ~, start := 32817, stop := 32864 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 32897, root := ~, start := 32889, stop := 32896 ), rec( content := " is a pe\ rmutation group, ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 32934, root := ~, start := 32922, stop := 32933 ), rec( content := " is an e\ dge or list of\n edges, and ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 32980, root := ~, start := 32972, stop := 32979 ), rec( content := " is a no\ n-negative integer such that ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 33025, root := ~, start := 33017, stop := 33024 ), rec( content := " fixes\n\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 1, 10, 3 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := \ "Text", name := "A", next := 33053, root := ~, start := 33045, stop := 33052 ) , rec( content := "\ ]", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "C", next := 33058, root := ~, start := 33036, stop := 33057 ), rec( content := " setwise\ , then\n this operation returns an immutable digraph with ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 33134, root := ~, start := 33126, stop := 33133 ), rec( content := " vertice\ s and the union\n of the orbits of the edges in ", count := [ 3, 1, 10, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ edges ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 33206, root := ~, start := 33192, stop := 33205 ), rec( content := " under t\ he action of the\n permutation group ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 33261, root := ~, start := 33253, stop := 33260 ), rec( content := ". An ed\ ge in this context is simply a pair of\n positive integers.\n ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "P", next := 33339, root := ~, start := 33335, stop := 33338 ), rec( content := "\n\n \ If the optional third argument ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "A", next := 33384, root := ~, start := 33376, stop := 33383 ), rec( content := " is not \ present, then the largest\n moved point of the permutation group ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "A", next := 33467, root := ~, start := 33459, stop := 33466 ), rec( content := " is used\ by default.", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 10, 5 ], mathmode := "Text", name := "P", next := 33491, root := ~, start := 33487, stop := 33490 ), rec( content := "\n\n \ ", count := [ 3, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := EdgeOrbitsDigraph(Group((1, 3), (1, 2)(3, 4)),\n> \ [[1, 2], [4, 5]], 5);\n\ngap> OutNeighbours(digraph);\n[ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4,\ 5 ], [ 1, 3, 5 ], [ ] ]\ngap> RepresentativeOutNeighbours(digraph);\n[ [ 2, \ 4, 5 ], [ ] ]\n", count := [ 3, 1, 10, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 6 ], mathmode := "Text", name := "Example", next := 33849, root := ~, start := 33497, stop := 33848 ), rec( content := "\n ", count := [ 3, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 33866, root := ~, start := 32868, stop := 33865 ) ], count := [ 3, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 33880, root := ~, start := 32747, stop := 33879 ), rec( content := "\n\n ", count := [ 3, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByInNeig\ hbours" ), content := 0, count := [ 3, 1, 11, 2 ], mathmode := "Text", name := "Oper", next := 33956, root := ~, start := 33901, stop := 33955 ), rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByInNeig\ hbors" ), content := 0, count := [ 3, 1, 11, 4 ], mathmode := "Text", name := "Oper", next := 34013, root := ~, start := 33959, stop := 34012 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Returns", next := 34045, root := ~, start := 34016, stop := 34044 ), rec( attributes := rec( ), content := [ rec( content := "\n If th\ e optional first argument ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 34106, root := ~, start := 34095, stop := 34105 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph be\ ing created will\n belong to. For example, if ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 34248, root := ~, start := 34237, stop := 34247 ), rec( content := " is ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Ref", next := 34282, root := ~, start := 34252, stop := 34281 ), rec( content := ",\n the\ n the digraph being created will be mutable, if ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 34348, root := ~, start := 34337, stop := 34347 ), rec( content := " is ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Ref", next := 34386, root := ~, start := 34352, stop := 34385 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 34469, root := ~, start := 34458, stop := 34468 ), rec( content := " is not \ present, then ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Ref", next := 34525, root := ~, start := 34491, stop := 34524 ), rec( content := " is used\ by default.", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "P", next := 34549, root := ~, start := 34545, stop := 34548 ), rec( content := "\n\n If\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "A", next := 34567, root := ~, start := 34556, stop := 34566 ), rec( content := " is a li\ st of lists of positive integers list the range\n ", count := [ 3, 1, 11, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. Length(", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := \ "Text", name := "A", next := 34652, root := ~, start := 34641, stop := 34651 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34658, root := ~, start := 34625, stop := 34657 ), rec( content := ", then t\ his function returns the\n digraph with vertices ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ ^0=", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "M", next := 34726, root := ~, start := 34715, stop := 34725 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. Length(", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := \ "Text", name := "A", next := 34753, root := ~, start := 34742, stop := 34752 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34759, root := ~, start := 34726, stop := 34758 ), rec( content := ", and\n \ edges corresponding to the entries of ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "A", next := 34816, root := ~, start := 34805, stop := 34815 ), rec( content := ". More p\ recisely, there is an\n edge with source vertex ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34880, root := ~, start := 34872, stop := 34879 ), rec( content := " and ran\ ge vertex ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34906, root := ~, start := 34898, stop := 34905 ), rec( content := " if ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34918, root := ~, start := 34910, stop := 34917 ), rec( content := " is list\ \n ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 6 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := \ "Text", name := "A", next := 34943, root := ~, start := 34932, stop := 34942 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34950, root := ~, start := 34929, stop := 34949 ), rec( content := ". ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "P", next := 34956, root := ~, start := 34952, stop := 34955 ), rec( content := "\n\n If\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 34971, root := ~, start := 34963, stop := 34970 ), rec( content := " occurs \ list ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 7 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := \ "Text", name := "A", next := 34998, root := ~, start := 34987, stop := 34997 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 35005, root := ~, start := 34984, stop := 35004 ), rec( content := " with mu\ ltiplicity ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 35032, root := ~, start := 35024, stop := 35031 ), rec( content := ",\n the\ n there are ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 35059, root := ~, start := 35051, stop := 35058 ), rec( content := " multipl\ e edges from ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 35088, root := ~, start := 35080, stop := 35087 ), rec( content := " to ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 35100, root := ~, start := 35092, stop := 35099 ), rec( content := ". ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "P", next := 35106, root := ~, start := 35102, stop := 35105 ), rec( content := "\n\n Se\ e ", count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "InNeigh\ bours" ), content := 0, count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "Ref", next := 35140, root := ~, start := 35114, stop := 35139 ), rec( content := ".\n\n \ ", count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphByInNeighbours([\n> [2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10]\ ,\n> [1], [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],\n> [1, 5, 9],\ [1, 2, 7, 8], [3, 5]]);\n\ngap>\ D := DigraphByInNeighbours([[2, 3, 2], [1], [1, 2, 3]]);\n\ngap> D := DigraphByInNeighbours(IsMutableDigra\ ph, \n> [[2, 3, 2], [1], [1, 2, 3]]);\n\n", count := [ 3, 1, 11, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 11, 9 ], mathmode := "Text", name := "Example", next := 35676, root := ~, start := 35147, stop := 35675 ), rec( content := "\n ", count := [ 3, 1, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Description", next := 35693, root := ~, start := 34048, stop := 35692 ) ], count := [ 3, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 35707, root := ~, start := 33886, stop := 35706 ), rec( content := "\n\n ", count := [ 3, 1, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "G[, gens]", Name := "CayleyDigraph" ), content := 0, count := [ 3, 1, 12, 2 ], mathmode := "Text", name := "Oper", next := 35772, root := ~, start := 35728, stop := 35771 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble digraph.", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 35815, root := ~, start := 35775, stop := 35814 ), rec( attributes := rec( ), content := [ rec( content := "\n Let\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 35848, root := ~, start := 35840, stop := 35847 ), rec( content := " be any \ group and let ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 35881, root := ~, start := 35870, stop := 35880 ), rec( content := " be a li\ st of elements of\n ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 35919, root := ~, start := 35911, stop := 35918 ), rec( content := ". This o\ peration returns an immutable digraph that corresponds to\n the Cayley grap\ h of ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 36017, root := ~, start := 36009, stop := 36016 ), rec( content := " with re\ spect\n ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 36046, root := ~, start := 36035, stop := 36045 ), rec( content := ". The ve\ rtices are the elements of ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 36089, root := ~, start := 36081, stop := 36088 ), rec( content := ". There \ exists an edge\n from the vertex ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 36140, root := ~, start := 36132, stop := 36139 ), rec( content := " to the \ vertex ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 36163, root := ~, start := 36155, stop := 36162 ), rec( content := " if and \ only if there exists\n a generator ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 36216, root := ~, start := 36208, stop := 36215 ), rec( content := " in ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 36231, root := ~, start := 36220, stop := 36230 ), rec( content := " such th\ at ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ * g = y", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 36258, root := ~, start := 36242, stop := 36257 ), rec( content := ". ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "P", next := 36264, root := ~, start := 36260, stop := 36263 ), rec( content := "\n\n \ ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( content := " FIXME W\ hat is the correspondence between vertices and elements? ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "A", next := 36391, root := ~, start := 36380, stop := 36390 ), rec( content := " is not \ present, then the\n generators of ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "A", next := 36443, root := ~, start := 36435, stop := 36442 ), rec( content := " are use\ d by default.", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "P", next := 36468, root := ~, start := 36464, stop := 36467 ), rec( content := "\n\n \ The digraph created by this operation belongs to the category ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "Ref", next := 36571, root := ~, start := 36536, stop := 36570 ), rec( content := ", the gr\ oup ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "A", next := 36591, root := ~, start := 36583, stop := 36590 ), rec( content := " can be \ recovered from the\n digraph using ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "GroupOf\ CayleyDigraph" ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "Ref", next := 36670, root := ~, start := 36636, stop := 36669 ), rec( content := ", and th\ e generators\n ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "A", next := 36706, root := ~, start := 36695, stop := 36705 ), rec( content := " can be \ obtained using ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Generat\ orsOfCayleyDigraph" ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "Ref", next := 36768, root := ~, start := 36729, stop := 36767 ), rec( content := ".", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 12, 6 ], mathmode := "Text", name := "P", next := 36773, root := ~, start := 36769, stop := 36772 ), rec( content := "\n\n \ Note that this function can only return an immutable digraph.\n\n ", count := [ 3, 1, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := DihedralGroup(8);\n\ngap> Cay\ leyDigraph(G);\n\ngap> G := Dihed\ ralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> CayleyDigraph(G);\ \n\ngap> digraph := CayleyDigraph\ (G, [()]);\n\ngap> GroupOfCayleyDi\ graph(digraph) = G;\ntrue\ngap> GeneratorsOfCayleyDigraph(digraph);\n[ () ]", count := [ 3, 1, 12, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 12, 7 ], mathmode := "Text", name := "Example", next := 37329, root := ~, start := 36846, stop := 37328 ), rec( content := "\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 37346, root := ~, start := 35818, stop := 37345 ) ], count := [ 3, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 37360, root := ~, start := 35713, stop := 37359 ), rec( content := "\n\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 37374, root := ~, start := 19314, stop := 37373 ), rec( content := "\n\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Changing represen\ tations", count := [ 3, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 37430, root := ~, start := 37387, stop := 37429 ), rec( content := "\n ", count := [ 3, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AsBinaryRelatio\ n" ), content := 0, count := [ 3, 2, 1, 2 ], mathmode := "Text", name := "Oper", next := 37495, root := ~, start := 37450, stop := 37494 ), rec( attributes := rec( ), content := [ rec( content := "A binary \ relation.", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "Returns", next := 37535, root := ~, start := 37498, stop := 37534 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "A", next := 37573, root := ~, start := 37559, stop := 37572 ), rec( content := " is a di\ graph with a positive number of vertices\n ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "M", next := 37634, root := ~, start := 37626, stop := 37633 ), rec( content := ", and no\ multiple edges, then this operation returns a binary\n relation on the poi\ nts ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1..n]", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "C", next := 37736, root := ~, start := 37723, stop := 37735 ), rec( content := ".\n T\ he pair ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "C", next := 37763, root := ~, start := 37751, stop := 37762 ), rec( content := " is in t\ he binary relation if and only if\n ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "C", next := 37821, root := ~, start := 37809, stop := 37820 ), rec( content := " is an e\ dge in ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "A", next := 37850, root := ~, start := 37836, stop := 37849 ), rec( content := ".\n " , count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 1, 4 ], mathmode := "Text", name := "P", next := 37860, root := ~, start := 37856, stop := 37859 ), rec( content := "\n", count := [ 3, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 2], [1, 2], [2], [3, 4]]);\n\ngap> AsBinaryRelation(D);\nBinary Relation on 4 points\n", count := [ 3, 2, 1, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 1, 5 ], mathmode := "Text", name := "Example", next := 38042, root := ~, start := 37861, stop := 38041 ), rec( content := "\n ", count := [ 3, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 38059, root := ~, start := 37538, stop := 38058 ) ], count := [ 3, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 38073, root := ~, start := 37435, stop := 38072 ), rec( content := "\n\n ", count := [ 3, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]trans[, \ n]", Name := "AsDigraph" ), content := 0, count := [ 3, 2, 2, 2 ], mathmode := "Text", name := "Oper", next := 38143, root := ~, start := 38094, stop := 38142 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "K", next := 38180, root := ~, start := 38169, stop := 38179 ), rec( content := ".", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 38191, root := ~, start := 38146, stop := 38190 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 38254, root := ~, start := 38243, stop := 38253 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 38400, root := ~, start := 38389, stop := 38399 ), rec( content := " is ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 38434, root := ~, start := 38404, stop := 38433 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 38502, root := ~, start := 38491, stop := 38501 ), rec( content := " is ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 38542, root := ~, start := 38506, stop := 38541 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 38627, root := ~, start := 38616, stop := 38626 ), rec( content := " is not \ present, then ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 38685, root := ~, start := 38649, stop := 38684 ), rec( content := " is used\ by default.", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "P", next := 38709, root := ~, start := 38705, stop := 38708 ), rec( content := "\n\n \ If ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 38730, root := ~, start := 38718, stop := 38729 ), rec( content := " is a tr\ ansformation, and ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 38764, root := ~, start := 38756, stop := 38763 ), rec( content := " is a no\ n-negative integer\n such that the restriction of ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 38836, root := ~, start := 38824, stop := 38835 ), rec( content := " to ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := \ "Text", name := "A", next := 38857, root := ~, start := 38849, stop := 38856 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 38862, root := ~, start := 38840, stop := 38861 ), rec( content := " defines\ \n a transformation of ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := \ "Text", name := "A", next := 38912, root := ~, start := 38904, stop := 38911 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 38917, root := ~, start := 38895, stop := 38916 ), rec( content := ", then " , count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sDigraph", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 38940, root := ~, start := 38924, stop := 38939 ), rec( content := "\n re\ turns the functional digraph with ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 38989, root := ~, start := 38981, stop := 38988 ), rec( content := " vertice\ s defined by\n ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 39026, root := ~, start := 39014, stop := 39025 ), rec( content := ". See ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsFunct\ ionalDigraph" ), content := 0, count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "Ref", next := 39065, root := ~, start := 39032, stop := 39064 ), rec( content := ".\n " , count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "P", next := 39075, root := ~, start := 39071, stop := 39074 ), rec( content := "\n\n \ Specifically, the digraph returned by ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sDigraph", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39135, root := ~, start := 39119, stop := 39134 ), rec( content := " has ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "A", next := 39149, root := ~, start := 39141, stop := 39148 ), rec( content := " edges:\\ n for each vertex ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39185, root := ~, start := 39177, stop := 39184 ), rec( content := " in ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := \ "Text", name := "A", next := 39206, root := ~, start := 39198, stop := 39205 ) , rec( content := "\ ]", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39211, root := ~, start := 39189, stop := 39210 ), rec( content := ", there \ is a unique edge\n with source ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39260, root := ~, start := 39252, stop := 39259 ), rec( content := "; this e\ dge has range ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ^", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "trans", count := [ 3, 2, 2, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := \ "Text", name := "A", next := 39299, root := ~, start := 39287, stop := 39298 ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39303, root := ~, start := 39282, stop := 39302 ), rec( content := ".\n " , count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "P", next := 39313, root := ~, start := 39309, stop := 39312 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "A", next := 39359, root := ~, start := 39351, stop := 39358 ), rec( content := " is not \ supplied, then the degree of\n the transformation ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "A", next := 39431, root := ~, start := 39419, stop := 39430 ), rec( content := " is used\ by default. If the restriction of\n ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "A", next := 39491, root := ~, start := 39479, stop := 39490 ), rec( content := " to ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 39512, root := ~, start := 39504, stop := 39511 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 39517, root := ~, start := 39495, stop := 39516 ), rec( content := " does no\ t define a transformation of\n ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 39575, root := ~, start := 39567, stop := 39574 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 39580, root := ~, start := 39558, stop := 39579 ), rec( content := ", then " , count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sDigraph(", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "trans", count := [ 3, 2, 2, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 39612, root := ~, start := 39600, stop := 39611 ) , rec( content := "\ , ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 39622, root := ~, start := 39614, stop := 39621 ) , rec( content := "\ )", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 39627, root := ~, start := 39587, stop := 39626 ), rec( content := "\n re\ turns ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "K", next := 39651, root := ~, start := 39640, stop := 39650 ), rec( content := ".\n\n \ ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> f := Transformation([4, 3, 3, 1, 7, 9, 10, 4, 2, 3]);\nTransformation( [\ 4, 3, 3, 1, 7, 9, 10, 4, 2, 3 ] )\ngap> AsDigraph(f);\n\ngap> AsDigraph(f, 4);\n\ngap> AsDigraph(f, 5);\nfail\ngap> b := BinaryRelationOnPoi\ nts(\n> [[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\nBinary Relation on 5 po\ ints\ngap> D := AsDigraph(b);\n\n\ ", count := [ 3, 2, 2, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 7 ], mathmode := "Text", name := "Example", next := 40144, root := ~, start := 39658, stop := 40143 ), rec( content := "\n ", count := [ 3, 2, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 40161, root := ~, start := 38194, stop := 40160 ) ], count := [ 3, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 40175, root := ~, start := 38079, stop := 40174 ), rec( content := "\n\n ", count := [ 3, 2, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "Graph" ), content := 0, count := [ 3, 2, 3, 2 ], mathmode := "Text", name := "Oper", next := 40230, root := ~, start := 40196, stop := 40229 ), rec( attributes := rec( ), content := [ rec( content := "A ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 3, 3 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 3, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ graph.", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 40276, root := ~, start := 40233, stop := 40275 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "A", next := 40314, root := ~, start := 40300, stop := 40313 ), rec( content := " is a mu\ table or immutable digraph without multiple edges, then this operation\n re\ turns a ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 3, 3 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 3, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ graph that is isomorphic to ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "A", next := 40465, root := ~, start := 40451, stop := 40464 ), rec( content := ". ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "P", next := 40471, root := ~, start := 40467, stop := 40470 ), rec( content := "\n\n \ If ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "A", next := 40494, root := ~, start := 40480, stop := 40493 ), rec( content := " is a mu\ ltidigraph, then since ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 3, 4 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 3, 4 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " does no\ t support\n multiple edges, the multiple edges will be reduced to a single \ edge in the\n result. In order words, for a multidigraph this operation wil\ l return the\n same as ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raph(DigraphRemoveAllMultipleEdges(", count := [ 3, 2, 3, 4 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "C", next := 40762, root := ~, start := 40719, stop := 40761 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "A", next := 40776, root := ~, start := 40762, stop := 40775 ), rec( attributes := rec( ), content := [ rec( content := ")\ )", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "C", next := 40785, root := ~, start := 40776, stop := 40784 ), rec( content := ".\n\n \ ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y)\ return Intersection(x, y) = []; end);;\ngap> Display(Petersen);\nrec(\n adja\ cencies := [ [ 3, 5, 8 ] ],\n group := \n Group( [ ( 1, 2, 3, 5, 7)( 4, 6, \ 8, 9,10), ( 2, 4)( 6, 9)( 7,10) \n ] ),\n isGraph := true,\n names := [ \ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], \n [ 2, 4 ], [ 1, 5 ], \ [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ],\n order := 10,\n representatives := [ 1 ],\n\ schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )\ngap> Digraph(Petersen\ );\n\ngap> Graph(last) = Peterse\ n;\ntrue", count := [ 3, 2, 3, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 5 ], mathmode := "Text", name := "Example", next := 41438, root := ~, start := 40792, stop := 41437 ), rec( content := "\n ", count := [ 3, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 41455, root := ~, start := 40279, stop := 41454 ) ], count := [ 3, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 41469, root := ~, start := 40181, stop := 41468 ), rec( content := "\n\n ", count := [ 3, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AsGraph" ), content := 0, count := [ 3, 2, 4, 2 ], mathmode := "Text", name := "Attr", next := 41526, root := ~, start := 41490, stop := 41525 ), rec( attributes := rec( ), content := [ rec( content := "A ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 4, 3 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 4, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ graph.", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 41572, root := ~, start := 41529, stop := 41571 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 41610, root := ~, start := 41596, stop := 41609 ), rec( content := " is a di\ graph, then this method returns the same as\n ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Graph" ), content := 0, count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 41685, root := ~, start := 41666, stop := 41684 ), rec( content := ", except\ that if ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 41716, root := ~, start := 41702, stop := 41715 ), rec( content := " is immu\ table, then the\n result will be stored as a mutable attribute of ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 41806, root := ~, start := 41792, stop := 41805 ), rec( content := ".\n I\ n this latter case, when ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sGraph(", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "C", next := 41853, root := ~, start := 41838, stop := 41852 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 41867, root := ~, start := 41853, stop := 41866 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "C", next := 41875, root := ~, start := 41867, stop := 41874 ), rec( content := " is call\ ed\n subsequently, the same ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object \ will be returned as before.\n\n ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], []]);\n\ngap> G := AsGraph(D);\nrec( adjacencies := [ [ 1, 2 ], [ 3 ], [ ] ], \ group := Group(()), \n isGraph := true, names := [ 1 .. 3 ], order := 3, \n \ representatives := [ 1, 2, 3 ], schreierVector := [ -1, -2, -3 ] )\n", count := [ 3, 2, 4, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 4, 4 ], mathmode := "Text", name := "Example", next := 42286, root := ~, start := 41959, stop := 42285 ), rec( content := "\n ", count := [ 3, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 42303, root := ~, start := 41575, stop := 42302 ) ], count := [ 3, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 42317, root := ~, start := 41475, stop := 42316 ), rec( content := "\n\n ", count := [ 3, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AsTransformatio\ n" ), content := 0, count := [ 3, 2, 5, 2 ], mathmode := "Text", name := "Attr", next := 42383, root := ~, start := 42338, stop := 42382 ), rec( attributes := rec( ), content := [ rec( content := "A transfo\ rmation, or ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "K", next := 42427, root := ~, start := 42416, stop := 42426 ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 42437, root := ~, start := 42386, stop := 42436 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "A", next := 42475, root := ~, start := 42461, stop := 42474 ), rec( content := " is a fu\ nctional digraph, then ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sTransformation", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "C", next := 42529, root := ~, start := 42506, stop := 42528 ), rec( content := "\n re\ turns the transformation which is defined by ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "A", next := 42595, root := ~, start := 42581, stop := 42594 ), rec( content := ". See " , count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsFunct\ ionalDigraph" ), content := 0, count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Ref", next := 42641, root := ~, start := 42602, stop := 42640 ), rec( content := ". Otherw\ ise,\n ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sTransformation(", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "C", next := 42682, root := ~, start := 42658, stop := 42681 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "A", next := 42696, root := ~, start := 42682, stop := 42695 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "C", next := 42704, root := ~, start := 42696, stop := 42703 ), rec( content := " returns\ ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "K", next := 42724, root := ~, start := 42713, stop := 42723 ), rec( content := ". ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "P", next := 42731, root := ~, start := 42727, stop := 42730 ), rec( content := "\n\n \ If ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "A", next := 42754, root := ~, start := 42740, stop := 42753 ), rec( content := " is a fu\ nctional digraph with ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "M", next := 42792, root := ~, start := 42784, stop := 42791 ), rec( content := " vertice\ s, then\n ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sTransformation(", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 42836, root := ~, start := 42812, stop := 42835 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "A", next := 42850, root := ~, start := 42836, stop := 42849 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 42858, root := ~, start := 42850, stop := 42857 ), rec( content := " will re\ turn the\n transformation ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 42902, root := ~, start := 42894, stop := 42901 ), rec( content := " of degr\ ee at most ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "M", next := 42929, root := ~, start := 42921, stop := 42928 ), rec( content := " where f\ or each ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ \n \\leq i \\leq n", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "M", next := 42973, root := ~, start := 42945, stop := 42972 ), rec( content := ", ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ^ f", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 42987, root := ~, start := 42975, stop := 42986 ), rec( content := " is equa\ l to the unique out-neighbour of\n vertex ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 43047, root := ~, start := 43039, stop := 43046 ), rec( content := " in ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "A", next := 43065, root := ~, start := 43051, stop := 43064 ), rec( content := ".\n\n \ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [3], [2]]);\n\ngap> AsTransformation(D);\nTransformation( [ 1, 3, 2 ] )\ngap> D := Cycl\ eDigraph(IsMutableDigraph, 3);\n\nga\ p> AsTransformation(D);\nTransformation( [ 2, 3, 1 ] )\ngap> AsPermutation(las\ t);\n(1,2,3)\ngap> D := Digraph([[2, 3], [], []]);\n\ngap> AsTransformation(D);\nfail", count := [ 3, 2, 5, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 5, 5 ], mathmode := "Text", name := "Example", next := 43531, root := ~, start := 43072, stop := 43530 ), rec( content := "\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 43548, root := ~, start := 42440, stop := 43547 ) ], count := [ 3, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 43562, root := ~, start := 42323, stop := 43561 ), rec( content := "\n\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 43576, root := ~, start := 37378, stop := 43575 ), rec( content := "\n\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "New digraphs from\ old", count := [ 3, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 43629, root := ~, start := 43589, stop := 43628 ), rec( content := "\n ", count := [ 3, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopy" ), content := 0, count := [ 3, 3, 1, 2 ], mathmode := "Text", name := "Oper", next := 43698, root := ~, start := 43649, stop := 43697 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opy" ), content := 0, count := [ 3, 3, 1, 4 ], mathmode := "Text", name := "Oper", next := 43748, root := ~, start := 43701, stop := 43747 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopySame\ Mutability" ), content := 0, count := [ 3, 3, 1, 6 ], mathmode := "Text", name := "Oper", next := 43805, root := ~, start := 43751, stop := 43804 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopy" ), content := 0, count := [ 3, 3, 1, 8 ], mathmode := "Text", name := "Oper", next := 43848, root := ~, start := 43808, stop := 43847 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "Returns", next := 43880, root := ~, start := 43851, stop := 43879 ), rec( attributes := rec( ), content := [ rec( content := "\n Eac\ h of these operations returns a new copy of ", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "A", next := 43962, root := ~, start := 43948, stop := 43961 ), rec( content := ",\n o\ f the appropriate mutability, retaining none of the attributes or\n propert\ ies of ", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "A", next := 44067, root := ~, start := 44053, stop := 44066 ), rec( content := ". ", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "P", next := 44073, root := ~, start := 44069, stop := 44072 ), rec( content := "\n\n \ ", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphCopy", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "C", next := 44097, root := ~, start := 44079, stop := 44096 ), rec( content := " is a sy\ nonym for ", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCopySameMutability", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "C", next := 44147, root := ~, start := 44115, stop := 44146 ), rec( content := ".\n", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(10);\n\ngap>\ DigraphCopy(D) = D;\ntrue\ngap> IsIdenticalObj(DigraphCopy(D), D);\nfalse\nga\ p> DigraphMutableCopy(D);\n\n", count := [ 3, 3, 1, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 1, 11 ], mathmode := "Text", name := "Example", next := 44401, root := ~, start := 44149, stop := 44400 ), rec( content := "\n ", count := [ 3, 3, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "Description", next := 44418, root := ~, start := 43883, stop := 44417 ) ], count := [ 3, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 44432, root := ~, start := 43634, stop := 44431 ), rec( content := "\n\n ", count := [ 3, 3, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopyIfImmutable" ), content := 0, count := [ 3, 3, 2, 2 ], mathmode := "Text", name := "Oper", next := 44513, root := ~, start := 44453, stop := 44512 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopyIfMutable" ), content := 0, count := [ 3, 3, 2, 4 ], mathmode := "Text", name := "Oper", next := 44574, root := ~, start := 44516, stop := 44573 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfMutable" ), content := 0, count := [ 3, 3, 2, 6 ], mathmode := "Text", name := "Oper", next := 44633, root := ~, start := 44577, stop := 44632 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfImmutable" ), content := 0, count := [ 3, 3, 2, 8 ], mathmode := "Text", name := "Oper", next := 44694, root := ~, start := 44636, stop := 44693 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "Returns", next := 44726, root := ~, start := 44697, stop := 44725 ), rec( attributes := rec( ), content := [ rec( content := "\n Eac\ h of these operations returns either the original argument\n ", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "A", next := 44827, root := ~, start := 44813, stop := 44826 ), rec( content := ", or a n\ ew copy of ", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "A", next := 44860, root := ~, start := 44846, stop := 44859 ), rec( content := " of the \ appropriate\n mutability, according to the mutability of ", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "A", next := 44941, root := ~, start := 44927, stop := 44940 ), rec( content := ".\n", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> C := CycleDigraph(10);\n\ngap>\ D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C = D;\ntrue\ngap> D := D\ igraphImmutableCopyIfMutable(C);\n\n\ gap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntrue\ngap> D := DigraphMutableC\ opyIfMutable(C);\n\ngap> IsMutableDi\ graph(D);\nfalse\ngap> D := DigraphMutableCopyIfImmutable(C);\n\ngap> IsMutableDigraph(D);\ntrue\ngap> C := Cycl\ eDigraph(IsMutableDigraph, 10);\n\ \ngap> D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntrue\ngap> D :\ = DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C = D;\ntrue\ngap> D := Digraph\ MutableCopyIfMutable(C);\n\ngap> I\ sMutableDigraph(D);\ntrue\ngap> D := DigraphMutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap>\ IsMutableDigraph(D);\ntrue", count := [ 3, 3, 2, 10 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 10 ], mathmode := "Text", name := "Example", next := 46203, root := ~, start := 44943, stop := 46202 ), rec( content := "\n ", count := [ 3, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "Description", next := 46220, root := ~, start := 44729, stop := 46219 ) ], count := [ 3, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 46234, root := ~, start := 44438, stop := 46233 ), rec( content := "\n\n ", count := [ 3, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, verts", Name := "InducedSubdigra\ ph" ), content := 0, count := [ 3, 3, 3, 2 ], mathmode := "Text", name := "Oper", next := 46308, root := ~, start := 46255, stop := 46307 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 46340, root := ~, start := 46311, stop := 46339 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46378, root := ~, start := 46364, stop := 46377 ), rec( content := " is a di\ graph, and ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46409, root := ~, start := 46397, stop := 46408 ), rec( content := " is a su\ bset of the\n vertices of ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46459, root := ~, start := 46445, stop := 46458 ), rec( content := ", then t\ his operation returns a digraph\n constructed from ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46534, root := ~, start := 46520, stop := 46533 ), rec( content := " by reta\ ining precisely those vertices in\n ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46592, root := ~, start := 46580, stop := 46591 ), rec( content := ", and th\ ose edges whose source and range vertices are both\n contained in ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46680, root := ~, start := 46668, stop := 46679 ), rec( content := ". ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "P", next := 46686, root := ~, start := 46682, stop := 46685 ), rec( content := "\n\n \ The vertices of the induced subdigraph are\n ", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1..Length(verts)]", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "C", next := 46764, root := ~, start := 46739, stop := 46763 ), rec( content := " but the\ original vertex labels can be\n accessed via ", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "Ref", next := 46853, root := ~, start := 46820, stop := 46852 ), rec( content := ". ", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "P", next := 46859, root := ~, start := 46855, stop := 46858 ), rec( content := "\n\n \ If ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "A", next := 46882, root := ~, start := 46868, stop := 46881 ), rec( content := " belongs\ to ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "Ref", next := 46924, root := ~, start := 46894, stop := 46923 ), rec( content := ", then\n\ ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "A", next := 46949, root := ~, start := 46935, stop := 46948 ), rec( content := " is modi\ fied in place. If ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "A", next := 46989, root := ~, start := 46975, stop := 46988 ), rec( content := " belongs\ to ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "Ref", next := 47039, root := ~, start := 47001, stop := 47038 ), rec( content := ", a new \ immutable digraph containing the\n appropriate\n vertices and edges is r\ eturned.", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 3, 6 ], mathmode := "Text", name := "P", next := 47135, root := ~, start := 47131, stop := 47134 ), rec( content := "\n\n \ ", count := [ 3, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 1, 2, 3, 4, 4], [1, 3, 4], [3, 1], [1, 1]]);\n\ngap> InducedSubdigraph(D, [1, 3, 4\ ]);\n\ngap> DigraphVertices(l\ ast);\n[ 1 .. 3 ]\ngap> D := DigraphMutableCopy(D);\n\ngap> new := InducedSubdigraph(D, [1, 3, 4]);\n\ngap> D = new;\ntrue\n", count := [ 3, 3, 3, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 3, 7 ], mathmode := "Text", name := "Example", next := 47613, root := ~, start := 47141, stop := 47612 ), rec( content := "\n ", count := [ 3, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 47630, root := ~, start := 46343, stop := 47629 ) ], count := [ 3, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 47644, root := ~, start := 46240, stop := 47643 ), rec( content := "\n\n ", count := [ 3, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "ReducedDigraph"\ ), content := 0, count := [ 3, 3, 4, 2 ], mathmode := "Text", name := "Oper", next := 47708, root := ~, start := 47665, stop := 47707 ), rec( attributes := rec( Arg := "digraph", Name := "ReducedDigraphA\ ttr" ), content := 0, count := [ 3, 3, 4, 4 ], mathmode := "Text", name := "Attr", next := 47758, root := ~, start := 47711, stop := 47757 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Returns", next := 47790, root := ~, start := 47761, stop := 47789 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a digraph isomorphic to the subdigraph of\n ", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "A", next := 47893, root := ~, start := 47879, stop := 47892 ), rec( content := " induced\ by the set of non-isolated vertices,\n i.e. the set of those vertices of " , count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "A", next := 47991, root := ~, start := 47977, stop := 47990 ), rec( content := " which a\ re the source or\n range of some edge in ", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "A", next := 48056, root := ~, start := 48042, stop := 48055 ), rec( content := ".\n S\ ee ", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Induced\ Subdigraph" ), content := 0, count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Ref", next := 48097, root := ~, start := 48066, stop := 48096 ), rec( content := ".\n " , count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "P", next := 48107, root := ~, start := 48103, stop := 48106 ), rec( content := "\n\n \ The ordering of the remaining vertices of ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "A", next := 48169, root := ~, start := 48155, stop := 48168 ), rec( content := " is pres\ erved, as are\n the labels of the remaining vertices and edges;\n see ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "Ref", next := 48284, root := ~, start := 48251, stop := 48283 ), rec( content := " and ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ EdgeLabels" ), content := 0, count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "Ref", next := 48320, root := ~, start := 48289, stop := 48319 ), rec( content := ".\n T\ his can allow one to match a vertex in the reduced digraph to the\n corresp\ onding vertex in ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "A", next := 48435, root := ~, start := 48421, stop := 48434 ), rec( content := ".\n " , count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "P", next := 48445, root := ~, start := 48441, stop := 48444 ), rec( content := "\n\n \ If ", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "A", next := 48468, root := ~, start := 48454, stop := 48467 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the isolated \ vertices of the mutable digraph ", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "A", next := 48599, root := ~, start := 48585, stop := 48598 ), rec( content := " are\n \ removed in-place.\n ", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [], [], [1, 4], []]);\n\ngap> R := ReducedDigraph(D);\n\ngap> OutNeighbours(R);\n[ [ 1, 2 ], [ ], [ 1, 3 ] ]\ngap> Di\ graphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 4, 1 ], [ 4, 4 ] ]\ngap> DigraphEdges(\ R);\n[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 ] ]\ngap> DigraphVertexLabel(R, 3)\ ;\n4\ngap> DigraphVertexLabel(R, 2);\n2\ngap> D := Digraph(IsMutableDigraph, [\ [], [3], [2]]);\n\ngap> ReducedDigra\ ph(D);\n\ngap> D;\n\n", count := [ 3, 3, 4, 8 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 8 ], mathmode := "Text", name := "Example", next := 49292, root := ~, start := 48630, stop := 49291 ), rec( content := "\n ", count := [ 3, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Description", next := 49309, root := ~, start := 47793, stop := 49308 ) ], count := [ 3, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 49323, root := ~, start := 47650, stop := 49322 ), rec( content := "\n\n ", count := [ 3, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraph" ), content := 0, count := [ 3, 3, 5, 2 ], mathmode := "Text", name := "Oper", next := 49399, root := ~, start := 49344, stop := 49398 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphAttr" ), content := 0, count := [ 3, 3, 5, 4 ], mathmode := "Text", name := "Attr", next := 49461, root := ~, start := 49402, stop := 49460 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoops" ), content := 0, count := [ 3, 3, 5, 6 ], mathmode := "Text", name := "Oper", next := 49531, root := ~, start := 49464, stop := 49530 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoopsAttr" ), content := 0, count := [ 3, 3, 5, 8 ], mathmode := "Text", name := "Attr", next := 49605, root := ~, start := 49534, stop := 49604 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "Returns", next := 49637, root := ~, start := 49608, stop := 49636 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "A", next := 49675, root := ~, start := 49661, stop := 49674 ), rec( content := " is a di\ graph, then ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximalSymmetricSubdigraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "C", next := 49728, root := ~, start := 49695, stop := 49727 ), rec( content := "\n re\ turns a symmetric digraph without multiple edges which has the same\n verte\ x set as ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "A", next := 49835, root := ~, start := 49821, stop := 49834 ), rec( content := ", and wh\ ose edge list is formed from\n ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "A", next := 49890, root := ~, start := 49876, stop := 49889 ), rec( content := " by igno\ ring the multiplicity of edges, and by ignoring\n edges ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,v]", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "C", next := 49968, root := ~, start := 49956, stop := 49967 ), rec( content := " for whi\ ch there does not exist an edge ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v,u]", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "C", next := 50020, root := ~, start := 50008, stop := 50019 ), rec( content := ".\n " , count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "P", next := 50030, root := ~, start := 50026, stop := 50029 ), rec( content := "\n\n \ The digraph returned by ", count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximalSymmetricSubdigraphWithoutLoops", count := [ 3, 3, 5, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "C", next := 50105, root := ~, start := 50060, stop := 50104 ), rec( content := " is\n \ the same, except that loops are removed.", count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "P", next := 50157, root := ~, start := 50153, stop := 50156 ), rec( content := "\n \n\ If ", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "A", next := 50184, root := ~, start := 50170, stop := 50183 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "A", next := 50290, root := ~, start := 50276, stop := 50289 ), rec( content := " is chan\ ged in-place into such\n a digraph described above.", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "P", next := 50355, root := ~, start := 50351, stop := 50354 ), rec( content := "\n\n \ See ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "Ref", next := 50397, root := ~, start := 50365, stop := 50396 ), rec( content := ", ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "Ref", next := 50427, root := ~, start := 50399, stop := 50426 ), rec( content := ", and\n \ ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "Ref", next := 50466, root := ~, start := 50437, stop := 50465 ), rec( content := " for mor\ e information.\n ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsSymmetricDigraph(D) and IsMultiDigraph(D);\ \ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ngap> \ S := MaximalSymmetricSubdigraph(D);\n\ngap> IsSymmetricDigraph(S) and not IsMultiDigraph(S);\ntrue\nga\ p> OutNeighbours(S);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\ngap> D := CycleDigraph(I\ sMutableDigraph, 3);\n\ngap> Maximal\ SymmetricSubdigraph(D);\n\ngap> D;\n\n", count := [ 3, 3, 5, 13 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 5, 13 ], mathmode := "Text", name := "Example", next := 51168, root := ~, start := 50493, stop := 51167 ), rec( content := "\n ", count := [ 3, 3, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "Description", next := 51185, root := ~, start := 49640, stop := 51184 ) ], count := [ 3, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 51199, root := ~, start := 49329, stop := 51198 ), rec( content := "\n\n ", count := [ 3, 3, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "MaximalAntiSymm\ etricSubdigraph" ), content := 0, count := [ 3, 3, 6, 2 ], mathmode := "Text", name := "Oper", next := 51279, root := ~, start := 51220, stop := 51278 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalAntiSymm\ etricSubdigraphAttr" ), content := 0, count := [ 3, 3, 6, 4 ], mathmode := "Text", name := "Attr", next := 51345, root := ~, start := 51282, stop := 51344 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "Returns", next := 51377, root := ~, start := 51348, stop := 51376 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51415, root := ~, start := 51401, stop := 51414 ), rec( content := " is a di\ graph, then ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximalAntiSymmetricSubdigraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51472, root := ~, start := 51435, stop := 51471 ), rec( content := "\n re\ turns an anti-symmetric subdigraph of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51531, root := ~, start := 51517, stop := 51530 ), rec( content := " formed \ by\n retaining the vertices of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51586, root := ~, start := 51572, stop := 51585 ), rec( content := ", discar\ ding any duplicate edges,\n and discarding any edge ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51660, root := ~, start := 51648, stop := 51659 ), rec( content := " of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51678, root := ~, start := 51664, stop := 51677 ), rec( content := " where " , count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ > j", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51697, root := ~, start := 51685, stop := 51696 ), rec( content := "\n an\ d the reverse edge ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j,i]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51735, root := ~, start := 51723, stop := 51734 ), rec( content := " is an e\ dge of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51764, root := ~, start := 51750, stop := 51763 ), rec( content := ".\n I\ n other words, for every\n symmetric pair of edges ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51836, root := ~, start := 51824, stop := 51835 ), rec( content := " and ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j,i]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51853, root := ~, start := 51841, stop := 51852 ), rec( content := " in ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51871, root := ~, start := 51857, stop := 51870 ), rec( content := ", where \ \n ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51892, root := ~, start := 51884, stop := 51891 ), rec( content := " and ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51905, root := ~, start := 51897, stop := 51904 ), rec( content := " are dis\ tinct, it discards the the edge\n ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ \\max(i,j),\\min(i,j)]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "M", next := 51977, root := ~, start := 51949, stop := 51976 ), rec( content := ".\n " , count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "P", next := 51987, root := ~, start := 51983, stop := 51986 ), rec( content := "\n\n \ If ", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "A", next := 52010, root := ~, start := 51996, stop := 52009 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "A", next := 52116, root := ~, start := 52102, stop := 52115 ), rec( content := " is chan\ ged in-place. ", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "P", next := 52142, root := ~, start := 52138, stop := 52141 ), rec( content := "\n\n \ See ", count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAntis\ ymmetricDigraph" ), content := 0, count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "Ref", next := 52188, root := ~, start := 52152, stop := 52187 ), rec( content := " for mor\ e information.\n ", count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsAntiSymmetricDigraph(D) and IsMultiDigraph\ (D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ng\ ap> D := MaximalAntiSymmetricSubdigraph(D);\n\ngap> IsAntiSymmetricDigraph(D) and not IsMultiDigra\ ph(D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ]\ngap> D :\ = Digraph(IsMutableDigraph, [[2], [1]]);\n\ngap> MaximalAntiSymmetricSubdigraph(D);\n\ngap> D;\n\n", count := [ 3, 3, 6, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 6, 8 ], mathmode := "Text", name := "Example", next := 52918, root := ~, start := 52215, stop := 52917 ), rec( content := "\n ", count := [ 3, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "Description", next := 52935, root := ~, start := 51380, stop := 52934 ) ], count := [ 3, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 52949, root := ~, start := 51205, stop := 52948 ), rec( content := "\n\n ", count := [ 3, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingForest" ), content := 0, count := [ 3, 3, 7, 2 ], mathmode := "Text", name := "Oper", next := 53023, root := ~, start := 52970, stop := 53022 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingForestAttr" ), content := 0, count := [ 3, 3, 7, 4 ], mathmode := "Text", name := "Attr", next := 53083, root := ~, start := 53026, stop := 53082 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTree" ), content := 0, count := [ 3, 3, 7, 6 ], mathmode := "Text", name := "Oper", next := 53137, root := ~, start := 53086, stop := 53136 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTreeAttr" ), content := 0, count := [ 3, 3, 7, 8 ], mathmode := "Text", name := "Attr", next := 53195, root := ~, start := 53140, stop := 53194 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "K", next := 53232, root := ~, start := 53221, stop := 53231 ), rec( content := ".", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Returns", next := 53243, root := ~, start := 53198, stop := 53242 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "A", next := 53281, root := ~, start := 53267, stop := 53280 ), rec( content := " is a di\ graph with at least one vertex, then\n ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningForest", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "C", next := 53361, root := ~, start := 53330, stop := 53360 ), rec( content := " returns\ an undirected spanning forest of\n ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "A", next := 53421, root := ~, start := 53407, stop := 53420 ), rec( content := ", otherw\ ise this attribute returns ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "K", next := 53467, root := ~, start := 53456, stop := 53466 ), rec( content := ". See ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningForest" ), content := 0, count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Ref", next := 53520, root := ~, start := 53473, stop := 53519 ), rec( content := " for the\ definition of an undirected\n spanning forest.", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "P", next := 53581, root := ~, start := 53577, stop := 53580 ), rec( content := "\n\n \ If ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "A", next := 53604, root := ~, start := 53590, stop := 53603 ), rec( content := " is a di\ graph with at least one vertex and whose ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraph" ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "Ref", next := 53699, root := ~, start := 53653, stop := 53698 ), rec( content := " is conn\ ected (see ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "Ref", next := 53757, root := ~, start := 53718, stop := 53756 ), rec( content := "), then \ ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningTree", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "C", next := 53794, root := ~, start := 53765, stop := 53793 ), rec( content := " returns\ \n an undirected spanning tree of ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "A", next := 53852, root := ~, start := 53838, stop := 53851 ), rec( content := ", otherw\ ise this attribute\n returns ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "K", next := 53902, root := ~, start := 53891, stop := 53901 ), rec( content := ". See " , count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningTree" ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "Ref", next := 53948, root := ~, start := 53909, stop := 53947 ), rec( content := " for the\ \n definition of an undirected spanning tree.", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "P", next := 54007, root := ~, start := 54003, stop := 54006 ), rec( content := "\n\n \ If ", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "A", next := 54030, root := ~, start := 54016, stop := 54029 ), rec( content := " is immu\ table, then an immutable digraph is returned.\n Otherwise, the mutable digr\ aph ", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "A", next := 54133, root := ~, start := 54119, stop := 54132 ), rec( content := " is chan\ ged in-place into an\n undirected spanning tree of ", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "A", next := 54208, root := ~, start := 54194, stop := 54207 ), rec( content := ".", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "P", next := 54213, root := ~, start := 54209, stop := 54212 ), rec( content := "\n\n \ Note that for an immutable digraph that has known undirected spanning tree,\n \ the attribute ", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningTree", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "C", next := 54342, root := ~, start := 54313, stop := 54341 ), rec( content := " returns\ the same digraph as\n the attribute ", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningForest", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "C", next := 54420, root := ~, start := 54389, stop := 54419 ), rec( content := ".\n\n \ ", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]]);\n\ngap> UndirectedSpanningTree(D);\nfail\ngap> for\ est := UndirectedSpanningForest(D);\n\ngap> OutNeighbours(forest);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\nga\ p> IsUndirectedSpanningForest(D, forest);\ntrue\ngap> DigraphConnectedComponen\ ts(forest).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> DigraphConnectedComponents(Max\ imalSymmetricSubdigraph(D)).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> UndirectedSpa\ nningForest(MaximalSymmetricSubdigraph(D))\n> = forest;\ntrue\ngap> D := Compl\ eteDigraph(4);\n\ngap> tree := Und\ irectedSpanningTree(D);\n\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap> tree = UndirectedSpanni\ ngForest(D);\ntrue\ngap> UndirectedSpanningForest(EmptyDigraph(0));\nfail\ngap\ > D := PetersenGraph(IsMutableDigraph);\n\ngap> UndirectedSpanningTree(D);\n\ngap> D;\n\n", count := [ 3, 3, 7, 13 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 13 ], mathmode := "Text", name := "Example", next := 55582, root := ~, start := 54427, stop := 55581 ), rec( content := "\n ", count := [ 3, 3, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Description", next := 55599, root := ~, start := 53246, stop := 55598 ) ], count := [ 3, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 55613, root := ~, start := 52955, stop := 55612 ), rec( content := "\n\n ", count := [ 3, 3, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, p", Name := "QuotientDigraph\ " ), content := 0, count := [ 3, 3, 8, 2 ], mathmode := "Text", name := "Oper", next := 55681, root := ~, start := 55634, stop := 55680 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 55713, root := ~, start := 55684, stop := 55712 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55751, root := ~, start := 55737, stop := 55750 ), rec( content := " is a di\ graph, and ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55778, root := ~, start := 55770, stop := 55777 ), rec( content := " is a pa\ rtition of the vertices\n of ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55831, root := ~, start := 55817, stop := 55830 ), rec( content := ", then t\ his operation returns a digraph constructed by\n amalgamating all vertices \ of ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55933, root := ~, start := 55919, stop := 55932 ), rec( content := " which l\ ie in the same part of\n ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55976, root := ~, start := 55968, stop := 55975 ), rec( content := ".\n " , count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "P", next := 55986, root := ~, start := 55982, stop := 55985 ), rec( content := "\n\n \ A partition of the vertices of ", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "A", next := 56037, root := ~, start := 56023, stop := 56036 ), rec( content := " is a li\ st of non-empty\n disjoint lists, such that the union of all the sub-lists \ is equal\n to vertex set of ", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "A", next := 56166, root := ~, start := 56152, stop := 56165 ), rec( content := ". In par\ ticular, each vertex must appear\n in precisely one sub-list.\n ", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "P", next := 56246, root := ~, start := 56242, stop := 56245 ), rec( content := "\n\n \ The vertices of ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "A", next := 56282, root := ~, start := 56268, stop := 56281 ), rec( content := " in part\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56299, root := ~, start := 56291, stop := 56298 ), rec( content := " of ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "A", next := 56311, root := ~, start := 56303, stop := 56310 ), rec( content := " will be\ come\n vertex ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56343, root := ~, start := 56335, stop := 56342 ), rec( content := " in the \ quotient, and there exists some edge in ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "A", next := 56405, root := ~, start := 56391, stop := 56404 ), rec( content := "\n wi\ th source in part ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56438, root := ~, start := 56430, stop := 56437 ), rec( content := " and ran\ ge in part ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56465, root := ~, start := 56457, stop := 56464 ), rec( content := " if and \ only if there\n is an edge from ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56515, root := ~, start := 56507, stop := 56514 ), rec( content := " to ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56527, root := ~, start := 56519, stop := 56526 ), rec( content := " in the \ quotient.\n In particular, this means that the quotient of a digraph has no\ multiple edges.\n which was a change introduced in version 1.0.0 of the ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ .\n ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "P", next := 56715, root := ~, start := 56711, stop := 56714 ), rec( content := "\n\n \ If ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "A", next := 56738, root := ~, start := 56724, stop := 56737 ), rec( content := " belongs\ to ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "Ref", next := 56780, root := ~, start := 56750, stop := 56779 ), rec( content := ", then\n\ ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "A", next := 56805, root := ~, start := 56791, stop := 56804 ), rec( content := " is modi\ fied in place. If ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "A", next := 56845, root := ~, start := 56831, stop := 56844 ), rec( content := " belongs\ to ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "Ref", next := 56895, root := ~, start := 56857, stop := 56894 ), rec( content := ", a new \ immutable digraph with the above\n properties is returned.", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 7 ], mathmode := "Text", name := "P", next := 56967, root := ~, start := 56963, stop := 56966 ), rec( content := "\n\n \ ", count := [ 3, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 1], [4], [1], [1, 3, 4]]);\n\ngap> DigraphVertices(D);\n[ 1 .. 4 ]\ngap> DigraphEdges(D)\ ;\n[ [ 1, 2 ], [ 1, 1 ], [ 2, 4 ], [ 3, 1 ], [ 4, 1 ], [ 4, 3 ], \n [ 4, 4 ] \ ]\ngap> p := [[1], [2, 4], [3]];\n[ [ 1 ], [ 2, 4 ], [ 3 ] ]\ngap> quo := Quot\ ientDigraph(D, p);\n\ngap> Digraph\ Vertices(quo);\n[ 1 .. 3 ]\ngap> DigraphEdges(quo);\n[ [ 1, 1 ], [ 1, 2 ], [ 2\ , 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> QuotientDigraph(EmptyDigraph(0), [\ ]);\n\n", count := [ 3, 3, 8, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 8, 8 ], mathmode := "Text", name := "Example", next := 57581, root := ~, start := 56973, stop := 57580 ), rec( content := "\n ", count := [ 3, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 57598, root := ~, start := 55716, stop := 57597 ) ], count := [ 3, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 57612, root := ~, start := 55619, stop := 57611 ), rec( content := "\n\n ", count := [ 3, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphReverse"\ ), content := 0, count := [ 3, 3, 9, 2 ], mathmode := "Text", name := "Oper", next := 57676, root := ~, start := 57633, stop := 57675 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReverseA\ ttr" ), content := 0, count := [ 3, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 57726, root := ~, start := 57679, stop := 57725 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 57758, root := ~, start := 57729, stop := 57757 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ reverse of a digraph is the digraph formed by reversing the orientation\n \ of each of its edges, i.e. for every edge ", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "C", next := 57914, root := ~, start := 57901, stop := 57913 ), rec( content := " of a di\ graph, the\n reverse contains the corresponding edge ", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j, i]", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "C", next := 57990, root := ~, start := 57977, stop := 57989 ), rec( content := ".", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "P", next := 57995, root := ~, start := 57991, stop := 57994 ), rec( content := "\n\n \ ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphReverse", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "C", next := 58022, root := ~, start := 58001, stop := 58021 ), rec( content := " returns\ the reverse of the digraph ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "A", next := 58072, root := ~, start := 58058, stop := 58071 ), rec( content := ". If\n \ ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "A", next := 58095, root := ~, start := 58081, stop := 58094 ), rec( content := " is immu\ table, then a new immutable digraph is returned. Otherwise,\n the mutable d\ igraph ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "A", next := 58201, root := ~, start := 58187, stop := 58200 ), rec( content := " is chan\ ged in-place into its reverse.\n ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> DigraphReverse(D);\n\ngap> OutNeighbours(last);\n[ [ 2, 3, 4 ], [ 4, 5 \ ], [ 1, 2, 5 ], [ 4 ], [ 2, 5 ] ]\ngap> D := Digraph([[2, 4], [1], [4], [3, 4]\ ]);\n\ngap> DigraphEdges(D);\n[ [ \ 1, 2 ], [ 1, 4 ], [ 2, 1 ], [ 3, 4 ], [ 4, 3 ], [ 4, 4 ] ]\ngap> DigraphEdges(\ DigraphReverse(D));\n[ [ 1, 2 ], [ 2, 1 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ], [ 4, \ 4 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> Digra\ phReverse(D);\n\ngap> OutNeighbours(\ D);\n[ [ 3 ], [ 1 ], [ 2 ] ]\n", count := [ 3, 3, 9, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 7 ], mathmode := "Text", name := "Example", next := 59065, root := ~, start := 58244, stop := 59064 ), rec( content := "\n ", count := [ 3, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 59082, root := ~, start := 57761, stop := 59081 ) ], count := [ 3, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 59096, root := ~, start := 57618, stop := 59095 ), rec( content := "\n\n ", count := [ 3, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDual" ), content := 0, count := [ 3, 3, 10, 2 ], mathmode := "Text", name := "Oper", next := 59157, root := ~, start := 59117, stop := 59156 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphDualAttr\ " ), content := 0, count := [ 3, 3, 10, 4 ], mathmode := "Text", name := "Attr", next := 59204, root := ~, start := 59160, stop := 59203 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "Returns", next := 59236, root := ~, start := 59207, stop := 59235 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ual", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "E", next := 59272, root := ~, start := 59261, stop := 59271 ), rec( content := " of ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "A", next := 59290, root := ~, start := 59276, stop := 59289 ), rec( content := " has the\ same vertices as ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "A", next := 59330, root := ~, start := 59316, stop := 59329 ), rec( content := ",\n a\ nd there is an edge in the dual from ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 59382, root := ~, start := 59374, stop := 59381 ), rec( content := " to ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 59394, root := ~, start := 59386, stop := 59393 ), rec( content := "\n wh\ enever there is no edge from ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 59438, root := ~, start := 59430, stop := 59437 ), rec( content := " to ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 59450, root := ~, start := 59442, stop := 59449 ), rec( content := " in ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "A", next := 59468, root := ~, start := 59454, stop := 59467 ), rec( content := ".\n T\ he ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ual", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "E", next := 59489, root := ~, start := 59478, stop := 59488 ), rec( content := " is some\ times called the ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplement", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "E", next := 59531, root := ~, start := 59514, stop := 59530 ), rec( content := ".", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "P", next := 59536, root := ~, start := 59532, stop := 59535 ), rec( content := "\n\n \ ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphDual", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "C", next := 59560, root := ~, start := 59542, stop := 59559 ), rec( content := " returns\ the dual of the digraph ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "A", next := 59607, root := ~, start := 59593, stop := 59606 ), rec( content := ". If\n \ ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "A", next := 59630, root := ~, start := 59616, stop := 59629 ), rec( content := " is an i\ mmutable digraph, then a new immutable digraph is\n returned. Otherwise, th\ e mutable digraph ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "A", next := 59747, root := ~, start := 59733, stop := 59746 ), rec( content := " is chan\ ged in-place\n into its dual.\n ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3], [], [4, 6], [5], [],\n> [7, 8, 9], [], [], []]);\n\ \ngap> DigraphDual(D);\n\ngap> D := CycleDigraph(IsMutableDigraph, \ 3);\n\ngap> DigraphDual(D);\n\ngap> D;\n\n", count := [ 3, 3, 10, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 7 ], mathmode := "Text", name := "Example", next := 60213, root := ~, start := 59791, stop := 60212 ), rec( content := "\n ", count := [ 3, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "Description", next := 60230, root := ~, start := 59239, stop := 60229 ) ], count := [ 3, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 60244, root := ~, start := 59102, stop := 60243 ), rec( content := "\n\n ", count := [ 3, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSymmetri\ cClosure" ), content := 0, count := [ 3, 3, 11, 2 ], mathmode := "Text", name := "Oper", next := 60317, root := ~, start := 60265, stop := 60316 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSymmetri\ cClosureAttr" ), content := 0, count := [ 3, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 60376, root := ~, start := 60320, stop := 60375 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 60408, root := ~, start := 60379, stop := 60407 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "A", next := 60446, root := ~, start := 60432, stop := 60445 ), rec( content := " is a di\ graph, then this attribute gives the minimal\n symmetric digraph which has \ the same vertices and contains all the edges of\n ", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "A", next := 60597, root := ~, start := 60583, stop := 60596 ), rec( content := ".", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "P", next := 60602, root := ~, start := 60598, stop := 60601 ), rec( content := "\n \n\ A digraph is ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ymmetric", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "E", next := 60641, root := ~, start := 60625, stop := 60640 ), rec( content := " if its \ adjacency matrix \n ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "Ref", next := 60700, root := ~, start := 60671, stop := 60699 ), rec( content := " is symm\ etric. For a digraph with multiple\n edges this means that there are the sa\ me number of edges from a vertex\n ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 60830, root := ~, start := 60822, stop := 60829 ), rec( content := " to a ve\ rtex ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 60851, root := ~, start := 60843, stop := 60850 ), rec( content := " as ther\ e are from ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 60878, root := ~, start := 60870, stop := 60877 ), rec( content := " to ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 60890, root := ~, start := 60882, stop := 60889 ), rec( content := "; \n \ see ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "Ref", next := 60933, root := ~, start := 60901, stop := 60932 ), rec( content := ".", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "P", next := 60938, root := ~, start := 60934, stop := 60937 ), rec( content := "\n\n \ If ", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "A", next := 60961, root := ~, start := 60947, stop := 60960 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "A", next := 61067, root := ~, start := 61053, stop := 61066 ), rec( content := " is chan\ ged in-place into its\n symmetric closure.\n", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 3], [2, 4], [1], [3, 4]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> IsSymmetricDigraph(D);\ntrue\nga\ p> List(OutNeighbours(D), AsSet);\n[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, \ 3, 4 ] ]\ngap> D := Digraph([[2, 2], [1]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ \ 1, 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSymmetricClosure(D);\n\ngap> D;\n\n", count := [ 3, 3, 11, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 11, 8 ], mathmode := "Text", name := "Example", next := 61908, root := ~, start := 61120, stop := 61907 ), rec( content := "\n ", count := [ 3, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 61925, root := ~, start := 60411, stop := 61924 ) ], count := [ 3, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 61939, root := ~, start := 60250, stop := 61938 ), rec( content := "\n\n ", count := [ 3, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veClosure" ), content := 0, count := [ 3, 3, 12, 2 ], mathmode := "Text", name := "Oper", next := 62013, root := ~, start := 61960, stop := 62012 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veClosureAttr" ), content := 0, count := [ 3, 3, 12, 4 ], mathmode := "Text", name := "Attr", next := 62073, root := ~, start := 62016, stop := 62072 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosure" ), content := 0, count := [ 3, 3, 12, 6 ], mathmode := "Text", name := "Oper", next := 62138, root := ~, start := 62076, stop := 62137 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosureAttr" ), content := 0, count := [ 3, 3, 12, 8 ], mathmode := "Text", name := "Attr", next := 62207, root := ~, start := 62141, stop := 62206 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "Returns", next := 62239, root := ~, start := 62210, stop := 62238 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "A", next := 62277, root := ~, start := 62263, stop := 62276 ), rec( content := " is a di\ graph with no multiple edges, then these\n attributes return the (reflexive\ ) transitive closure of ", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "A", next := 62400, root := ~, start := 62386, stop := 62399 ), rec( content := ".", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "P", next := 62405, root := ~, start := 62401, stop := 62404 ), rec( content := "\n\n \ A digraph is ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eflexive", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "E", next := 62440, root := ~, start := 62424, stop := 62439 ), rec( content := " if it h\ as a loop at every vertex, and it is\n ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ransitive", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "E", next := 62506, root := ~, start := 62489, stop := 62505 ), rec( content := " if when\ ever ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "C", next := 62531, root := ~, start := 62519, stop := 62530 ), rec( content := " and ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j,k]", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "C", next := 62548, root := ~, start := 62536, stop := 62547 ), rec( content := " are edg\ es of\n ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "A", next := 62580, root := ~, start := 62566, stop := 62579 ), rec( content := ", ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,k]", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "C", next := 62594, root := ~, start := 62582, stop := 62593 ), rec( content := " is also\ an edge. The ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ reflexive)\n transitive closure", count := [ 3, 3, 12, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "E", next := 62660, root := ~, start := 62617, stop := 62659 ), rec( content := " of a di\ graph ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "A", next := 62688, root := ~, start := 62674, stop := 62687 ), rec( content := " is the \ least\n (reflexive and) transitive digraph containing ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "A", next := 62766, root := ~, start := 62752, stop := 62765 ), rec( content := ".", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "P", next := 62771, root := ~, start := 62767, stop := 62770 ), rec( content := "\n\n \ If ", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "A", next := 62794, root := ~, start := 62780, stop := 62793 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "A", next := 62900, root := ~, start := 62886, stop := 62899 ), rec( content := " is chan\ ged in-place into its\n (reflexive) transitive closure.", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "P", next := 62969, root := ~, start := 62965, stop := 62968 ), rec( content := "\n\n \ Let ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 62987, root := ~, start := 62979, stop := 62986 ), rec( content := " be the \ number of vertices of ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "A", next := 63031, root := ~, start := 63017, stop := 63030 ), rec( content := ", and le\ t\n ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 63053, root := ~, start := 63045, stop := 63052 ), rec( content := " be the \ number of edges. For an arbitrary digraph, these\n attributes will use a v\ ersion of the Floyd-Warshall algorithm, with\n complexity ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (n^3)", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 63211, root := ~, start := 63198, stop := 63210 ), rec( content := ".\n\n \ However, for a topologically sortable digraph [see ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "Ref", next := 63311, root := ~, start := 63269, stop := 63310 ), rec( content := "], these\ attributes will use methods\n with complexity ", count := [ 3, 3, 12, 12 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n + m \\cdot n)", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 63395, root := ~, start := 63368, stop := 63394 ), rec( content := " when th\ is is faster.", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 13 ], mathmode := "Text", name := "P", next := 63420, root := ~, start := 63416, stop := 63419 ), rec( content := "\n\n \ ", count := [ 3, 3, 12, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDiSparse6String(\".H`eOWR`Ul^\");\n\ngap> IsReflexiveDigraph(D) or IsTransitiveDigraph(D);\ \nfalse\ngap> OutNeighbours(D);\n[ [ 4, 6 ], [ 1, 3 ], [ ], [ 5 ], [ ], [ 7,\ 8, 9 ], [ ], [ ], \n [ ] ]\ngap> T := DigraphTransitiveClosure(D);\n\ngap> OutNeighbours(T);\n[\ [ 4, 6, 5, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ], [ ], [ 5 ], \n [ ], [ 7,\ 8, 9 ], [ ], [ ], [ ] ]\ngap> RT := DigraphReflexiveTransitiveClosure(D);\ \n\ngap> OutNeighbours(R\ T);\n[ [ 4, 6, 5, 7, 8, 9, 1 ], [ 1, 3, 4, 5, 6, 7, 8, 9, 2 ], [ 3 ], \n [ 5,\ 4 ], [ 5 ], [ 7, 8, 9, 6 ], [ 7 ], [ 8 ], [ 9 ] ]\ngap> D := CycleDigraph(IsM\ utableDigraph, 3);\n\ngap> DigraphRe\ flexiveTransitiveClosure(D);\n\ngap>\ D;\n\n", count := [ 3, 3, 12, 14 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 12, 14 ], mathmode := "Text", name := "Example", next := 64416, root := ~, start := 63426, stop := 64415 ), rec( content := "\n ", count := [ 3, 3, 12, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "Description", next := 64433, root := ~, start := 62242, stop := 64432 ) ], count := [ 3, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 64447, root := ~, start := 61945, stop := 64446 ), rec( content := "\n\n ", count := [ 3, 3, 12, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veReduction" ), content := 0, count := [ 3, 3, 13, 2 ], mathmode := "Text", name := "Oper", next := 64523, root := ~, start := 64468, stop := 64522 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veReductionAttr" ), content := 0, count := [ 3, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 64585, root := ~, start := 64526, stop := 64584 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReduction" ), content := 0, count := [ 3, 3, 13, 6 ], mathmode := "Text", name := "Oper", next := 64652, root := ~, start := 64588, stop := 64651 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReductionAttr" ), content := 0, count := [ 3, 3, 13, 8 ], mathmode := "Text", name := "Attr", next := 64723, root := ~, start := 64655, stop := 64722 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Returns", next := 64755, root := ~, start := 64726, stop := 64754 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "A", next := 64793, root := ~, start := 64779, stop := 64792 ), rec( content := " is a to\ pologically sortable digraph\n [see ", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Ref", next := 64875, root := ~, start := 64839, stop := 64874 ), rec( content := "]\n w\ ith no multiple edges, then these operations return the\n (reflexive) tran\ sitive reduction of ", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "A", next := 64993, root := ~, start := 64979, stop := 64992 ), rec( content := ".", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "P", next := 64998, root := ~, start := 64994, stop := 64997 ), rec( content := "\n\n \ The (reflexive) transitive reduction of such a digraph is the unique\n leas\ t subgraph such that the (reflexive) transitive closure of the\n subgraph i\ s equal to the (reflexive) transitive closure of ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "A", next := 65221, root := ~, start := 65207, stop := 65220 ), rec( content := "\n [s\ ee ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveClosure" ), content := 0, count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "Ref", next := 65278, root := ~, start := 65231, stop := 65277 ), rec( content := "].\n \ In order words, it is the least subgraph of ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "A", next := 65343, root := ~, start := 65329, stop := 65342 ), rec( content := " which\n\ retains the same reachability as ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "A", next := 65401, root := ~, start := 65387, stop := 65400 ), rec( content := ".", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "P", next := 65406, root := ~, start := 65402, stop := 65405 ), rec( content := "\n\n \ If ", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "A", next := 65429, root := ~, start := 65415, stop := 65428 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "A", next := 65535, root := ~, start := 65521, stop := 65534 ), rec( content := " is chan\ ged in-place into its\n (reflexive) transitive reduction.", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "P", next := 65606, root := ~, start := 65602, stop := 65605 ), rec( content := "\n\n \ Let ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "M", next := 65624, root := ~, start := 65616, stop := 65623 ), rec( content := " be the \ number of vertices of an arbitrary digraph, and let\n ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "M", next := 65700, root := ~, start := 65692, stop := 65699 ), rec( content := " be the \ number of edges.\n Then these operations\n use methods with complexity " , count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n + m \\cdot n)", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "M", next := 65810, root := ~, start := 65783, stop := 65809 ), rec( content := ".\n " , count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 13 ], mathmode := "Text", name := "P", next := 65820, root := ~, start := 65816, stop := 65819 ), rec( content := "\n ", count := [ 3, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 3], [3], [3]]);;\ngap> DigraphHasLoops(D);\ntrue\ng\ ap> D1 := DigraphReflexiveTransitiveReduction(D);\n\ngap> DigraphHasLoops(D1);\nfalse\ngap> OutNeighbours(D1);\n\ [ [ 2 ], [ 3 ], [ ] ]\ngap> D2 := DigraphTransitiveReduction(D);\n\ngap> DigraphHasLoops(D2);\ntrue\ngap> OutNe\ ighbours(D2);\n[ [ 2, 1 ], [ 3 ], [ 3 ] ]\ngap> DigraphReflexiveTransitiveClos\ ure(D)\n> = DigraphReflexiveTransitiveClosure(D1);\ntrue\ngap> DigraphTransit\ iveClosure(D)\n> = DigraphTransitiveClosure(D2);\ntrue\ngap> D := Digraph(IsM\ utableDigraph, [[1], [1], [1, 2, 3]]);\n\ngap> DigraphReflexiveTransitiveReduction(D);\n\ngap> D;\n\n", count := [ 3, 3, 13, 14 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 13, 14 ], mathmode := "Text", name := "Example", next := 66680, root := ~, start := 65825, stop := 66679 ), rec( content := "\n ", count := [ 3, 3, 13, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Description", next := 66697, root := ~, start := 64758, stop := 66696 ) ], count := [ 3, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 66711, root := ~, start := 64453, stop := 66710 ), rec( content := "\n\n ", count := [ 3, 3, 13, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, label \ ]", Name := "DigraphAddVertex" ), content := 0, count := [ 3, 3, 14, 2 ], mathmode := "Text", name := "Oper", next := 66787, root := ~, start := 66732, stop := 66786 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 66819, root := ~, start := 66790, stop := 66818 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ operation returns a digraph constructed from ", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "A", next := 66903, root := ~, start := 66889, stop := 66902 ), rec( content := " by addi\ ng a\n single new vertex, and no new edges. ", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "P", next := 66961, root := ~, start := 66957, stop := 66960 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abel", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "A", next := 67011, root := ~, start := 66999, stop := 67010 ), rec( content := " is a ", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object,\ \n then the new vertex will be labelled ", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abel", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "A", next := 67084, root := ~, start := 67072, stop := 67083 ), rec( content := ".\n " , count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "P", next := 67094, root := ~, start := 67090, stop := 67093 ), rec( content := "\n\n \ If ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 67117, root := ~, start := 67103, stop := 67116 ), rec( content := " belongs\ to ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "Ref", next := 67159, root := ~, start := 67129, stop := 67158 ), rec( content := ", then t\ he\n vertex is added directly to ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 67216, root := ~, start := 67202, stop := 67215 ), rec( content := ". If ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 67235, root := ~, start := 67221, stop := 67234 ), rec( content := " belongs\ to\n ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "Ref", next := 67283, root := ~, start := 67251, stop := 67282 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 67320, root := ~, start := 67306, stop := 67319 ), rec( content := " with\n \ the additional vertex is returned.", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 14, 6 ], mathmode := "Text", name := "P", next := 67368, root := ~, start := 67364, stop := 67367 ), rec( content := "\n\n \ ", count := [ 3, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(3);\n\n\ gap> new := DigraphAddVertex(D);\n\ \ngap> D = new;\nfalse\ngap> DigraphVertices(new);\n[ 1 .. 4 ]\ngap> new := Di\ graphAddVertex(D, Group([(1, 2)]));\n\ngap> DigraphVertexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]) ]\ngap> D := \ CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> new := DigraphAddVertex(D);\n\ngap> D = new;\ntrue\n", count := [ 3, 3, 14, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 14, 7 ], mathmode := "Text", name := "Example", next := 67973, root := ~, start := 67374, stop := 67972 ), rec( content := "\n ", count := [ 3, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 67990, root := ~, start := 66822, stop := 67989 ) ], count := [ 3, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 68004, root := ~, start := 66717, stop := 68003 ), rec( content := "\n\n ", count := [ 3, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, m", Label := "for a digraph \ and an integer", Name := "DigraphAddVertices" ), content := 0, count := [ 3, 3, 15, 2 ], mathmode := "Text", name := "Oper", next := 68116, root := ~, start := 68025, stop := 68115 ), rec( attributes := rec( Arg := "digraph, labels" , Label := "for a digraph \ and a list of labels", Name := "DigraphAddVertices" ), content := 0, count := [ 3, 3, 15, 4 ], mathmode := "Text", name := "Oper", next := 68221, root := ~, start := 68119, stop := 68220 ), rec( content := " The 3-argument \ version is deliberately undocumented. It is redundant,\n but it is inclu\ ded for backwards compatibility. ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "Returns", next := 68389, root := ~, start := 68360, stop := 68388 ), rec( attributes := rec( ), content := [ rec( content := "\n For\ a non-negative integer ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "A", next := 68445, root := ~, start := 68437, stop := 68444 ), rec( content := ", this o\ peration returns a digraph\n constructed from ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "A", next := 68515, root := ~, start := 68501, stop := 68514 ), rec( content := " by addi\ ng ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "A", next := 68534, root := ~, start := 68526, stop := 68533 ), rec( content := " new ver\ tices.\n ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "P", next := 68557, root := ~, start := 68553, stop := 68556 ), rec( content := "\n\n \ Otherwise, if ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "A", next := 68590, root := ~, start := 68577, stop := 68589 ), rec( content := " is a li\ st\n consisting of ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "C", next := 68627, root := ~, start := 68619, stop := 68626 ), rec( content := " ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ , then this operation returns a\n digraph constructed from ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "A", next := 68716, root := ~, start := 68702, stop := 68715 ), rec( content := " by addi\ ng ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "C", next := 68735, root := ~, start := 68727, stop := 68734 ), rec( content := " new ver\ tices,\n which are labelled according to this list.\n ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "P", next := 68805, root := ~, start := 68801, stop := 68804 ), rec( content := "\n\n \ If ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 68828, root := ~, start := 68814, stop := 68827 ), rec( content := " belongs\ to ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 68870, root := ~, start := 68840, stop := 68869 ), rec( content := ", then t\ he\n vertices are added directly to ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 68930, root := ~, start := 68916, stop := 68929 ), rec( content := ", which \ is changed in-place.\n If ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 68980, root := ~, start := 68966, stop := 68979 ), rec( content := " belongs\ to ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 69024, root := ~, start := 68992, stop := 69023 ), rec( content := ", then t\ he\n result is returned as an immutable digraph.\n ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 15, 8 ], mathmode := "Text", name := "P", next := 69091, root := ~, start := 69087, stop := 69090 ), rec( content := "\n\n \ ", count := [ 3, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(3);\n\n\ gap> new := DigraphAddVertices(D, 3);\n\ngap> DigraphVertices(new);\n[ 1 .. 6 ]\ngap> new := DigraphAddVertices(\ D, [Group([(1, 2)]), \"d\"]);\n\ng\ ap> DigraphVertexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]), \"d\" ]\ngap> Digr\ aphAddVertices(D, 0) = D;\ntrue\ngap> D := CompleteBipartiteDigraph(IsMutableD\ igraph, 2, 3);\n\ngap> new := Digra\ phAddVertices(D, 4);\n\ngap> D = ne\ w;\ntrue\n", count := [ 3, 3, 15, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 9 ], mathmode := "Text", name := "Example", next := 69740, root := ~, start := 69097, stop := 69739 ), rec( content := "\n ", count := [ 3, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "Description", next := 69757, root := ~, start := 68392, stop := 69756 ) ], count := [ 3, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 69771, root := ~, start := 68010, stop := 69770 ), rec( content := "\n\n ", count := [ 3, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Label := "for a digraph \ and an edge", Name := "DigraphAddEdge" ), content := 0, count := [ 3, 3, 16, 2 ], mathmode := "Text", name := "Oper", next := 69879, root := ~, start := 69792, stop := 69878 ), rec( attributes := rec( Arg := "digraph, src, ra\ n", Label := "for a digraph, source, and range", Name := "DigraphAddEdge" ), content := 0, count := [ 3, 3, 16, 4 ], mathmode := "Text", name := "Oper", next := 69980, root := ~, start := 69882, stop := 69979 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "Returns", next := 70012, root := ~, start := 69983, stop := 70011 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70047, root := ~, start := 70036, stop := 70046 ), rec( content := " is a pa\ ir of vertices of ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70087, root := ~, start := 70073, stop := 70086 ), rec( content := ", or ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70102, root := ~, start := 70092, stop := 70101 ), rec( content := " and\n \ ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70121, root := ~, start := 70111, stop := 70120 ), rec( content := " are ver\ tices of ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70152, root := ~, start := 70138, stop := 70151 ), rec( content := ",\n t\ hen this operation returns a digraph constructed from ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70227, root := ~, start := 70213, stop := 70226 ), rec( content := "\n by\ adding a new edge with source ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70276, root := ~, start := 70265, stop := 70275 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "C", next := 70286, root := ~, start := 70276, stop := 70285 ), rec( content := " [", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70298, root := ~, start := 70288, stop := 70297 ), rec( content := "] and ra\ nge\n ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70325, root := ~, start := 70314, stop := 70324 ), rec( attributes := rec( ), content := [ rec( content := "[\ 2]", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "C", next := 70335, root := ~, start := 70325, stop := 70334 ), rec( content := " [", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70347, root := ~, start := 70337, stop := 70346 ), rec( content := "].\n \ ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "P", next := 70358, root := ~, start := 70354, stop := 70357 ), rec( content := "\n \n\ If ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 70385, root := ~, start := 70371, stop := 70384 ), rec( content := " belongs\ to ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "Ref", next := 70427, root := ~, start := 70397, stop := 70426 ), rec( content := ", then t\ he edge\n is added directly to ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 70482, root := ~, start := 70468, stop := 70481 ), rec( content := ". If ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 70501, root := ~, start := 70487, stop := 70500 ), rec( content := " belongs\ to ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "Ref", next := 70551, root := ~, start := 70513, stop := 70550 ), rec( content := ", then a\ n immutable copy of ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 70593, root := ~, start := 70579, stop := 70592 ), rec( content := " with th\ e\n additional edge is returned. ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 16, 7 ], mathmode := "Text", name := "P", next := 70641, root := ~, start := 70637, stop := 70640 ), rec( content := "\n\n \ ", count := [ 3, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := Digraph([[2], [3], []]);\n\ngap> DigraphEdges(D1);\n[ [ 1, 2 ], [ 2, 3 ] ]\ngap> D2 := DigraphAddEdg\ e(D1, [3, 1]);\n\ngap> DigraphEdge\ s(D2);\n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D3 := DigraphAddEdge(D2, [2, 3]\ );\n\ngap> DigraphEdges(D3);\ \n[ [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D := CycleDigraph(IsMutable\ Digraph, 4);\n\ngap> new := DigraphA\ ddEdge(D, [1, 3]);\n\ngap> DigraphEd\ ges(new);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 3, 4 ], [ 4, 1 ] ]\ngap> D = new;\ \ntrue\n", count := [ 3, 3, 16, 8 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 8 ], mathmode := "Text", name := "Example", next := 71367, root := ~, start := 70647, stop := 71366 ), rec( content := "\n ", count := [ 3, 3, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "Description", next := 71384, root := ~, start := 70015, stop := 71383 ) ], count := [ 3, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 71398, root := ~, start := 69777, stop := 71397 ), rec( content := "\n\n ", count := [ 3, 3, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Name := "DigraphAddEdgeO\ rbit" ), content := 0, count := [ 3, 3, 17, 2 ], mathmode := "Text", name := "Oper", next := 71473, root := ~, start := 71419, stop := 71472 ), rec( attributes := rec( ), content := [ rec( content := "\n A n\ ew digraph.\n ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 71517, root := ~, start := 71476, stop := 71516 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a new digraph with the same vertices and edges as\n ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71629, root := ~, start := 71615, stop := 71628 ), rec( content := " and wit\ h additional edges consisting of the orbit of the\n edge ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71707, root := ~, start := 71696, stop := 71706 ), rec( content := " under t\ he action of the ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Ref", next := 71758, root := ~, start := 71732, stop := 71757 ), rec( content := " of\n \ ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71780, root := ~, start := 71766, stop := 71779 ), rec( content := ". If ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71797, root := ~, start := 71786, stop := 71796 ), rec( content := " is alre\ ady an edge in ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71834, root := ~, start := 71820, stop := 71833 ), rec( content := ", then\n\ ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71859, root := ~, start := 71845, stop := 71858 ), rec( content := " is retu\ rned unchanged. The argument ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71910, root := ~, start := 71896, stop := 71909 ), rec( content := " must be\ \n an immutable digraph.\n ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "P", next := 71953, root := ~, start := 71949, stop := 71952 ), rec( content := "\n\n \ An edge is simply a pair of vertices of ", count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "A", next := 72013, root := ~, start := 71999, stop := 72012 ), rec( content := ".\n\n \ ", count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := CayleyDigraph(DihedralGroup(8));\n\ngap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n\ngap> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ],\ [ 1, 4 ], [ 2, 1 ], [ 2, 8 ], [ 2, 6 ], \n [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4\ , 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 \ ], [ 6, 2 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ]\ \ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ],\ [ 2, 8 ], \n [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], \n \ [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], \n [ 5, 8 ], [ 5\ , 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 \ ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := Digraph\ RemoveEdgeOrbit(gr2, [1, 8]);\n\n\ gap> gr3 = gr1;\ntrue\n", count := [ 3, 3, 17, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 5 ], mathmode := "Text", name := "Example", next := 72989, root := ~, start := 72020, stop := 72988 ), rec( content := "\n ", count := [ 3, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 73006, root := ~, start := 71520, stop := 73005 ) ], count := [ 3, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 73020, root := ~, start := 71404, stop := 73019 ), rec( content := "\n\n ", count := [ 3, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edges", Name := "DigraphAddEdges\ " ), content := 0, count := [ 3, 3, 18, 2 ], mathmode := "Text", name := "Oper", next := 73092, root := ~, start := 73041, stop := 73091 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 73124, root := ~, start := 73095, stop := 73123 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73160, root := ~, start := 73148, stop := 73159 ), rec( content := " is a (p\ ossibly empty) list of pairs of vertices of\n ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73230, root := ~, start := 73216, stop := 73229 ), rec( content := ", then t\ his operation returns a digraph constructed from\n ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73305, root := ~, start := 73291, stop := 73304 ), rec( content := " by addi\ ng the edges specified by ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73351, root := ~, start := 73339, stop := 73350 ), rec( content := ". More\\ n precisely, for every ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "C", next := 73395, root := ~, start := 73384, stop := 73394 ), rec( content := " in ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73411, root := ~, start := 73399, stop := 73410 ), rec( content := ", a new \ edge\n will be added with source ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge[1]", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "C", next := 73468, root := ~, start := 73454, stop := 73467 ), rec( content := " and ran\ ge ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges[2]", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "C", next := 73494, root := ~, start := 73479, stop := 73493 ), rec( content := ".\n " , count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "P", next := 73504, root := ~, start := 73500, stop := 73503 ), rec( content := "\n\n \ If an edge is included in ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 73548, root := ~, start := 73536, stop := 73547 ), rec( content := " with mu\ ltiplicity ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "C", next := 73575, root := ~, start := 73567, stop := 73574 ), rec( content := ",\n t\ hen it will be added ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "C", next := 73611, root := ~, start := 73603, stop := 73610 ), rec( content := " times.\\ n\n If ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 73641, root := ~, start := 73627, stop := 73640 ), rec( content := " belongs\ to ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "Ref", next := 73683, root := ~, start := 73653, stop := 73682 ), rec( content := ", then t\ he edges\n are added directly to ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 73740, root := ~, start := 73726, stop := 73739 ), rec( content := ". If ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 73759, root := ~, start := 73745, stop := 73758 ), rec( content := " belongs\ to ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "Ref", next := 73809, root := ~, start := 73771, stop := 73808 ), rec( content := ", then t\ he result is returned as an immutable\n digraph.\n ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 18, 5 ], mathmode := "Text", name := "P", next := 73876, root := ~, start := 73872, stop := 73875 ), rec( content := "\n\n \ ", count := [ 3, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> func := function(n)\n> local source, range, i;\n> source := [];\n> ra\ nge := [];\n> for i in [1 .. n - 2] do\n> Add(source, i);\n> Add(range\ , i + 1);\n> od;\n> return Digraph(n, source, range);\n> end;;\ngap> D := fu\ nc(1024);\n\ngap> new := Dig\ raphAddEdges(D,\n> [[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);\n\ngap> D = new;\nfalse\ngap>\ D2 := DigraphMutableCopy(func(1024));\n\ngap> new := DigraphAddEdges(D2,\n> [[1023, 1024], [1, 1024], [1023\ , 1024], [1024, 1]]);\n\n\ gap> D2 = new;\ntrue\n", count := [ 3, 3, 18, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 6 ], mathmode := "Text", name := "Example", next := 74613, root := ~, start := 73882, stop := 74612 ), rec( content := "\n ", count := [ 3, 3, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 74630, root := ~, start := 73127, stop := 74629 ) ], count := [ 3, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 74644, root := ~, start := 73026, stop := 74643 ), rec( content := "\n\n ", count := [ 3, 3, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, v", Name := "DigraphRemoveVe\ rtex" ), content := 0, count := [ 3, 3, 19, 2 ], mathmode := "Text", name := "Oper", next := 74716, root := ~, start := 74665, stop := 74715 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "Returns", next := 74748, root := ~, start := 74719, stop := 74747 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 74780, root := ~, start := 74772, stop := 74779 ), rec( content := " is a ve\ rtex of ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 74810, root := ~, start := 74796, stop := 74809 ), rec( content := ", then t\ his operation returns a\n digraph constructed from ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 74885, root := ~, start := 74871, stop := 74884 ), rec( content := " by remo\ ving vertex ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 74913, root := ~, start := 74905, stop := 74912 ), rec( content := ",\n a\ long with any edge whose source or range vertex is ", count := [ 3, 3, 19, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 74979, root := ~, start := 74971, stop := 74978 ), rec( content := ".", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "P", next := 74984, root := ~, start := 74980, stop := 74983 ), rec( content := "\n\n \ If ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "A", next := 75007, root := ~, start := 74993, stop := 75006 ), rec( content := " has ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "C", next := 75020, root := ~, start := 75012, stop := 75019 ), rec( content := " vertice\ s, then the vertices of the returned\n digraph are ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1..n-1]", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "C", next := 75096, root := ~, start := 75081, stop := 75095 ), rec( content := ", but th\ e original labels can be\n accessed via ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "Ref", next := 75179, root := ~, start := 75146, stop := 75178 ), rec( content := ". ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "P", next := 75185, root := ~, start := 75181, stop := 75184 ), rec( content := "\n\n \ If ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 75208, root := ~, start := 75194, stop := 75207 ), rec( content := " belongs\ to ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "Ref", next := 75250, root := ~, start := 75220, stop := 75249 ), rec( content := ", then t\ he\n vertex is removed directly from ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 75311, root := ~, start := 75297, stop := 75310 ), rec( content := ". If ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 75330, root := ~, start := 75316, stop := 75329 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "Ref", next := 75378, root := ~, start := 75346, stop := 75377 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 75415, root := ~, start := 75401, stop := 75414 ), rec( content := "\n wi\ thout the vertex is returned.", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 19, 6 ], mathmode := "Text", name := "P", next := 75455, root := ~, start := 75451, stop := 75454 ), rec( content := "\n\n \ ", count := [ 3, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"a\", \ \"b\", \"c\", \"c\"],\n> [\"b\", \"c\", \"a\", \"a\", \"c\"])\ ;\n\ngap> DigraphVertexLabels(D);\ \n[ \"a\", \"b\", \"c\" ]\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1\ ], [ 3, 1 ], [ 3, 3 ] ]\ngap> new := DigraphRemoveVertex(D, 2);\n\ngap> DigraphVertexLabels(new);\n[ \"a\", \"c\ \" ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> new := DigraphRemoveVertex(D, 1);\n\ngap> DigraphVertexLabels(D);\n[ 2, 3, 4, 5 ]\ngap> D =\ new;\ntrue\n", count := [ 3, 3, 19, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 7 ], mathmode := "Text", name := "Example", next := 76149, root := ~, start := 75461, stop := 76148 ), rec( content := "\n ", count := [ 3, 3, 19, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 76166, root := ~, start := 74751, stop := 76165 ) ], count := [ 3, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 76180, root := ~, start := 74650, stop := 76179 ), rec( content := "\n\n ", count := [ 3, 3, 19, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, verts", Name := "DigraphRemoveVe\ rtices" ), content := 0, count := [ 3, 3, 20, 2 ], mathmode := "Text", name := "Oper", next := 76258, root := ~, start := 76201, stop := 76257 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "Returns", next := 76290, root := ~, start := 76261, stop := 76289 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76326, root := ~, start := 76314, stop := 76325 ), rec( content := " is a (p\ ossibly empty) duplicate-free list of vertices of\n ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76402, root := ~, start := 76388, stop := 76401 ), rec( content := ", then t\ his operation returns a digraph constructed from\n ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76477, root := ~, start := 76463, stop := 76476 ), rec( content := " by remo\ ving every vertex in ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76518, root := ~, start := 76506, stop := 76517 ), rec( content := ", along \ with any edge\n whose source or range vertex is in ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76591, root := ~, start := 76579, stop := 76590 ), rec( content := ".", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "P", next := 76596, root := ~, start := 76592, stop := 76595 ), rec( content := "\n\n \ If ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "A", next := 76619, root := ~, start := 76605, stop := 76618 ), rec( content := " has ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "C", next := 76632, root := ~, start := 76624, stop := 76631 ), rec( content := " vertice\ s, then the vertices of the new\n digraph are ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. n-Length(", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "vert\ s", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := \ "Text", name := "A", next := 76718, root := ~, start := 76706, stop := 76717 ) , rec( content := "\ )]", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "C", next := 76724, root := ~, start := 76688, stop := 76723 ), rec( content := ", but th\ e\n original labels can be accessed via ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "Ref", next := 76807, root := ~, start := 76774, stop := 76806 ), rec( content := ". ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "P", next := 76813, root := ~, start := 76809, stop := 76812 ), rec( content := "\n\n \ If ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 76836, root := ~, start := 76822, stop := 76835 ), rec( content := " belongs\ to ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "Ref", next := 76878, root := ~, start := 76848, stop := 76877 ), rec( content := ", then t\ he\n vertices are removed directly from ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 76942, root := ~, start := 76928, stop := 76941 ), rec( content := ". If ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 76961, root := ~, start := 76947, stop := 76960 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "Ref", next := 77009, root := ~, start := 76977, stop := 77008 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 77046, root := ~, start := 77032, stop := 77045 ), rec( content := "\n wi\ thout the vertices is returned.", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 20, 6 ], mathmode := "Text", name := "P", next := 77088, root := ~, start := 77084, stop := 77087 ), rec( content := "\n\n \ ", count := [ 3, 3, 20, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> SetDigraphVertexLabels(D, [\"a\", \"b\ \", \"c\", \"d\", \"e\"]);\ngap> new := DigraphRemoveVertices(D, [2, 4]);\n\ngap> DigraphVertexLabels(new);\n[ \ \"a\", \"c\", \"e\" ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> new := DigraphRemoveVertices(D, [1, 3]\ );\n\ngap> DigraphVertexLabels(D);\n[\ 2, 4, 5 ]\ngap> D = new;\ntrue\n", count := [ 3, 3, 20, 7 ], name := "PCDATA" , root := ~ ) ], count := [ 3, 3, 20, 7 ], mathmode := "Text", name := "Example", next := 77671, root := ~, start := 77094, stop := 77670 ), rec( content := "\n ", count := [ 3, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 77688, root := ~, start := 76293, stop := 77687 ) ], count := [ 3, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 77702, root := ~, start := 76186, stop := 77701 ), rec( content := "\n\n ", count := [ 3, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Label := "for a digraph \ and an edge", Name := "DigraphRemoveEdge" ), content := 0, count := [ 3, 3, 21, 2 ], mathmode := "Text", name := "Oper", next := 77813, root := ~, start := 77723, stop := 77812 ), rec( attributes := rec( Arg := "digraph, src, ra\ n", Label := "for a digraph, source, and range", Name := "DigraphRemoveEdge" ) , content := 0, count := [ 3, 3, 21, 4 ], mathmode := "Text", name := "Oper", next := 77917, root := ~, start := 77816, stop := 77916 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "Returns", next := 77949, root := ~, start := 77920, stop := 77948 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 77987, root := ~, start := 77973, stop := 77986 ), rec( content := " is a di\ graph with no multiple edges and ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78039, root := ~, start := 78028, stop := 78038 ), rec( content := " is a\n \ pair of vertices of ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78083, root := ~, start := 78069, stop := 78082 ), rec( content := ", or ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78098, root := ~, start := 78088, stop := 78097 ), rec( content := " and ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78113, root := ~, start := 78103, stop := 78112 ), rec( content := " are\n \ vertices of ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78148, root := ~, start := 78134, stop := 78147 ), rec( content := ", then t\ his operation returns a digraph\n constructed from ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78223, root := ~, start := 78209, stop := 78222 ), rec( content := " by remo\ ving the edge specified by\n ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78273, root := ~, start := 78262, stop := 78272 ), rec( content := " or ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ src, ran]", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78294, root := ~, start := 78277, stop := 78293 ), rec( content := ". ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "P", next := 78300, root := ~, start := 78296, stop := 78299 ), rec( content := " \n\n \ If ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 78325, root := ~, start := 78311, stop := 78324 ), rec( content := " belongs\ to ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "Ref", next := 78373, root := ~, start := 78337, stop := 78372 ), rec( content := ", then t\ he edge is removed directly from\n ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 78432, root := ~, start := 78418, stop := 78431 ), rec( content := ". If ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 78451, root := ~, start := 78437, stop := 78450 ), rec( content := " belongs\ to ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "Ref", next := 78501, root := ~, start := 78463, stop := 78500 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 78538, root := ~, start := 78524, stop := 78537 ), rec( content := " without\ \n the edge is returned.\n ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "P", next := 78581, root := ~, start := 78577, stop := 78580 ), rec( content := "\n\n \ Note that if ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78614, root := ~, start := 78600, stop := 78613 ), rec( content := " belongs\ to ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "Ref", next := 78658, root := ~, start := 78626, stop := 78657 ), rec( content := ",\n t\ hen a new copy of ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78697, root := ~, start := 78683, stop := 78696 ), rec( content := " will be\ returned even if ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78734, root := ~, start := 78723, stop := 78733 ), rec( content := " or \n \ ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ src, ran]", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78760, root := ~, start := 78743, stop := 78759 ), rec( content := " does no\ t define an edge of ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78802, root := ~, start := 78788, stop := 78801 ), rec( content := ".", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 21, 8 ], mathmode := "Text", name := "P", next := 78807, root := ~, start := 78803, stop := 78806 ), rec( content := "\n\n \ ", count := [ 3, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdge(D, [250000, 1]);\n\ngap> new := DigraphRemoveEdge(D, [25000, 2]);;\nga\ p> new = D;\ntrue\ngap> IsIdenticalObj(new, D);\nfalse\ngap> D := DigraphMutab\ leCopy(D);;\ngap> new := DigraphRemoveEdge(D, 2500, 2);;\ngap> IsIdenticalObj(\ new, D);\ntrue\n", count := [ 3, 3, 21, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 9 ], mathmode := "Text", name := "Example", next := 79237, root := ~, start := 78813, stop := 79236 ), rec( content := "\n ", count := [ 3, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "Description", next := 79254, root := ~, start := 77952, stop := 79253 ) ], count := [ 3, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 79268, root := ~, start := 77708, stop := 79267 ), rec( content := "\n\n ", count := [ 3, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Name := "DigraphRemoveEd\ geOrbit" ), content := 0, count := [ 3, 3, 22, 2 ], mathmode := "Text", name := "Oper", next := 79346, root := ~, start := 79289, stop := 79345 ), rec( attributes := rec( ), content := [ rec( content := "\n A n\ ew digraph.\n ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Returns", next := 79390, root := ~, start := 79349, stop := 79389 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a new digraph with the same vertices as\n ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79492, root := ~, start := 79478, stop := 79491 ), rec( content := " and wit\ h the orbit of the edge ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79535, root := ~, start := 79524, stop := 79534 ), rec( content := " (under \ the action\n of the ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Ref", next := 79591, root := ~, start := 79565, stop := 79590 ), rec( content := " of ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79609, root := ~, start := 79595, stop := 79608 ), rec( content := ") remove\ d.\n If ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79638, root := ~, start := 79627, stop := 79637 ), rec( content := " is not \ an edge in ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79671, root := ~, start := 79657, stop := 79670 ), rec( content := ", then " , count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79692, root := ~, start := 79678, stop := 79691 ), rec( content := " is\n \ returned unchanged. The argument ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79747, root := ~, start := 79733, stop := 79746 ), rec( content := " must be\ an immutable\n digraph.\n ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "P", next := 79790, root := ~, start := 79786, stop := 79789 ), rec( content := "\n\n \ An edge is simply a pair of vertices of ", count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "A", next := 79850, root := ~, start := 79836, stop := 79849 ), rec( content := ".\n " , count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := CayleyDigraph(DihedralGroup(8));\n\ngap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n\ngap> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ],\ [ 1, 4 ], [ 2, 1 ], [ 2, 8 ], [ 2, 6 ], \n [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4\ , 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 \ ], [ 6, 2 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ]\ \ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ],\ [ 2, 8 ], \n [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], \n \ [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], \n [ 5, 8 ], [ 5\ , 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 \ ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := Digraph\ RemoveEdgeOrbit(gr2, [1, 8]);\n\n\ gap> gr3 = gr1;\ntrue\n", count := [ 3, 3, 22, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 5 ], mathmode := "Text", name := "Example", next := 80825, root := ~, start := 79856, stop := 80824 ), rec( content := "\n ", count := [ 3, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 80842, root := ~, start := 79393, stop := 80841 ) ], count := [ 3, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 80856, root := ~, start := 79274, stop := 80855 ), rec( content := "\n\n ", count := [ 3, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edges", Name := "DigraphRemoveEd\ ges" ), content := 0, count := [ 3, 3, 23, 2 ], mathmode := "Text", name := "Oper", next := 80931, root := ~, start := 80877, stop := 80930 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "Returns", next := 80963, root := ~, start := 80934, stop := 80962 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ one of the following holds:\n ", count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 3, 3, 23, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "digraph", count := [ 3, 3, 23, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], mathmode := "Te\ xt", name := "A", next := 81061, root := ~, start := 81047, stop := 81060 ), rec( cont\ ent := " is a digraph with no multiple edges, and\n ", count := [ 3, 3,\ 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "edges", count := [ 3, 3, 23, 5 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], math\ mode := "Text", name := "A", next := 81123, root := ~, start := 81111, stop :=\ 81122 ), rec( content := " is a list of pairs of vertices of ", count := [ 3,\ 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "digraph", count := [ 3, 3, 23, 5 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ] , math\ mode := "Text", name := "A", next := 81172, root := ~, start := 81158, stop :=\ 81171 ), rec( content := ", or\n ", count := [ 3, 3, 23, 5 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], name := "Ite\ m", next := 81190, root := ~, start := 81032, stop := 81189 ), rec( attributes :\ = rec( ), content := [ rec( content := " ", count := [ 3, 3, 23, 7 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "digraph", count := [ 3, 3, 23, 7 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 7 ], mathmode := "Text\ ", name := "A", next := 81218, root := ~, start := 81204, stop := 81217 ), rec( cont\ ent := " is a digraph and ", count := [ 3, 3, 23, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "edges", count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "A", next\ := 81248, root := ~, start := 81236, stop := 81247 ), rec( content := " is an\ empty list ", count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 3, 23, 7 ], name := "Item", next := 81273, root := ~, start := 811\ 97, stop := 81272 ) ], count := [ 3, 3, 23, 4 ], mathmode := "Text", name := "List", next := 81285, root := ~, start := 81019, stop := 81284 ), rec( content := "\n\n \ then this operation returns a digraph constructed from ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "A", next := 81360, root := ~, start := 81346, stop := 81359 ), rec( content := "\n by\ removing all of the edges specified by ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "A", next := 81419, root := ~, start := 81407, stop := 81418 ), rec( content := "\n (s\ ee ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and an edge", Oper := "DigraphRemoveEdge" ), content := 0, count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "Ref", next := 81494, root := ~, start := 81429, stop := 81493 ), rec( content := ").\n \ ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "P", next := 81505, root := ~, start := 81501, stop := 81504 ), rec( content := "\n\n \ If ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 81528, root := ~, start := 81514, stop := 81527 ), rec( content := " belongs\ to ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "Ref", next := 81570, root := ~, start := 81540, stop := 81569 ), rec( content := ", then t\ he edge\n is removed directly from ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 81629, root := ~, start := 81615, stop := 81628 ), rec( content := ". If ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 81648, root := ~, start := 81634, stop := 81647 ), rec( content := " belongs\ to ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "Ref", next := 81698, root := ~, start := 81660, stop := 81697 ), rec( content := ", the ed\ ge is removed from an immutable copy of\n ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 81764, root := ~, start := 81750, stop := 81763 ), rec( content := " and thi\ s new digraph is returned.", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "P", next := 81802, root := ~, start := 81798, stop := 81801 ), rec( content := "\n\n \ Note that if ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 81833, root := ~, start := 81821, stop := 81832 ), rec( content := " is empt\ y, then this operation\n will always return ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 81901, root := ~, start := 81887, stop := 81900 ), rec( content := " rather \ than a copy. Also, if any element\n of ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 81962, root := ~, start := 81950, stop := 81961 ), rec( content := " is inva\ lid (i.e. does not define an edge of ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 82021, root := ~, start := 82007, stop := 82020 ), rec( content := ")\n t\ hen that element will simply be ignored.\n\n ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdges(D, [[250000, 1]]);\n\ngap> D := DigraphMutableCopy(D);\n\ngap> new := DigraphRemoveEdges(D, [[1\ , 2], [2, 3], [3, 100]]);\n\ngap> new = D;\ntrue\n", count := [ 3, 3, 23, 12 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 12 ], mathmode := "Text", name := "Example", next := 82509, root := ~, start := 82074, stop := 82508 ), rec( content := "\n ", count := [ 3, 3, 23, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 82526, root := ~, start := 80966, stop := 82525 ) ], count := [ 3, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 82540, root := ~, start := 80862, stop := 82539 ), rec( content := "\n\n ", count := [ 3, 3, 23, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveLo\ ops" ), content := 0, count := [ 3, 3, 24, 2 ], mathmode := "Text", name := "Oper", next := 82608, root := ~, start := 82561, stop := 82607 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveLo\ opsAttr" ), content := 0, count := [ 3, 3, 24, 4 ], mathmode := "Text", name := "Attr", next := 82662, root := ~, start := 82611, stop := 82661 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "Returns", next := 82694, root := ~, start := 82665, stop := 82693 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "A", next := 82732, root := ~, start := 82718, stop := 82731 ), rec( content := " is a di\ graph, then this operation returns a\n digraph constructed from ", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "A", next := 82821, root := ~, start := 82807, stop := 82820 ), rec( content := " by remo\ ving every loop. A loop\n is an edge with equal source and range.", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "P", next := 82900, root := ~, start := 82896, stop := 82899 ), rec( content := "\n\n \ If ", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "A", next := 82923, root := ~, start := 82909, stop := 82922 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the loops are\ removed from the mutable digraph ", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "A", next := 83056, root := ~, start := 83042, stop := 83055 ), rec( content := "\n in\ -place.\n ", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 4], [1, 4], [3, 4], [1, 4, 5], [1, 5]]);\n\ngap> DigraphRemoveLoops(D);\n\ngap> D := Digraph(IsMutableDigraph, [[1, 2]\ , [1]]);\n\ngap> DigraphRemoveLoops(\ D);\n\ngap> D;\n\n", count := [ 3, 3, 24, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 7 ], mathmode := "Text", name := "Example", next := 83510, root := ~, start := 83075, stop := 83509 ), rec( content := "\n ", count := [ 3, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "Description", next := 83527, root := ~, start := 82697, stop := 83526 ) ], count := [ 3, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 83541, root := ~, start := 82546, stop := 83540 ), rec( content := "\n\n ", count := [ 3, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveAl\ lMultipleEdges" ), content := 0, count := [ 3, 3, 25, 2 ], mathmode := "Text", name := "Oper", next := 83620, root := ~, start := 83562, stop := 83619 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveAl\ lMultipleEdgesAttr" ), content := 0, count := [ 3, 3, 25, 4 ], mathmode := "Text", name := "Attr", next := 83685, root := ~, start := 83623, stop := 83684 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "Returns", next := 83717, root := ~, start := 83688, stop := 83716 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "A", next := 83755, root := ~, start := 83741, stop := 83754 ), rec( content := " is a di\ graph, then this operation returns a\n digraph constructed from ", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "A", next := 83844, root := ~, start := 83830, stop := 83843 ), rec( content := " by remo\ ving all multiple edges.\n The result is the largest subdigraph of ", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "A", next := 83935, root := ~, start := 83921, stop := 83934 ), rec( content := " which d\ oes not\n contain multiple edges.", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "P", next := 83982, root := ~, start := 83978, stop := 83981 ), rec( content := "\n\n \ If ", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "A", next := 84005, root := ~, start := 83991, stop := 84004 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the multiple \ edges of the mutable digraph ", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "A", next := 84133, root := ~, start := 84119, stop := 84132 ), rec( content := " are\n \ removed in-place.\n ", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := Digraph([[1, 2, 3, 2], [1, 1, 3], [2, 2, 2]]);\n\ngap> D2 := DigraphRemoveAllMultipleEdges(D1)\ ;\n\ngap> OutNeighbours(D2);\n[ [ \ 1, 2, 3 ], [ 1, 3 ], [ 2 ] ]\ngap> D := Digraph(IsMutableDigraph, [[2, 2], [1]\ ]);\n\ngap> DigraphRemoveAllMul\ tipleEdges(D);\n\ngap> D;\n\n", count := [ 3, 3, 25, 7 ], name := "PCDAT\ A", root := ~ ) ], count := [ 3, 3, 25, 7 ], mathmode := "Text", name := "Example", next := 84686, root := ~, start := 84164, stop := 84685 ), rec( content := "\n ", count := [ 3, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "Description", next := 84703, root := ~, start := 83720, stop := 84702 ) ], count := [ 3, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 84717, root := ~, start := 83547, stop := 84716 ), rec( content := "\n\n ", count := [ 3, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edges", Label := "for a digraph \ and a list of edges", Name := "DigraphReverseEdges" ), content := 0, count := [ 3, 3, 26, 2 ], mathmode := "Text", name := "Oper", next := 84839, root := ~, start := 84738, stop := 84838 ), rec( attributes := rec( Arg := "digraph, edge", Label := "for a digraph \ and an edge", Name := "DigraphReverseEdge" ), content := 0, count := [ 3, 3, 26, 4 ], mathmode := "Text", name := "Oper", next := 84933, root := ~, start := 84842, stop := 84932 ), rec( attributes := rec( Arg := "digraph, src, ra\ n", Label := "for a digraph, source, and range", Name := "DigraphReverseEdge" ), content := 0, count := [ 3, 3, 26, 6 ], mathmode := "Text", name := "Oper", next := 85038, root := ~, start := 84936, stop := 85037 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "Returns", next := 85070, root := ~, start := 85041, stop := 85069 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85108, root := ~, start := 85094, stop := 85107 ), rec( content := " is a di\ graph without multiple edges, and ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85162, root := ~, start := 85150, stop := 85161 ), rec( content := " is a\n \ list of pairs of vertices of ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85215, root := ~, start := 85201, stop := 85214 ), rec( content := " (the en\ tries of each pair\n corresponding to the source and the range of an edge, \ respectively),\n then ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphReverseEdges", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "C", next := 85350, root := ~, start := 85324, stop := 85349 ), rec( content := " returns\ a digraph constructed from\n ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85404, root := ~, start := 85390, stop := 85403 ), rec( content := " by reve\ rsing the orientation of every edge specified by\n ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85477, root := ~, start := 85465, stop := 85476 ), rec( content := ". If onl\ y one edge is to be reversed, then\n ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphReverseEdge", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "C", next := 85549, root := ~, start := 85524, stop := 85548 ), rec( content := " can be \ used instead. In this case, the second\n argument should just be a single v\ ertex-pair, or the second and third\n arguments should be the source and ra\ nge of an edge respectively. ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "P", next := 85744, root := ~, start := 85740, stop := 85743 ), rec( content := "\n\n \ Note that even though ", count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "A", next := 85786, root := ~, start := 85772, stop := 85785 ), rec( content := " cannot \ have multiple edges, the\n output may have multiple edges. ", count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "P", next := 85859, root := ~, start := 85855, stop := 85858 ), rec( content := "\n\n \ If ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 85882, root := ~, start := 85868, stop := 85881 ), rec( content := " belongs\ to ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "Ref", next := 85924, root := ~, start := 85894, stop := 85923 ), rec( content := ", then t\ he edges\n are reversed in ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 85975, root := ~, start := 85961, stop := 85974 ), rec( content := ". If ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 85994, root := ~, start := 85980, stop := 85993 ), rec( content := " belongs\ to ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "Ref", next := 86044, root := ~, start := 86006, stop := 86043 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 86081, root := ~, start := 86067, stop := 86080 ), rec( content := " with th\ e\n specified edges reversed is returned.", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 26, 10 ], mathmode := "Text", name := "P", next := 86136, root := ~, start := 86132, stop := 86135 ), rec( content := "\n ", count := [ 3, 3, 26, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDiSparse6String(\".Tg?i@s?t_e?_qEsC\");\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 7 ], [\ 1, 8 ], [ 5, 21 ], [ 7, 19 ], [ 9, 1 ], \n [ 11, 2 ], [ 21, 1 ] ]\ngap> new \ := DigraphReverseEdge(D, [7, 19]);\n\ngap> DigraphEdges(new);\n[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 9, \ 1 ], [ 11, 2 ], \n [ 19, 7 ], [ 21, 1 ] ]\ngap> D2 := DigraphMutableCopy(new)\ ;;\ngap> new := DigraphReverseEdges(D2, [[19, 7]]);;\ngap> D2 = new;\ntrue\nga\ p> D = new;\ntrue\n", count := [ 3, 3, 26, 11 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 11 ], mathmode := "Text", name := "Example", next := 86719, root := ~, start := 86141, stop := 86718 ), rec( content := "\n ", count := [ 3, 3, 26, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "Description", next := 86736, root := ~, start := 85073, stop := 86735 ) ], count := [ 3, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 86750, root := ~, start := 84723, stop := 86749 ), rec( content := "\n\n ", count := [ 3, 3, 26, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, ...", Label := "for an\n ar\ bitrary number of digraphs", Name := "DigraphDisjointUnion" ), content := 0, count := [ 3, 3, 27, 2 ], mathmode := "Text", name := "Func", next := 86872, root := ~, start := 86771, stop := 86871 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphDisjointUnion" ), content := 0, count := [ 3, 3, 27, 4 ], mathmode := "Text", name := "Func", next := 86952, root := ~, start := 86875, stop := 86951 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "Returns", next := 86984, root := ~, start := 86955, stop := 86983 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "A", next := 87036, root := ~, start := 87027, stop := 87035 ), rec( content := ", ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "A", next := 87047, root := ~, start := 87038, stop := 87046 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDisjointUnion", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "C", next := 87104, root := ~, start := 87077, stop := 87103 ), rec( content := " returns\ their disjoint union.\n\n In the second form, if ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "A", next := 87174, root := ~, start := 87163, stop := 87173 ), rec( content := " is a no\ n-empty list of digraphs, then\n ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDisjointUnion", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "C", next := 87244, root := ~, start := 87217, stop := 87243 ), rec( content := " returns\ the disjoint union of the digraphs\n contained in the list. ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 6 ], mathmode := "Text", name := "P", next := 87320, root := ~, start := 87316, stop := 87319 ), rec( content := "\n\n \ For a disjoint union of digraphs, the vertex set is the disjoint union of\n \ the vertex sets, and the edge list is the disjoint union of the edge lists.\n\ ", count := [ 3, 3, 27, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "P", next := 87488, root := ~, start := 87484, stop := 87487 ), rec( content := "\n\n \ More specifically, for a collection of digraphs ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87551, root := ~, start := 87542, stop := 87550 ), rec( content := ", ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87562, root := ~, start := 87553, stop := 87561 ), rec( content := ",\n " , count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87578, root := ~, start := 87568, stop := 87577 ), rec( content := ", the di\ sjoint union with have\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87638, root := ~, start := 87613, stop := 87637 ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87647, root := ~, start := 87638, stop := 87646 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87655, root := ~, start := 87647, stop := 87654 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87664, root := ~, start := 87656, stop := 87663 ), rec( content := "\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87694, root := ~, start := 87669, stop := 87693 ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87703, root := ~, start := 87694, stop := 87702 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87711, root := ~, start := 87703, stop := 87710 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87720, root := ~, start := 87712, stop := 87719 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87731, root := ~, start := 87721, stop := 87730 ), rec( content := " vertice\ s.\n The edges of ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87768, root := ~, start := 87759, stop := 87767 ), rec( content := " will re\ main unchanged, whilst the edges of the\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87828, root := ~, start := 87820, stop := 87827 ), rec( content := "th digra\ ph, ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87848, root := ~, start := 87840, stop := 87847 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87858, root := ~, start := 87848, stop := 87857 ), rec( content := ", will b\ e changed so that they\n belong to the vertices of the disjoint union corre\ sponding to\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87967, root := ~, start := 87959, stop := 87966 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87977, root := ~, start := 87967, stop := 87976 ), rec( content := ". In pa\ rticular, the edges of ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 88016, root := ~, start := 88008, stop := 88015 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88026, root := ~, start := 88016, stop := 88025 ), rec( content := " will\n \ have their source and range increased by\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88106, root := ~, start := 88081, stop := 88105 ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 88115, root := ~, start := 88106, stop := 88114 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88123, root := ~, start := 88115, stop := 88122 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88132, root := ~, start := 88124, stop := 88131 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88143, root := ~, start := 88133, stop := 88142 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88152, root := ~, start := 88144, stop := 88151 ), rec( content := "\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88182, root := ~, start := 88157, stop := 88181 ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 88190, root := ~, start := 88182, stop := 88189 ), rec( attributes := rec( ), content := [ rec( content := "[\ i-1])", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88203, root := ~, start := 88190, stop := 88202 ), rec( content := ".\n\n \ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "P", next := 88214, root := ~, start := 88210, stop := 88213 ), rec( content := "\n\n \ Note that previously set ", count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "Ref", next := 88278, root := ~, start := 88245, stop := 88277 ), rec( content := " will be\ lost.", count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "P", next := 88296, root := ~, start := 88292, stop := 88295 ), rec( content := "\n\n \ If the first digraph ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88332, root := ~, start := 88323, stop := 88331 ), rec( content := " [", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88348, root := ~, start := 88334, stop := 88347 ), rec( content := "] belong\ s to ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "Ref", next := 88395, root := ~, start := 88361, stop := 88394 ), rec( content := ", then " , count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88411, root := ~, start := 88402, stop := 88410 ), rec( content := " [", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88427, root := ~, start := 88413, stop := 88426 ), rec( content := "] is mod\ ified in\n place to contain the appropriate vertices and edges. If ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88518, root := ~, start := 88504, stop := 88517 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "Ref", next := 88566, root := ~, start := 88534, stop := 88565 ), rec( content := ", a new \ immutable digraph containing the\n appropriate vertices and edges is return\ ed.\n", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := CycleDigraph(3);\n\ngap> \ OutNeighbours(D1);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> D2 := CompleteDigraph(3);\n<\ immutable complete digraph with 3 vertices>\ngap> OutNeighbours(D2);\n[ [ 2, 3\ ], [ 1, 3 ], [ 1, 2 ] ]\ngap> union := DigraphDisjointUnion(D1, D2);\n\ngap> OutNeighbours(union);\n[ [ 2 ], [ \ 3 ], [ 1 ], [ 5, 6 ], [ 4, 6 ], [ 4, 5 ] ]\n", count := [ 3, 3, 27, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 10 ], mathmode := "Text", name := "Example", next := 89108, root := ~, start := 88655, stop := 89107 ), rec( content := "\n ", count := [ 3, 3, 27, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "Description", next := 89125, root := ~, start := 86987, stop := 89124 ) ], count := [ 3, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 89139, root := ~, start := 86756, stop := 89138 ), rec( content := "\n\n ", count := [ 3, 3, 27, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphEdgeUnion" ), content := 0, count := [ 3, 3, 28, 2 ], mathmode := "Text", name := "Func", next := 89255, root := ~, start := 89160, stop := 89254 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphEdgeUnion" ), content := 0, count := [ 3, 3, 28, 4 ], mathmode := "Text", name := "Func", next := 89331, root := ~, start := 89258, stop := 89330 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "Returns", next := 89363, root := ~, start := 89334, stop := 89362 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "A", next := 89415, root := ~, start := 89406, stop := 89414 ), rec( content := ", ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "A", next := 89426, root := ~, start := 89417, stop := 89425 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeUnion", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "C", next := 89479, root := ~, start := 89456, stop := 89478 ), rec( content := " returns\ their edge union.\n\n In the second form, if ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "A", next := 89545, root := ~, start := 89534, stop := 89544 ), rec( content := " is a no\ n-empty list of digraphs, then\n ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeUnion", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "C", next := 89611, root := ~, start := 89588, stop := 89610 ), rec( content := " returns\ the edge union of the digraphs contained\n in the list. ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "P", next := 89683, root := ~, start := 89679, stop := 89682 ), rec( content := "\n\n \ The vertex set of the edge union of a collection of digraphs is the\n ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ nion", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 89773, root := ~, start := 89761, stop := 89772 ), rec( content := " of the \ vertex sets, whilst the edge list of the edge union is\n the ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oncatenation", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 89864, root := ~, start := 89844, stop := 89863 ), rec( content := " of the \ edge lists. The number of vertices of the\n edge union is equal to the ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximum", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 89960, root := ~, start := 89946, stop := 89959 ), rec( content := " number \ of vertices of one of the\n digraphs, whilst the number of edges of the edg\ e union will equal the\n ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ um", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 90082, root := ~, start := 90072, stop := 90081 ), rec( content := " of the \ number of edges of each digraph. ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "P", next := 90127, root := ~, start := 90123, stop := 90126 ), rec( content := "\n\n \ Note that previously set ", count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "Ref", next := 90191, root := ~, start := 90158, stop := 90190 ), rec( content := " will be\ lost.\n ", count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "P", next := 90214, root := ~, start := 90210, stop := 90213 ), rec( content := "\n\n \ If the first digraph ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90250, root := ~, start := 90241, stop := 90249 ), rec( content := " [", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90266, root := ~, start := 90252, stop := 90265 ), rec( content := "] belong\ s to ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "Ref", next := 90313, root := ~, start := 90279, stop := 90312 ), rec( content := ", then " , count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90329, root := ~, start := 90320, stop := 90328 ), rec( content := " [", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90345, root := ~, start := 90331, stop := 90344 ), rec( content := "] is mod\ ified in\n place to contain the appropriate vertices and edges. If ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90436, root := ~, start := 90422, stop := 90435 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "Ref", next := 90484, root := ~, start := 90452, stop := 90483 ), rec( content := ", a new \ immutable digraph containing the\n appropriate vertices and edges is return\ ed.\n", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(10);\n\ngap>\ DigraphEdgeUnion(D, D);\n\ \ngap> D1 := Digraph([[2], [1]]);\n\ngap> D2 := Digraph([[2, 3], [2], [1]]);\n\ngap> union := DigraphEdgeUnion(D1, D2);\n\ngap> OutNeighbours(union);\n[ [ 2, 2, 3 ], [ 1, 2 ],\ [ 1 ] ]\ngap> union = DigraphByEdges(\n> Concatenation(DigraphEdges(D1), Digr\ aphEdges(D2)));\ntrue\n", count := [ 3, 3, 28, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 9 ], mathmode := "Text", name := "Example", next := 91157, root := ~, start := 90573, stop := 91156 ), rec( content := "\n ", count := [ 3, 3, 28, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "Description", next := 91174, root := ~, start := 89366, stop := 91173 ) ], count := [ 3, 3, 28, 1 ], mathmode := "Text", name := "ManSection", next := 91188, root := ~, start := 89145, stop := 91187 ), rec( content := "\n\n ", count := [ 3, 3, 28, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphJoin" ), content := 0, count := [ 3, 3, 29, 2 ], mathmode := "Text", name := "Func", next := 91299, root := ~, start := 91209, stop := 91298 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphJoin" ), content := 0, count := [ 3, 3, 29, 4 ], mathmode := "Text", name := "Func", next := 91370, root := ~, start := 91302, stop := 91369 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "Returns", next := 91402, root := ~, start := 91373, stop := 91401 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "A", next := 91454, root := ~, start := 91445, stop := 91453 ), rec( content := ", ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "A", next := 91465, root := ~, start := 91456, stop := 91464 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphJoin", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "C", next := 91513, root := ~, start := 91495, stop := 91512 ), rec( content := " returns\ their join.\n\n In the second form, if ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "A", next := 91573, root := ~, start := 91562, stop := 91572 ), rec( content := " is a no\ n-empty list of digraphs,\n then ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphJoin", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "C", next := 91634, root := ~, start := 91616, stop := 91633 ), rec( content := " returns\ the join of the digraphs contained in the\n list. ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "P", next := 91699, root := ~, start := 91695, stop := 91698 ), rec( content := "\n\n \ The join of a collection of digraphs ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 91751, root := ~, start := 91742, stop := 91750 ), rec( content := ", ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 91762, root := ~, start := 91753, stop := 91761 ), rec( content := ", ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "C", next := 91774, root := ~, start := 91764, stop := 91773 ), rec( content := " is\n \ formed by first taking the ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DisjointUnion", Label := "for a list of digraphs" ), content := 0, count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "Ref", next := 91880, root := ~, start := 91809, stop := 91879 ), rec( content := " of the \ collection.\n\n In the disjoint union, if ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ \\neq j", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "M", next := 91946, root := ~, start := 91931, stop := 91945 ), rec( content := " then th\ ere are no edges between\n vertices corresponding to digraphs ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 92026, root := ~, start := 92018, stop := 92025 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "C", next := 92036, root := ~, start := 92026, stop := 92035 ), rec( content := " and\n \ ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 92053, root := ~, start := 92045, stop := 92052 ), rec( attributes := rec( ), content := [ rec( content := "[\ j]", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "C", next := 92063, root := ~, start := 92053, stop := 92062 ), rec( content := " in the \ collection; the join is created by including\n all such edges.", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 7 ], mathmode := "Text", name := "P", next := 92139, root := ~, start := 92135, stop := 92138 ), rec( content := "\n\n \ For example, the join of two empty digraphs is a complete bipartite\n digra\ ph.", count := [ 3, 3, 29, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "P", next := 92229, root := ~, start := 92225, stop := 92228 ), rec( content := "\n\n \ Note that previously set ", count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "Ref", next := 92293, root := ~, start := 92260, stop := 92292 ), rec( content := " will be\ lost.\n ", count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "P", next := 92316, root := ~, start := 92312, stop := 92315 ), rec( content := "\n\n \ If the first digraph ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92352, root := ~, start := 92343, stop := 92351 ), rec( content := " [", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92368, root := ~, start := 92354, stop := 92367 ), rec( content := "] belong\ s to ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "Ref", next := 92415, root := ~, start := 92381, stop := 92414 ), rec( content := ", then " , count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92431, root := ~, start := 92422, stop := 92430 ), rec( content := " [", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92447, root := ~, start := 92433, stop := 92446 ), rec( content := "] is mod\ ified in\n place to contain the appropriate vertices and edges. If ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92538, root := ~, start := 92524, stop := 92537 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "Ref", next := 92586, root := ~, start := 92554, stop := 92585 ), rec( content := ", a new \ immutable digraph containing the\n appropriate vertices and edges is return\ ed.\n", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(3);\n\n\ gap> IsCompleteDigraph(DigraphJoin(D, D));\ntrue\ngap> D2 := CycleDigraph(3);\ \n\ngap> DigraphJoin(D, D2);\n\n", count := [ 3, 3, 29, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 29, 10 ], mathmode := "Text", name := "Example", next := 92971, root := ~, start := 92675, stop := 92970 ), rec( content := "\n ", count := [ 3, 3, 29, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "Description", next := 92988, root := ~, start := 91405, stop := 92987 ) ], count := [ 3, 3, 29, 1 ], mathmode := "Text", name := "ManSection", next := 93002, root := ~, start := 91194, stop := 93001 ), rec( content := "\n\n ", count := [ 3, 3, 29, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "gr1, gr2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphCartesianProduct" ), content := 0, count := [ 3, 3, 30, 2 ], mathmode := "Text", name := "Func", next := 93127, root := ~, start := 93023, stop := 93126 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphCartesianProduct" ), content := 0, count := [ 3, 3, 30, 4 ], mathmode := "Text", name := "Func", next := 93214, root := ~, start := 93130, stop := 93213 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "Returns", next := 93246, root := ~, start := 93217, stop := 93245 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "A", next := 93299, root := ~, start := 93289, stop := 93298 ), rec( content := ", ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "A", next := 93311, root := ~, start := 93301, stop := 93310 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "C", next := 93371, root := ~, start := 93341, stop := 93370 ), rec( content := " returns\ a digraph isomorphic to their\n Cartesian product.\n ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "P", next := 93441, root := ~, start := 93437, stop := 93440 ), rec( content := "\n\n \ In the second form, if ", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "A", next := 93481, root := ~, start := 93470, stop := 93480 ), rec( content := " is a no\ n-empty list of digraphs,\n then ", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "C", next := 93554, root := ~, start := 93524, stop := 93553 ), rec( content := " returns\ a digraph isomorphic to the \n Cartesian product of the digraphs contained\ in the list.\n ", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "P", next := 93661, root := ~, start := 93657, stop := 93660 ), rec( content := "\n \n \ Mathematically, the Cartesian product of two digraphs ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93730, root := ~, start := 93722, stop := 93729 ), rec( content := ", ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93740, root := ~, start := 93732, stop := 93739 ), rec( content := "\n is\ a digraph with vertex set ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ artesian(DigraphVertices(G),\n DigraphVertices(H))", count := [ 3, 3, 30, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93834, root := ~, start := 93774, stop := 93833 ), rec( content := " such th\ at there is an edge from ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, u']", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93881, root := ~, start := 93867, stop := 93880 ), rec( content := " to\n \ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v, v']", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93903, root := ~, start := 93889, stop := 93902 ), rec( content := " iff ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ u = v ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93922, root := ~, start := 93908, stop := 93921 ), rec( content := " and the\ re is an edge from ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93958, root := ~, start := 93949, stop := 93957 ), rec( content := " to\n \ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ '", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93975, root := ~, start := 93966, stop := 93974 ), rec( content := " in ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93987, root := ~, start := 93979, stop := 93986 ), rec( content := " or ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ u' = v'", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94006, root := ~, start := 93991, stop := 94005 ), rec( content := " and the\ re is an edge from\n ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94045, root := ~, start := 94037, stop := 94044 ), rec( content := " to ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94057, root := ~, start := 94049, stop := 94056 ), rec( content := " in ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94069, root := ~, start := 94061, stop := 94068 ), rec( content := ".\n " , count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "P", next := 94079, root := ~, start := 94075, stop := 94078 ), rec( content := "\n\n \ Due to technical reasons, the digraph ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94131, root := ~, start := 94123, stop := 94130 ), rec( content := " returne\ d by\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94178, root := ~, start := 94148, stop := 94177 ), rec( content := " has ver\ tex set\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. DigraphNrVertices(G)*DigraphNrVertices(H)]", count := [ 3, 3, 30, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94254, root := ~, start := 94198, stop := 94253 ), rec( content := " instead\ , and the\n exact\n method of encoding pairs of vertices into integers \ is implementation\n specific. The original vertex pair can be somewhat reg\ ained by using\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ CartesianProductProjections" ), content := 0, count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "Ref", next := 94482, root := ~, start := 94434, stop := 94481 ), rec( content := ". In add\ ition,\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "Ref", next := 94534, root := ~, start := 94501, stop := 94533 ), rec( content := " are pre\ served: if vertex pair\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,u']", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94582, root := ~, start := 94569, stop := 94581 ), rec( content := " was enc\ oded as ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94606, root := ~, start := 94598, stop := 94605 ), rec( content := " then th\ e vertex label of ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94640, root := ~, start := 94632, stop := 94639 ), rec( content := "\n wi\ ll be the pair of vertex labels of ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94690, root := ~, start := 94682, stop := 94689 ), rec( content := " and ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94704, root := ~, start := 94695, stop := 94703 ), rec( content := " i.e.\n \ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabel(D,i) = [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H\ ,u')]", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94804, root := ~, start := 94714, stop := 94803 ), rec( content := ".\n " , count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "P", next := 94814, root := ~, start := 94810, stop := 94813 ), rec( content := "\n\n \ As the Cartesian product is associative, the Cartesian product of a collection\ \n of digraphs ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "A", next := 94926, root := ~, start := 94916, stop := 94925 ), rec( content := ", ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "A", next := 94938, root := ~, start := 94928, stop := 94937 ), rec( content := ", ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "C", next := 94950, root := ~, start := 94940, stop := 94949 ), rec( content := " is comp\ uted in the obvious \n fashion. \n ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 10 ], mathmode := "Text", name := "P", next := 95001, root := ~, start := 94997, stop := 95000 ), rec( content := "\n\n", count := [ 3, 3, 30, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := ChainDigraph(4);\n\ngap> \ gr2 := CycleDigraph(3);\n\ngap> gr3 :\ = DigraphCartesianProduct(gr, gr2);\n\ngap> IsIsomorphicDigraph(gr3, \n> Digraph([[2, 5], [3, 6], [4, 7], [8],\ \n> [6, 9], [7, 10], [8, 11], [12],\n> [10, 1], [11, 2], [1\ 2, 3], [4]]));\ntrue\n", count := [ 3, 3, 30, 11 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 11 ], mathmode := "Text", name := "Example", next := 95435, root := ~, start := 95003, stop := 95434 ), rec( content := "\n ", count := [ 3, 3, 30, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "Description", next := 95452, root := ~, start := 93249, stop := 95451 ) ], count := [ 3, 3, 30, 1 ], mathmode := "Text", name := "ManSection", next := 95466, root := ~, start := 93008, stop := 95465 ), rec( content := "\n\n ", count := [ 3, 3, 30, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "gr1, gr2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphDirectProduct" ), content := 0, count := [ 3, 3, 31, 2 ], mathmode := "Text", name := "Func", next := 95588, root := ~, start := 95487, stop := 95587 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphDirectProduct" ), content := 0, count := [ 3, 3, 31, 4 ], mathmode := "Text", name := "Func", next := 95672, root := ~, start := 95591, stop := 95671 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "Returns", next := 95704, root := ~, start := 95675, stop := 95703 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "A", next := 95757, root := ~, start := 95747, stop := 95756 ), rec( content := ", ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "A", next := 95769, root := ~, start := 95759, stop := 95768 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "C", next := 95826, root := ~, start := 95799, stop := 95825 ), rec( content := " returns\ a digraph isomorphic to their direct\n product.\n ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "P", next := 95893, root := ~, start := 95889, stop := 95892 ), rec( content := "\n\n \ In the second form, if ", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "A", next := 95933, root := ~, start := 95922, stop := 95932 ), rec( content := " is a no\ n-empty list of digraphs, then\n ", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "C", next := 96003, root := ~, start := 95976, stop := 96002 ), rec( content := " returns\ a digraph isomorphic to the direct\n product of the digraphs contained in \ the list.\n ", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "P", next := 96106, root := ~, start := 96102, stop := 96105 ), rec( content := "\n\n \ Mathematically, the direct product of two digraphs ", count := [ 3, 3, 31, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96171, root := ~, start := 96163, stop := 96170 ), rec( content := ", ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96181, root := ~, start := 96173, stop := 96180 ), rec( content := " is a\n \ digraph with vertex set ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ artesian(DigraphVertices(G),\n DigraphVertices(H))", count := [ 3, 3, 31, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96275, root := ~, start := 96215, stop := 96274 ), rec( content := " such th\ at there is an edge from ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, u']", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96322, root := ~, start := 96308, stop := 96321 ), rec( content := "\n to\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v, v']", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96344, root := ~, start := 96330, stop := 96343 ), rec( content := " iff the\ re is an edge from ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96379, root := ~, start := 96371, stop := 96378 ), rec( content := " to ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96391, root := ~, start := 96383, stop := 96390 ), rec( content := " in\n \ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96407, root := ~, start := 96399, stop := 96406 ), rec( content := " and an \ edge from ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96434, root := ~, start := 96425, stop := 96433 ), rec( content := " to ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ '", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96447, root := ~, start := 96438, stop := 96446 ), rec( content := " in ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96459, root := ~, start := 96451, stop := 96458 ), rec( content := ". \n \ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "P", next := 96470, root := ~, start := 96466, stop := 96469 ), rec( content := " \n\n \ Due to technical reasons, the digraph ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 96523, root := ~, start := 96515, stop := 96522 ), rec( content := " returne\ d by\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 96567, root := ~, start := 96540, stop := 96566 ), rec( content := " has ver\ tex set\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. DigraphNrVertices(G)*DigraphNrVertices(H)]", count := [ 3, 3, 31, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 96643, root := ~, start := 96587, stop := 96642 ), rec( content := " instead\ , and the\n exact method of encoding pairs of vertices into integers is imp\ lementation\n specific. The original vertex pair can be somewhat regained \ by using\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ DirectProductProjections" ), content := 0, count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "Ref", next := 96863, root := ~, start := 96818, stop := 96862 ), rec( content := ". In add\ ition\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "Ref", next := 96914, root := ~, start := 96881, stop := 96913 ), rec( content := " are pre\ served: if vertex pair\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,u']", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 96962, root := ~, start := 96949, stop := 96961 ), rec( content := " was enc\ oded as ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 96986, root := ~, start := 96978, stop := 96985 ), rec( content := " then th\ e vertex label of ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97020, root := ~, start := 97012, stop := 97019 ), rec( content := "\n wi\ ll be the pair of vertex labels of ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97070, root := ~, start := 97062, stop := 97069 ), rec( content := " and ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97084, root := ~, start := 97075, stop := 97083 ), rec( content := " i.e.\n \ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabel(D,i) = [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H,u\ ')]", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97182, root := ~, start := 97094, stop := 97181 ), rec( content := ".\n " , count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "P", next := 97192, root := ~, start := 97188, stop := 97191 ), rec( content := "\n\n \ As the direct product is associative, the direct product of a collection \n \ of digraphs ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "A", next := 97298, root := ~, start := 97288, stop := 97297 ), rec( content := ", ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "A", next := 97310, root := ~, start := 97300, stop := 97309 ), rec( content := ", ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "C", next := 97322, root := ~, start := 97312, stop := 97321 ), rec( content := " is comp\ uted in the obvious \n fashion. \n ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 10 ], mathmode := "Text", name := "P", next := 97373, root := ~, start := 97369, stop := 97372 ), rec( content := "\n\n", count := [ 3, 3, 31, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := ChainDigraph(4);\n\ngap> \ gr2 := CycleDigraph(3);\n\ngap> gr3 :\ = DigraphDirectProduct(gr, gr2);\n\ngap> IsIsomorphicDigraph(gr3, \n> Digraph([[6], [7], [8], [], \n> \ [10], [11], [12], [],\n> [2], [3], [4], []]));\ntrue\n", count := [ 3, 3, 31, 11 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 11 ], mathmode := "Text", name := "Example", next := 97770, root := ~, start := 97375, stop := 97769 ), rec( content := "\n ", count := [ 3, 3, 31, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "Description", next := 97787, root := ~, start := 95707, stop := 97786 ) ], count := [ 3, 3, 31, 1 ], mathmode := "Text", name := "ManSection", next := 97801, root := ~, start := 95472, stop := 97800 ), rec( content := "\n\n ", count := [ 3, 3, 31, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphCartesia\ nProductProjections" ), content := 0, count := [ 3, 3, 32, 2 ], mathmode := "Text", name := "Attr", next := 97885, root := ~, start := 97822, stop := 97884 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "Returns", next := 97933, root := ~, start := 97888, stop := 97932 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "A", next := 97971, root := ~, start := 97957, stop := 97970 ), rec( content := " is a Ca\ rtesian product digraph, \n ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph = DigraphCartesianProduct(gr_1, gr_2, ... )", count := [ 3, 3, 32, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98067, root := ~, start := 98009, stop := 98066 ), rec( content := ", then\n\ ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProductProjections", count := [ 3, 3, 32, 3 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98119, root := ~, start := 98078, stop := 98118 ), rec( content := " returns\ a list ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98146, root := ~, start := 98135, stop := 98145 ), rec( content := "\n su\ ch that ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98175, root := ~, start := 98161, stop := 98174 ), rec( content := " is the \ projection onto the ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98211, root := ~, start := 98203, stop := 98210 ), rec( content := "-th\n \ coordinate of the product.\n ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "P", next := 98254, root := ~, start := 98250, stop := 98253 ), rec( content := "\n\n \ A projection is an idempotent endomorphism of ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "A", next := 98320, root := ~, start := 98306, stop := 98319 ), rec( content := ". If\n \ ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1, gr2, ...", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "C", next := 98349, root := ~, start := 98329, stop := 98348 ), rec( content := " are all\ loopless digraphs, then the induced \n subdigraph of ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "A", next := 98427, root := ~, start := 98413, stop := 98426 ), rec( content := " on the \ image of ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "C", next := 98458, root := ~, start := 98444, stop := 98457 ), rec( content := " is \n \ isomorphic to ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r_i", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "C", next := 98492, root := ~, start := 98481, stop := 98491 ), rec( content := ".\n " , count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "P", next := 98502, root := ~, start := 98498, stop := 98501 ), rec( content := "\n\n \ Currently this attribute is only set upon creating an immutable \n digraph \ via ", count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "C", next := 98619, root := ~, start := 98589, stop := 98618 ), rec( content := " and the\ re is no way \n of calculating it for other digraphs.\n ", count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "P", next := 98691, root := ~, start := 98687, stop := 98690 ), rec( content := "\n\n \ For more information see\n ", count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ CartesianProduct", Label := "for a list of digraphs" ), content := 0, count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "Ref", next := 98794, root := ~, start := 98726, stop := 98793 ), rec( content := " \n\n \ ", count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphCartesianProduct(ChainDigraph(3), CycleDigraph(4),\n> Digrap\ h([[2], [2]]));;\ngap> HasDigraphCartesianProductProjections(D);\ntrue\ngap> p\ roj := DigraphCartesianProductProjections(D);; Length(proj);\n3\ngap> IsIdempo\ tent(proj[2]);\ntrue\ngap> RankOfTransformation(proj[3]);\n2\ngap> S := ImageS\ etOfTransformation(proj[2]);;\ngap> IsIsomorphicDigraph(CycleDigraph(4), Induc\ edSubdigraph(D, S));\ntrue\n", count := [ 3, 3, 32, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 7 ], mathmode := "Text", name := "Example", next := 99237, root := ~, start := 98801, stop := 99236 ), rec( content := "\n ", count := [ 3, 3, 32, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "Description", next := 99254, root := ~, start := 97936, stop := 99253 ) ], count := [ 3, 3, 32, 1 ], mathmode := "Text", name := "ManSection", next := 99268, root := ~, start := 97807, stop := 99267 ), rec( content := "\n\n ", count := [ 3, 3, 32, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDirectPr\ oductProjections" ), content := 0, count := [ 3, 3, 33, 2 ], mathmode := "Text", name := "Attr", next := 99349, root := ~, start := 99289, stop := 99348 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "Returns", next := 99397, root := ~, start := 99352, stop := 99396 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "A", next := 99435, root := ~, start := 99421, stop := 99434 ), rec( content := " is a di\ rect product digraph, \n ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph = DigraphDirectProduct(gr_1, gr_2, ... )", count := [ 3, 3, 33, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99525, root := ~, start := 99470, stop := 99524 ), rec( content := ", then\n\ ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProductProjections", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99574, root := ~, start := 99536, stop := 99573 ), rec( content := " returns\ a list ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99601, root := ~, start := 99590, stop := 99600 ), rec( content := "\n su\ ch that ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99630, root := ~, start := 99616, stop := 99629 ), rec( content := " is the \ projection onto the ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99666, root := ~, start := 99658, stop := 99665 ), rec( content := "-th\n \ coordinate of the product.\n ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "P", next := 99709, root := ~, start := 99705, stop := 99708 ), rec( content := "\n\n \ A projection is an idempotent endomorphism of ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "A", next := 99775, root := ~, start := 99761, stop := 99774 ), rec( content := ". If\n \ ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1, gr2, ...", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "C", next := 99804, root := ~, start := 99784, stop := 99803 ), rec( content := " are all\ loopless digraphs, then the image \n of ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "A", next := 99869, root := ~, start := 99855, stop := 99868 ), rec( content := " under " , count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "C", next := 99890, root := ~, start := 99876, stop := 99889 ), rec( content := " is isom\ orphic to ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r_i", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "C", next := 99919, root := ~, start := 99908, stop := 99918 ), rec( content := ".\n " , count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "P", next := 99929, root := ~, start := 99925, stop := 99928 ), rec( content := "\n\n \ Currently this attribute is only set upon creating an immutable\n digraph v\ ia ", count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "C", next := 100042, root := ~, start := 100015, stop := 100041 ), rec( content := " and the\ re is no way of \n calculating it for other digraphs.\n ", count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "P", next := 100114, root := ~, start := 100110, stop := 100113 ), rec( content := "\n\n \ For more information, see\n ", count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DirectProduct", Label := "for a list of digraphs" ), content := 0, count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "Ref", next := 100215, root := ~, start := 100150, stop := 100214 ), rec( content := " \n\n \ ", count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphDirectProduct(ChainDigraph(3), CycleDigraph(4),\n> Digraph([\ [2], [2]]));;\ngap> HasDigraphDirectProductProjections(D);\ntrue\ngap> proj :=\ DigraphDirectProductProjections(D);; Length(proj);\n3\ngap> IsIdempotent(proj\ [2]);\ntrue\ngap> RankOfTransformation(proj[3]);\n2\ngap> P := DigraphRemoveAl\ lMultipleEdges(\n> ReducedDigraph(OnDigraphs(D, proj[2])));; \ngap> IsIsomorph\ icDigraph(CycleDigraph(4), P);\ntrue\n", count := [ 3, 3, 33, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 33, 7 ], mathmode := "Text", name := "Example", next := 100667, root := ~, start := 100222, stop := 100666 ), rec( content := "\n ", count := [ 3, 3, 33, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "Description", next := 100684, root := ~, start := 99400, stop := 100683 ) ], count := [ 3, 3, 33, 1 ], mathmode := "Text", name := "ManSection", next := 100698, root := ~, start := 99274, stop := 100697 ), rec( content := "\n\n ", count := [ 3, 3, 33, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "LineDigraph" ), content := 0, count := [ 3, 3, 34, 2 ], mathmode := "Text", name := "Oper", next := 100759, root := ~, start := 100719, stop := 100758 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeDigraph" ), content := 0, count := [ 3, 3, 34, 4 ], mathmode := "Text", name := "Oper", next := 100802, root := ~, start := 100762, stop := 100801 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "Returns", next := 100834, root := ~, start := 100805, stop := 100833 ), rec( attributes := rec( ), content := [ rec( content := "\n Giv\ en a digraph ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "A", next := 100885, root := ~, start := 100871, stop := 100884 ), rec( content := ", the op\ eration returns the digraph\n obtained by associating a vertex with each ed\ ge of ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "A", next := 100990, root := ~, start := 100976, stop := 100989 ), rec( content := ", and\n \ creating an edge from a vertex ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 101039, root := ~, start := 101031, stop := 101038 ), rec( content := " to a ve\ rtex ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 101060, root := ~, start := 101052, stop := 101059 ), rec( content := " if and \ only\n if the terminal vertex of the edge associated with ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 101136, root := ~, start := 101128, stop := 101135 ), rec( content := " is the \ start\n vertex of the edge associated with ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 101197, root := ~, start := 101189, stop := 101196 ), rec( content := ".", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "P", next := 101202, root := ~, start := 101198, stop := 101201 ), rec( content := "\n\n \ Note that the returned digraph is always a new immutable digraph, and the\n \ argument ", count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "A", next := 101309, root := ~, start := 101295, stop := 101308 ), rec( content := " is neve\ r modified.\n ", count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> LineDigraph(CompleteDigraph(3));\n\ngap> LineDigraph(ChainDigraph(3));\n\n", count := [ 3, 3, 34, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 7 ], mathmode := "Text", name := "Example", next := 101528, root := ~, start := 101333, stop := 101527 ), rec( content := "\n ", count := [ 3, 3, 34, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "Description", next := 101545, root := ~, start := 100837, stop := 101544 ) ], count := [ 3, 3, 34, 1 ], mathmode := "Text", name := "ManSection", next := 101559, root := ~, start := 100704, stop := 101558 ), rec( content := "\n\n ", count := [ 3, 3, 34, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "LineUndirectedD\ igraph" ), content := 0, count := [ 3, 3, 35, 2 ], mathmode := "Text", name := "Oper", next := 101630, root := ~, start := 101580, stop := 101629 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeUndirectedD\ igraph" ), content := 0, count := [ 3, 3, 35, 4 ], mathmode := "Text", name := "Oper", next := 101683, root := ~, start := 101633, stop := 101682 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "Returns", next := 101715, root := ~, start := 101686, stop := 101714 ), rec( attributes := rec( ), content := [ rec( content := "\n Giv\ en a symmetric digraph ", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "A", next := 101776, root := ~, start := 101762, stop := 101775 ), rec( content := ", the op\ eration returns the\n symmetric digraph obtained by associating a vertex wi\ th each edge of\n ", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "A", next := 101895, root := ~, start := 101881, stop := 101894 ), rec( content := ", ignori\ ng directions and multiplicites, and adding an edge\n between two vertices \ if and only if the corresponding edges have a vertex in\n common.", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "P", next := 102051, root := ~, start := 102047, stop := 102050 ), rec( content := "\n\n \ Note that the returned digraph is always a new immutable digraph, and the\n \ argument ", count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "A", next := 102158, root := ~, start := 102144, stop := 102157 ), rec( content := " is neve\ r modified.\n ", count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> LineUndirectedDigraph(CompleteDigraph(3));\n\ngap> LineUndirectedDigraph(DigraphSymmetricClosure(ChainDigr\ aph(3)));\n\n", count := [ 3, 3, 35, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 35, 7 ], mathmode := "Text", name := "Example", next := 102422, root := ~, start := 102182, stop := 102421 ), rec( content := "\n ", count := [ 3, 3, 35, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "Description", next := 102439, root := ~, start := 101718, stop := 102438 ) ], count := [ 3, 3, 35, 1 ], mathmode := "Text", name := "ManSection", next := 102453, root := ~, start := 101565, stop := 102452 ), rec( content := "\n\n ", count := [ 3, 3, 35, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DoubleDigraph" ), content := 0, count := [ 3, 3, 36, 2 ], mathmode := "Text", name := "Oper", next := 102516, root := ~, start := 102474, stop := 102515 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "Returns", next := 102548, root := ~, start := 102519, stop := 102547 ), rec( attributes := rec( ), content := [ rec( content := "\n Let\ ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "A", next := 102587, root := ~, start := 102573, stop := 102586 ), rec( content := " be a di\ graph with vertex set ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 102625, root := ~, start := 102617, stop := 102624 ), rec( content := ". This f\ unction\n returns the double digraph of ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "A", next := 102689, root := ~, start := 102675, stop := 102688 ), rec( content := ". The ve\ rtex set of the double\n digraph is the orginal vertex set together with a \ duplicate. The edges are\n ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_1, v_2]", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 102820, root := ~, start := 102803, stop := 102819 ), rec( content := " and ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_2, v_1]", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 102842, root := ~, start := 102825, stop := 102841 ), rec( content := " if and \ only if ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 102871, root := ~, start := 102858, stop := 102870 ), rec( content := " is an\n\ edge in ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "A", next := 102904, root := ~, start := 102890, stop := 102903 ), rec( content := ", togeth\ er with the original edges and their\n duplicates.", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "P", next := 102968, root := ~, start := 102964, stop := 102967 ), rec( content := "\n\n \ If ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "A", next := 102991, root := ~, start := 102977, stop := 102990 ), rec( content := " is muta\ ble, then ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "A", next := 103023, root := ~, start := 103009, stop := 103022 ), rec( content := " is modi\ fied in-place. If\n ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "A", next := 103067, root := ~, start := 103053, stop := 103066 ), rec( content := " is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gamma := Digraph([[2], [3], [1]]);\n\ngap> DoubleDigraph(gamma);\n\n", count := [ 3, 3, 36, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 5 ], mathmode := "Text", name := "Example", next := 103353, root := ~, start := 103163, stop := 103352 ), rec( content := "\n ", count := [ 3, 3, 36, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "Description", next := 103370, root := ~, start := 102551, stop := 103369 ) ], count := [ 3, 3, 36, 1 ], mathmode := "Text", name := "ManSection", next := 103384, root := ~, start := 102459, stop := 103383 ), rec( content := "\n\n ", count := [ 3, 3, 36, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "BipartiteDouble\ Digraph" ), content := 0, count := [ 3, 3, 37, 2 ], mathmode := "Text", name := "Oper", next := 103456, root := ~, start := 103405, stop := 103455 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "Returns", next := 103488, root := ~, start := 103459, stop := 103487 ), rec( attributes := rec( ), content := [ rec( content := "\n Let\ ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "A", next := 103527, root := ~, start := 103513, stop := 103526 ), rec( content := " be a di\ graph with vertex set ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 103565, root := ~, start := 103557, stop := 103564 ), rec( content := ". This f\ unction\n returns the bipartite double digraph of ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "A", next := 103639, root := ~, start := 103625, stop := 103638 ), rec( content := ". The ve\ rtex set of\n the double digraph is the original vertex set together with a\ duplicate. The\n edges are ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_1, v_2]", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 103771, root := ~, start := 103754, stop := 103770 ), rec( content := " and ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_2, v_1]", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 103793, root := ~, start := 103776, stop := 103792 ), rec( content := " if and \ only if ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,\n v]", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 103828, root := ~, start := 103809, stop := 103827 ), rec( content := " is an e\ dge in ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "A", next := 103857, root := ~, start := 103843, stop := 103856 ), rec( content := ". The re\ sulting graph is bipartite,\n since the orignal edges are not included in t\ he resulting digraph.", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "P", next := 103967, root := ~, start := 103963, stop := 103966 ), rec( content := "\n\n \ If ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "A", next := 103990, root := ~, start := 103976, stop := 103989 ), rec( content := " is muta\ ble, then ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "A", next := 104022, root := ~, start := 104008, stop := 104021 ), rec( content := " is modi\ fied in-place. If\n ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "A", next := 104066, root := ~, start := 104052, stop := 104065 ), rec( content := " is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gamma := Digraph([[2], [3], [1]]);\n\ngap> BipartiteDoubleDigraph(gamma);\n\n", count := [ 3, 3, 37, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 5 ], mathmode := "Text", name := "Example", next := 104360, root := ~, start := 104162, stop := 104359 ), rec( content := "\n ", count := [ 3, 3, 37, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "Description", next := 104377, root := ~, start := 103491, stop := 104376 ) ], count := [ 3, 3, 37, 1 ], mathmode := "Text", name := "ManSection", next := 104391, root := ~, start := 103390, stop := 104390 ), rec( content := "\n\n ", count := [ 3, 3, 37, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAddAllLo\ ops" ), content := 0, count := [ 3, 3, 38, 2 ], mathmode := "Text", name := "Oper", next := 104459, root := ~, start := 104412, stop := 104458 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphAddAllLo\ opsAttr" ), content := 0, count := [ 3, 3, 38, 4 ], mathmode := "Text", name := "Attr", next := 104513, root := ~, start := 104462, stop := 104512 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "Returns", next := 104545, root := ~, start := 104516, stop := 104544 ), rec( attributes := rec( ), content := [ rec( content := "\n For\ a digraph ", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "A", next := 104594, root := ~, start := 104580, stop := 104593 ), rec( content := " this op\ eration returns a new digraph constructed\n from ", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "A", next := 104667, root := ~, start := 104653, stop := 104666 ), rec( content := ", such t\ hat a loop is added for every vertex which did not\n have a loop in ", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "A", next := 104759, root := ~, start := 104745, stop := 104758 ), rec( content := ".", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "P", next := 104764, root := ~, start := 104760, stop := 104763 ), rec( content := "\n\n \ If ", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "A", next := 104787, root := ~, start := 104773, stop := 104786 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the loops are\ added to the loopless vertices of the mutable digraph\n ", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "A", next := 104945, root := ~, start := 104931, stop := 104944 ), rec( content := " in-plac\ e.\n ", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := EmptyDigraph(13);\n\ngap>\ D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], \ [ 7 ], [ 8 ], [ 9 ], \n [ 10 ], [ 11 ], [ 12 ], [ 13 ] ]\ngap> D := Digraph([\ [1, 2, 3], [1, 3], [1]]);\n\ngap> \ D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n[ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 1, 3 ] ]\ngap> D \ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphAddAllLoops(D);\n\n\ gap> D;\n\n", count := [ 3, 3, 38, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 38, 7 ], mathmode := "Text", name := "Example", next := 105727, root := ~, start := 104960, stop := 105726 ), rec( content := "\n ", count := [ 3, 3, 38, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "Description", next := 105744, root := ~, start := 104548, stop := 105743 ) ], count := [ 3, 3, 38, 1 ], mathmode := "Text", name := "ManSection", next := 105758, root := ~, start := 104397, stop := 105757 ), rec( content := "\n\n ", count := [ 3, 3, 38, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i", Label := "for digraph an\ d int", Name := "DistanceDigraph" ), content := 0, count := [ 3, 3, 39, 2 ], mathmode := "Text", name := "Oper", next := 105862, root := ~, start := 105779, stop := 105861 ), rec( attributes := rec( Arg := "digraph, list", Label := "for digraph an\ d list", Name := "DistanceDigraph" ), content := 0, count := [ 3, 3, 39, 4 ], mathmode := "Text", name := "Oper", next := 105952, root := ~, start := 105865, stop := 105951 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Returns", next := 105984, root := ~, start := 105955, stop := 105983 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ first argument is a digraph, the second argument is a non-negative\n integ\ er or a list of positive integers. This operation returns a digraph on\n th\ e same set of vertices as ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 106202, root := ~, start := 106188, stop := 106201 ), rec( content := ", with t\ wo vertices being adjacent\n if and only if the distance between them in ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 106299, root := ~, start := 106285, stop := 106298 ), rec( content := " equals \ ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 106315, root := ~, start := 106307, stop := 106314 ), rec( content := "\n or\ is a number in ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 106349, root := ~, start := 106338, stop := 106348 ), rec( content := ". See " , count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and two vertices", Oper := "DigraphShortestDistance" ), content := 0, count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Ref", next := 106436, root := ~, start := 106356, stop := 106435 ), rec( content := ". ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "P", next := 106442, root := ~, start := 106438, stop := 106441 ), rec( content := "\n\n \ If ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "A", next := 106465, root := ~, start := 106451, stop := 106464 ), rec( content := " is muta\ ble, then ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "A", next := 106497, root := ~, start := 106483, stop := 106496 ), rec( content := " is modi\ fied in-place. If\n ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "A", next := 106541, root := ~, start := 106527, stop := 106540 ), rec( content := " is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := DigraphFromSparse6String(\n> \":]n?AL`BC_DEbEF`GIaGHdIJeGKcKL\ _@McDHfILaBJfHMjKM\");\n\ngap> D\ istanceDigraph(digraph, 1);\n\ng\ ap> DistanceDigraph(digraph, [1, 2]);\n\n", count := [ 3, 3, 39, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 7 ], mathmode := "Text", name := "Example", next := 106981, root := ~, start := 106637, stop := 106980 ), rec( content := "\n ", count := [ 3, 3, 39, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Description", next := 106998, root := ~, start := 105987, stop := 106997 ) ], count := [ 3, 3, 39, 1 ], mathmode := "Text", name := "ManSection", next := 107012, root := ~, start := 105764, stop := 107011 ), rec( content := "\n\n ", count := [ 3, 3, 39, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, k", Name := "DigraphClosure"\ ), content := 0, count := [ 3, 3, 40, 2 ], mathmode := "Text", name := "Oper", next := 107079, root := ~, start := 107033, stop := 107078 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Returns", next := 107111, root := ~, start := 107082, stop := 107110 ), rec( attributes := rec( ), content := [ rec( content := "\n Giv\ en a symmetric loopless digraph with no multiple edges ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 107204, root := ~, start := 107190, stop := 107203 ), rec( content := ",\n t\ he ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "k", count := [ 3, 3, 40, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := \ "Text", name := "A", next := 107225, root := ~, start := 107217, stop := 10722\ 4 ), rec( content := "-closure of ", count := [ 3, 3, 40, 3 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "digraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 1072\ 51, root := ~, start := 107237, stop := 107250 ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "E", next := 107255, root := ~, start := 107214, stop := 107254 ), rec( content := " is defi\ ned to be the unique\n smallest symmetric loopless digraph ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "C", next := 107332, root := ~, start := 107324, stop := 107331 ), rec( content := " with no\ multiple edges on the\n vertices of ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 107393, root := ~, start := 107379, stop := 107392 ), rec( content := " that co\ ntains all the edges of ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 107439, root := ~, start := 107425, stop := 107438 ), rec( content := "\n an\ d satsifies the property that the sum of the degrees of every two\n non-adj\ acenct vertices in ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "C", next := 107550, root := ~, start := 107542, stop := 107549 ), rec( content := " is less\ than ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 107572, root := ~, start := 107564, stop := 107571 ), rec( content := ". See ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 107614, root := ~, start := 107578, stop := 107613 ), rec( content := ", ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 107645, root := ~, start := 107616, stop := 107644 ), rec( content := ", ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 107679, root := ~, start := 107647, stop := 107678 ), rec( content := ", and ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 107716, root := ~, start := 107685, stop := 107715 ), rec( content := ". ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "P", next := 107722, root := ~, start := 107718, stop := 107721 ), rec( content := "\n\n \ The operation ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphClosure", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "C", next := 107763, root := ~, start := 107742, stop := 107762 ), rec( content := " returns\ the ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "A", next := 107784, root := ~, start := 107776, stop := 107783 ), rec( content := "-closure\ of\n ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "A", next := 107814, root := ~, start := 107800, stop := 107813 ), rec( content := ".\n " , count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(6);\n\n\ gap> D := DigraphRemoveEdges(D, [[1, 2], [2, 1]]);\n\ngap> closure := DigraphClosure(D, 6);\n\ngap> IsCompleteDigraph(closure);\ntrue\n", count := [ 3, 3, 40, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 40, 5 ], mathmode := "Text", name := "Example", next := 108146, root := ~, start := 107820, stop := 108145 ), rec( content := "\n ", count := [ 3, 3, 40, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Description", next := 108163, root := ~, start := 107114, stop := 108162 ) ], count := [ 3, 3, 40, 1 ], mathmode := "Text", name := "ManSection", next := 108177, root := ~, start := 107018, stop := 108176 ), rec( content := "\n\n ", count := [ 3, 3, 40, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphMycielsk\ ian" ), content := 0, count := [ 3, 3, 41, 2 ], mathmode := "Text", name := "Oper", next := 108245, root := ~, start := 108198, stop := 108244 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMycielsk\ ianAttr" ), content := 0, count := [ 3, 3, 41, 4 ], mathmode := "Text", name := "Attr", next := 108299, root := ~, start := 108248, stop := 108298 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "Returns", next := 108331, root := ~, start := 108302, stop := 108330 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "A", next := 108369, root := ~, start := 108355, stop := 108368 ), rec( content := " is a sy\ mmetric digraph, then\n ", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphMycielskian", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "C", next := 108428, root := ~, start := 108403, stop := 108427 ), rec( content := " returns\ the Mycielskian of ", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "A", next := 108470, root := ~, start := 108456, stop := 108469 ), rec( content := ".", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "P", next := 108475, root := ~, start := 108471, stop := 108474 ), rec( content := "\n \n\ The Mycielskian of a symmetric digraph is a larger symmetric digraph\n \ constructed from it, which has a larger chromatic number. For further\n inf\ ormation, see ", count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://en.wikipedia.org/wiki/Mycielskian", count := [ 3, 3, 41, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "URL", next := 108701, root := ~, start := 108649, stop := 108700 ), rec( content := ".", count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "P", next := 108706, root := ~, start := 108702, stop := 108705 ), rec( content := "\n\n \ If ", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "A", next := 108729, root := ~, start := 108715, stop := 108728 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "A", next := 108835, root := ~, start := 108821, stop := 108834 ), rec( content := " is chan\ ged in-place into\n its Mycielskian.", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 41, 8 ], mathmode := "Text", name := "P", next := 108885, root := ~, start := 108881, stop := 108884 ), rec( content := "\n\n \ ", count := [ 3, 3, 41, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(2);\n\ngap> C\ hromaticNumber(D);\n2\ngap> D := DigraphMycielskian(D);\n\ngap> ChromaticNumber(D);\n3\ngap> D := DigraphMyciel\ skian(D);\n\ngap> ChromaticNumbe\ r(D);\n4\ngap> D := CompleteBipartiteDigraph(IsMutable, 2, 3);\n\ngap> DigraphMycielskian(D);\n\ngap> D;\n\n", count := [ 3, 3, 41, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 9 ], mathmode := "Text", name := "Example", next := 109455, root := ~, start := 108891, stop := 109454 ), rec( content := "\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "Description", next := 109472, root := ~, start := 108334, stop := 109471 ) ], count := [ 3, 3, 41, 1 ], mathmode := "Text", name := "ManSection", next := 109486, root := ~, start := 108183, stop := 109485 ), rec( content := "\n\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 109500, root := ~, start := 43580, stop := 109499 ), rec( content := "\n\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Random digraphs", count := [ 3, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 109547, root := ~, start := 109513, stop := 109546 ), rec( content := "\n ", count := [ 3, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n[, p]", Name := "RandomDigraph" ), content := 0, count := [ 3, 4, 1, 2 ], mathmode := "Text", name := "Oper", next := 109616, root := ~, start := 109567, stop := 109615 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 109648, root := ~, start := 109619, stop := 109647 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 109711, root := ~, start := 109700, stop := 109710 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 109857, root := ~, start := 109846, stop := 109856 ), rec( content := " is ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Ref", next := 109891, root := ~, start := 109861, stop := 109890 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 109959, root := ~, start := 109948, stop := 109958 ), rec( content := " is ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Ref", next := 109999, root := ~, start := 109963, stop := 109998 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 110084, root := ~, start := 110073, stop := 110083 ), rec( content := " is not \ present, then ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Ref", next := 110144, root := ~, start := 110106, stop := 110143 ), rec( content := " is used\ by default.", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "P", next := 110168, root := ~, start := 110164, stop := 110167 ), rec( content := "\n\n \ If ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "A", next := 110185, root := ~, start := 110177, stop := 110184 ), rec( content := " is a po\ sitive integer, then this function returns a random\n digraph with ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "A", next := 110270, root := ~, start := 110262, stop := 110269 ), rec( content := " vertice\ s and without multiple edges. The result\n may or may not have loops.", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "P", next := 110353, root := ~, start := 110349, stop := 110352 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 110399, root := ~, start := 110391, stop := 110398 ), rec( content := " is a fl\ oat with value\n ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ \\leq ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "M", next := 110440, root := ~, start := 110426, stop := 110439 ), rec( content := " ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ p ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 110451, root := ~, start := 110441, stop := 110450 ), rec( content := " ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ \\leq 1", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "M", next := 110466, root := ~, start := 110452, stop := 110465 ), rec( content := ", then a\ n edge will exist between each\n pair of vertices with probability approxim\ ately ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 110565, root := ~, start := 110557, stop := 110564 ), rec( content := ".\n I\ f ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 110582, root := ~, start := 110574, stop := 110581 ), rec( content := " is not \ specified, then a random probability will be assumed\n (chosen with uniform\ probability).\n ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomDigraph(1000);\n\ngap> RandomDigraph(10000, 0.023);\n\ngap> RandomDigraph(IsMutableDigraph, 1000, 1 / 2);\n\n", count := [ 3, 4, 1, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 4, 1, 6 ], mathmode := "Text", name := "Log", next := 110980, root := ~, start := 110686, stop := 110979 ), rec( content := "\n ", count := [ 3, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 110997, root := ~, start := 109651, stop := 110996 ) ], count := [ 3, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 111011, root := ~, start := 109552, stop := 111010 ), rec( content := "\n\n ", count := [ 3, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "n[, m]", Name := "RandomMultiDigr\ aph" ), content := 0, count := [ 3, 4, 2, 2 ], mathmode := "Text", name := "Oper", next := 111078, root := ~, start := 111032, stop := 111077 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 111110, root := ~, start := 111081, stop := 111109 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111142, root := ~, start := 111134, stop := 111141 ), rec( content := " is a po\ sitive integer, then this function returns a random\n digraph with ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111227, root := ~, start := 111219, stop := 111226 ), rec( content := " vertice\ s. If the optional second argument\n ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111282, root := ~, start := 111274, stop := 111281 ), rec( content := " is a po\ sitive integer, then the digraph will have ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111341, root := ~, start := 111333, stop := 111340 ), rec( content := "\n ed\ ges. If ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111364, root := ~, start := 111356, stop := 111363 ), rec( content := " is not \ specified, then the number of edges will be\n chosen randomly (with uniform\ probability) from the range\n ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "C", next := 111495, root := ~, start := 111482, stop := 111494 ), rec( attributes := rec( ), content := [ rec( content := "{\ n \\choose 2}", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "M", next := 111515, root := ~, start := 111495, stop := 111514 ), rec( attributes := rec( ), content := [ rec( content := "]\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "C", next := 111523, root := ~, start := 111515, stop := 111522 ), rec( content := ".\n " , count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 2, 4 ], mathmode := "Text", name := "P", next := 111533, root := ~, start := 111529, stop := 111532 ), rec( content := "\n\n \ The method used by this function chooses each edge from the set of\n all po\ ssible edges with uniform probability. No effort is made to avoid\n creatin\ g multiple edges, so it is possible (but not guaranteed) that the\n result \ will have multiple edges. The result may or may not have loops.", count := [ 3, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 2, 5 ], mathmode := "Text", name := "P", next := 111837, root := ~, start := 111833, stop := 111836 ), rec( content := "\n ", count := [ 3, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomMultiDigraph(1000);\n\ngap> RandomMultiDigraph(1000, 950);\n\n", count := [ 3, 4, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 6 ], mathmode := "Text", name := "Log", next := 112046, root := ~, start := 111842, stop := 112045 ), rec( content := "\n ", count := [ 3, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 112063, root := ~, start := 111113, stop := 112062 ) ], count := [ 3, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 112077, root := ~, start := 111017, stop := 112076 ), rec( content := "\n\n ", count := [ 3, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "RandomTournamen\ t" ), content := 0, count := [ 3, 4, 3, 2 ], mathmode := "Text", name := "Oper", next := 112145, root := ~, start := 112098, stop := 112144 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Returns", next := 112177, root := ~, start := 112148, stop := 112176 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 112240, root := ~, start := 112229, stop := 112239 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 112386, root := ~, start := 112375, stop := 112385 ), rec( content := " is ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Ref", next := 112420, root := ~, start := 112390, stop := 112419 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 112488, root := ~, start := 112477, stop := 112487 ), rec( content := " is ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Ref", next := 112528, root := ~, start := 112492, stop := 112527 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 112613, root := ~, start := 112602, stop := 112612 ), rec( content := " is not \ present, then ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Ref", next := 112673, root := ~, start := 112635, stop := 112672 ), rec( content := " is used\ by default.", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "P", next := 112697, root := ~, start := 112693, stop := 112696 ), rec( content := "\n\n \ If ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "A", next := 112714, root := ~, start := 112706, stop := 112713 ), rec( content := " is a no\ n-negative integer, this function returns a random\n tournament with ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "A", next := 112801, root := ~, start := 112793, stop := 112800 ), rec( content := " vertice\ s. See ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTourn\ ament" ), content := 0, count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "Ref", next := 112842, root := ~, start := 112816, stop := 112841 ), rec( content := ". ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 3, 5 ], mathmode := "Text", name := "P", next := 112848, root := ~, start := 112844, stop := 112847 ), rec( content := "\n ", count := [ 3, 4, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomTournament(10);\n\ngap> Ran\ domTournament(IsMutableDigraph, 10);\n\n", count := [ 3, 4, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 6 ], mathmode := "Text", name := "Log", next := 113040, root := ~, start := 112853, stop := 113039 ), rec( content := "\n ", count := [ 3, 4, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Description", next := 113057, root := ~, start := 112180, stop := 113056 ) ], count := [ 3, 4, 3, 1 ], mathmode := "Text", name := "ManSection", next := 113071, root := ~, start := 112083, stop := 113070 ), rec( content := "\n\n ", count := [ 3, 4, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "n", Name := "RandomLattice" ), content := 0, count := [ 3, 4, 4, 2 ], mathmode := "Text", name := "Oper", next := 113128, root := ~, start := 113092, stop := 113127 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Returns", next := 113160, root := ~, start := 113131, stop := 113159 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 113223, root := ~, start := 113212, stop := 113222 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 113369, root := ~, start := 113358, stop := 113368 ), rec( content := " is ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Ref", next := 113403, root := ~, start := 113373, stop := 113402 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 113471, root := ~, start := 113460, stop := 113470 ), rec( content := " is ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Ref", next := 113511, root := ~, start := 113475, stop := 113510 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 113596, root := ~, start := 113585, stop := 113595 ), rec( content := " is not \ present, then ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Ref", next := 113656, root := ~, start := 113618, stop := 113655 ), rec( content := " is used\ by default.", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "P", next := 113680, root := ~, start := 113676, stop := 113679 ), rec( content := "\n\n \ If ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "A", next := 113697, root := ~, start := 113689, stop := 113696 ), rec( content := " is a po\ sitive integer, this function return a random lattice\n with ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 113776, root := ~, start := 113768, stop := 113775 ), rec( content := " vertice\ s, where it is guaranteed that ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 113823, root := ~, start := 113815, stop := 113822 ), rec( content := " is betw\ een\n ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "A", next := 113847, root := ~, start := 113839, stop := 113846 ), rec( content := " and ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ * ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 4, 4, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := \ "Text", name := "A", next := 113867, root := ~, start := 113859, stop := 11386\ 6 ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 113871, root := ~, start := 113852, stop := 113870 ), rec( content := ". See ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsLatti\ ceDigraph" ), content := 0, count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "Ref", next := 113907, root := ~, start := 113877, stop := 113906 ), rec( content := ". ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 4, 5 ], mathmode := "Text", name := "P", next := 113913, root := ~, start := 113909, stop := 113912 ), rec( content := "\n ", count := [ 3, 4, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomLattice(10);\n\ngap> RandomLattice(IsMutableDigraph, 10);\n\n", count := [ 3, 4, 4, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 6 ], mathmode := "Text", name := "Log", next := 114108, root := ~, start := 113918, stop := 114107 ), rec( content := "\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Description", next := 114125, root := ~, start := 113163, stop := 114124 ) ], count := [ 3, 4, 4, 1 ], mathmode := "Text", name := "ManSection", next := 114139, root := ~, start := 113077, stop := 114138 ), rec( content := "\n\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 114153, root := ~, start := 109504, stop := 114152 ) , rec( content := "\n\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Standard examples\ ", count := [ 3, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 114202, root := ~, start := 114166, stop := 114201 ), rec( content := "\n ", count := [ 3, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "ChainDigraph" ) , content := 0, count := [ 3, 5, 1, 2 ], mathmode := "Text", name := "Oper", next := 114265, root := ~, start := 114222, stop := 114264 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 114297, root := ~, start := 114268, stop := 114296 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "A", next := 114329, root := ~, start := 114321, stop := 114328 ), rec( content := " is a po\ sitive integer, this function returns a chain\n with ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "A", next := 114400, root := ~, start := 114392, stop := 114399 ), rec( content := " vertice\ s and ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "n", count := [ 3, 5, 1, 3 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := \ "Text", name := "A", next := 114425, root := ~, start := 114417, stop := 11442\ 4 ), rec( content := " - 1", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114433, root := ~, start := 114414, stop := 114432 ), rec( content := " edges.\\ n Specifically,\n for each vertex ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114487, root := ~, start := 114479, stop := 114486 ), rec( content := " (with " , count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114502, root := ~, start := 114494, stop := 114501 ), rec( content := " < ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114516, root := ~, start := 114508, stop := 114515 ), rec( content := "), there\ is a directed\n edge with source ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114568, root := ~, start := 114560, stop := 114567 ), rec( content := " and ran\ ge ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ + 1", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114591, root := ~, start := 114579, stop := 114590 ), rec( content := ". ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "P", next := 114597, root := ~, start := 114593, stop := 114596 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 114645, root := ~, start := 114634, stop := 114644 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 114791, root := ~, start := 114780, stop := 114790 ), rec( content := " is ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "Ref", next := 114825, root := ~, start := 114795, stop := 114824 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 114893, root := ~, start := 114882, stop := 114892 ), rec( content := " is ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "Ref", next := 114933, root := ~, start := 114897, stop := 114932 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 115018, root := ~, start := 115007, stop := 115017 ), rec( content := " is not \ present, then ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "Ref", next := 115078, root := ~, start := 115040, stop := 115077 ), rec( content := " is used\ by default.", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "P", next := 115102, root := ~, start := 115098, stop := 115101 ), rec( content := "\n\n \ The ", count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveClosure" ), content := 0, count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 115159, root := ~, start := 115112, stop := 115158 ), rec( content := " of a ch\ ain represents a\n total order.\n ", count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ChainDigraph(42);\n\ngap> Chai\ nDigraph(IsMutableDigraph, 10);\n\n\ ", count := [ 3, 5, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 6 ], mathmode := "Text", name := "Example", next := 115388, root := ~, start := 115205, stop := 115387 ), rec( content := "\n ", count := [ 3, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 115405, root := ~, start := 114300, stop := 115404 ) ], count := [ 3, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 115419, root := ~, start := 114207, stop := 115418 ), rec( content := "\n\n ", count := [ 3, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "CompleteDigraph\ " ), content := 0, count := [ 3, 5, 2, 2 ], mathmode := "Text", name := "Oper", next := 115486, root := ~, start := 115440, stop := 115485 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 115518, root := ~, start := 115489, stop := 115517 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "A", next := 115550, root := ~, start := 115542, stop := 115549 ), rec( content := " is a no\ n-negative integer, this function returns the complete\n digraph with ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "A", next := 115638, root := ~, start := 115630, stop := 115637 ), rec( content := " vertice\ s. See ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsCompl\ eteDigraph" ), content := 0, count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 115684, root := ~, start := 115653, stop := 115683 ), rec( content := ". ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "P", next := 115690, root := ~, start := 115686, stop := 115689 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 115738, root := ~, start := 115727, stop := 115737 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 115884, root := ~, start := 115873, stop := 115883 ), rec( content := " is ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 115918, root := ~, start := 115888, stop := 115917 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 115986, root := ~, start := 115975, stop := 115985 ), rec( content := " is ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 116026, root := ~, start := 115990, stop := 116025 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 116111, root := ~, start := 116100, stop := 116110 ), rec( content := " is not \ present, then ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 116171, root := ~, start := 116133, stop := 116170 ), rec( content := " is used\ by default.\n ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CompleteDigraph(20);\n\ngap\ > CompleteDigraph(IsMutableDigraph, 10);\n\n", count := [ 3, 5, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 5 ], mathmode := "Text", name := "Example", next := 116389, root := ~, start := 116196, stop := 116388 ), rec( content := "\n ", count := [ 3, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 116406, root := ~, start := 115521, stop := 116405 ) ], count := [ 3, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 116420, root := ~, start := 115425, stop := 116419 ), rec( content := "\n\n ", count := [ 3, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]m, n", Name := "CompleteBiparti\ teDigraph" ), content := 0, count := [ 3, 5, 3, 2 ], mathmode := "Text", name := "Oper", next := 116499, root := ~, start := 116441, stop := 116498 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 116531, root := ~, start := 116502, stop := 116530 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ omplete bipartite digraph is a digraph whose vertices\n can be partitioned \ into two non-empty vertex sets, such there exists\n a unique edge with sour\ ce ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 116720, root := ~, start := 116712, stop := 116719 ), rec( content := " and ran\ ge ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 116739, root := ~, start := 116731, stop := 116738 ), rec( content := " if and \ only if\n ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 116767, root := ~, start := 116759, stop := 116766 ), rec( content := " and ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 116780, root := ~, start := 116772, stop := 116779 ), rec( content := " lie in \ different vertex sets. ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "P", next := 116815, root := ~, start := 116811, stop := 116814 ), rec( content := "\n\n \ If ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 116832, root := ~, start := 116824, stop := 116831 ), rec( content := " and ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 116845, root := ~, start := 116837, stop := 116844 ), rec( content := " are pos\ itive integers, this function\n returns the complete bipartite digraph with\ vertex sets of sizes\n ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 116964, root := ~, start := 116956, stop := 116963 ), rec( content := " (contai\ ning the vertices ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. m]", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "C", next := 117005, root := ~, start := 116990, stop := 117004 ), rec( content := ") and\n \ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 117023, root := ~, start := 117015, stop := 117022 ), rec( content := " (contai\ ning the vertices ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ m + 1 .. m + n]", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "C", next := 117072, root := ~, start := 117049, stop := 117071 ), rec( content := ").\n \ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "P", next := 117083, root := ~, start := 117079, stop := 117082 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 117131, root := ~, start := 117120, stop := 117130 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 117277, root := ~, start := 117266, stop := 117276 ), rec( content := " is ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "Ref", next := 117311, root := ~, start := 117281, stop := 117310 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 117379, root := ~, start := 117368, stop := 117378 ), rec( content := " is ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "Ref", next := 117419, root := ~, start := 117383, stop := 117418 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 117504, root := ~, start := 117493, stop := 117503 ), rec( content := " is not \ present, then ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "Ref", next := 117564, root := ~, start := 117526, stop := 117563 ), rec( content := " is used\ by default.\n\n ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CompleteBipartiteDigraph(2, 3);\n\ngap> CompleteBipartiteDigraph(IsMutableDigraph\ , 3, 2);\n\n", count := [ 3, 5, 3, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 3, 6 ], mathmode := "Text", name := "Example", next := 117828, root := ~, start := 117590, stop := 117827 ), rec( content := "\n ", count := [ 3, 5, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 117845, root := ~, start := 116534, stop := 117844 ) ], count := [ 3, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 117859, root := ~, start := 116426, stop := 117858 ), rec( content := "\n\n ", count := [ 3, 5, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]orders", Name := "CompleteMultipa\ rtiteDigraph" ), content := 0, count := [ 3, 5, 4, 2 ], mathmode := "Text", name := "Oper", next := 117949, root := ~, start := 117884, stop := 117948 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "Returns", next := 117983, root := ~, start := 117954, stop := 117982 ), rec( attributes := rec( ), content := [ rec( content := "\n F\ or a list ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rders", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "A", next := 118032, root := ~, start := 118019, stop := 118031 ), rec( content := " of ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 118044, root := ~, start := 118036, stop := 118043 ), rec( content := " positiv\ e integers,\n this function returns the digraph containing ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 118123, root := ~, start := 118115, stop := 118122 ), rec( content := " indepen\ dent sets\n of vertices of orders ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "l", count := [ 3, 5, 4, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := \ "Text", name := "A", next := 118181, root := ~, start := 118173, stop := 11818\ 0 ), rec( content := "[1] .. ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := \ "Text", name := "A", next := 118196, root := ~, start := 118188, stop := 11819\ 5 ), rec( content := "[n]]", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 118204, root := ~, start := 118169, stop := 118203 ), rec( content := ". Moreov\ er,\n each vertex is adjacent to every other not contained in the\n \ same independent set.\n ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "P", next := 118320, root := ~, start := 118316, stop := 118319 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 118370, root := ~, start := 118359, stop := 118369 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digrap\ h being created will\n belong to. For example, if ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 118520, root := ~, start := 118509, stop := 118519 ), rec( content := " is ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "Ref", next := 118554, root := ~, start := 118524, stop := 118553 ), rec( content := ",\n \ then the digraph being created will be mutable, if ", count := [ 3, 5, 4, 4 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 118624, root := ~, start := 118613, stop := 118623 ), rec( content := " is ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "Ref", next := 118666, root := ~, start := 118628, stop := 118665 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 118753, root := ~, start := 118742, stop := 118752 ), rec( content := " is not \ present, then ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "Ref", next := 118815, root := ~, start := 118775, stop := 118814 ), rec( content := " is used\ by default.\n ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CompleteMultipartiteDigraph([5, 4, 2]);\n\ngap> CompleteMultipartiteDigraph(IsMuta\ bleDigraph, [5, 4, 2]);\n\n", count := [ 3, 5, 4, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 4, 5 ], mathmode := "Text", name := "Example", next := 119096, root := ~, start := 118842, stop := 119095 ), rec( content := "\n ", count := [ 3, 5, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "Description", next := 119113, root := ~, start := 117988, stop := 119112 ) ], count := [ 3, 5, 4, 1 ], mathmode := "Text", name := "ManSection", next := 119127, root := ~, start := 117867, stop := 119126 ), rec( content := "\n\n ", count := [ 3, 5, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "CycleDigraph" ) , content := 0, count := [ 3, 5, 5, 2 ], mathmode := "Text", name := "Oper", next := 119191, root := ~, start := 119148, stop := 119190 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "Returns", next := 119223, root := ~, start := 119194, stop := 119222 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "A", next := 119255, root := ~, start := 119247, stop := 119254 ), rec( content := " is a po\ sitive integer, this function returns a ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ycle", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "E", next := 119315, root := ~, start := 119303, stop := 119314 ), rec( content := "\n di\ graph with ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "A", next := 119341, root := ~, start := 119333, stop := 119340 ), rec( content := " vertice\ s and ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "A", next := 119363, root := ~, start := 119355, stop := 119362 ), rec( content := " edges. \ Specifically,\n for each vertex ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119413, root := ~, start := 119405, stop := 119412 ), rec( content := " (with " , count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119428, root := ~, start := 119420, stop := 119427 ), rec( content := " < ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119442, root := ~, start := 119434, stop := 119441 ), rec( content := "), there\ is a directed\n edge with source ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119494, root := ~, start := 119486, stop := 119493 ), rec( content := " and ran\ ge ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ + 1", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119517, root := ~, start := 119505, stop := 119516 ), rec( content := ". In add\ ition, there is\n an edge with source ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119573, root := ~, start := 119565, stop := 119572 ), rec( content := " and ran\ ge ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119592, root := ~, start := 119584, stop := 119591 ), rec( content := ". ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "P", next := 119598, root := ~, start := 119594, stop := 119597 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 119646, root := ~, start := 119635, stop := 119645 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 119792, root := ~, start := 119781, stop := 119791 ), rec( content := " is ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "Ref", next := 119826, root := ~, start := 119796, stop := 119825 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 119894, root := ~, start := 119883, stop := 119893 ), rec( content := " is ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "Ref", next := 119934, root := ~, start := 119898, stop := 119933 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 120019, root := ~, start := 120008, stop := 120018 ), rec( content := " is not \ present, then ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "Ref", next := 120077, root := ~, start := 120041, stop := 120076 ), rec( content := " is used\ by default.\n ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CycleDigraph(1);\n\ngap> CycleD\ igraph(123);\n\ngap> CycleDigraph(I\ sMutableDigraph, 10);\n\n", count := [ 3, 5, 5, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 5, 5 ], mathmode := "Text", name := "Example", next := 120350, root := ~, start := 120100, stop := 120349 ), rec( content := "\n ", count := [ 3, 5, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "Description", next := 120367, root := ~, start := 119226, stop := 120366 ) ], count := [ 3, 5, 5, 1 ], mathmode := "Text", name := "ManSection", next := 120381, root := ~, start := 119133, stop := 120380 ), rec( content := "\n\n ", count := [ 3, 5, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "EmptyDigraph" ) , content := 0, count := [ 3, 5, 6, 2 ], mathmode := "Text", name := "Oper", next := 120445, root := ~, start := 120402, stop := 120444 ), rec( attributes := rec( Arg := "[filt, ]n", Name := "NullDigraph" ), content := 0, count := [ 3, 5, 6, 4 ], mathmode := "Text", name := "Oper", next := 120490, root := ~, start := 120448, stop := 120489 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "Returns", next := 120522, root := ~, start := 120493, stop := 120521 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "A", next := 120554, root := ~, start := 120546, stop := 120553 ), rec( content := " is a no\ n-negative integer, this function returns the\n ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ mpty", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "E", next := 120624, root := ~, start := 120612, stop := 120623 ), rec( content := " or ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ull", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "E", next := 120639, root := ~, start := 120628, stop := 120638 ), rec( content := " digraph\ with ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "A", next := 120661, root := ~, start := 120653, stop := 120660 ), rec( content := " vertice\ s. An empty\n digraph is one with no edges.\n ", count := [ 3, 5, 6, 5 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "P", next := 120723, root := ~, start := 120719, stop := 120722 ), rec( content := "\n \ \n If the optional first argument ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 120777, root := ~, start := 120766, stop := 120776 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 120923, root := ~, start := 120912, stop := 120922 ), rec( content := " is ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "Ref", next := 120957, root := ~, start := 120927, stop := 120956 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 121025, root := ~, start := 121014, stop := 121024 ), rec( content := " is ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "Ref", next := 121065, root := ~, start := 121029, stop := 121064 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 121150, root := ~, start := 121139, stop := 121149 ), rec( content := " is not \ present, then ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "Ref", next := 121210, root := ~, start := 121172, stop := 121209 ), rec( content := " is used\ by default.", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "P", next := 121234, root := ~, start := 121230, stop := 121233 ), rec( content := "\n\n \ ", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ ullDigraph", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "C", next := 121258, root := ~, start := 121240, stop := 121257 ), rec( content := " is a sy\ nonym for ", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ mptyDigraph", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "C", next := 121295, root := ~, start := 121276, stop := 121294 ), rec( content := ".\n\n \ ", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> EmptyDigraph(20);\n\ngap> Null\ Digraph(10);\n\ngap> EmptyDigraph(Is\ MutableDigraph, 10);\n", count := [ 3, 5, 6, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 6, 8 ], mathmode := "Text", name := "Example", next := 121546, root := ~, start := 121302, stop := 121545 ), rec( content := "\n ", count := [ 3, 5, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "Description", next := 121563, root := ~, start := 120525, stop := 121562 ) ], count := [ 3, 5, 6, 1 ], mathmode := "Text", name := "ManSection", next := 121577, root := ~, start := 120387, stop := 121576 ), rec( content := "\n\n ", count := [ 3, 5, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n, k", Name := "JohnsonDigraph"\ ), content := 0, count := [ 3, 5, 7, 2 ], mathmode := "Text", name := "Oper", next := 121646, root := ~, start := 121598, stop := 121645 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "Returns", next := 121678, root := ~, start := 121649, stop := 121677 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "A", next := 121710, root := ~, start := 121702, stop := 121709 ), rec( content := " and ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "A", next := 121723, root := ~, start := 121715, stop := 121722 ), rec( content := " are non\ -negative integers, then this operation\n returns a symmetric digraph which\ corresponds to the undirected ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ ohnson\n graph", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "E", next := 121863, root := ~, start := 121839, stop := 121862 ), rec( content := " ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ (n, k)", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "M", next := 121878, root := ~, start := 121864, stop := 121877 ), rec( content := ". ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "P", next := 121884, root := ~, start := 121880, stop := 121883 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 121932, root := ~, start := 121921, stop := 121931 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 122078, root := ~, start := 122067, stop := 122077 ), rec( content := " is ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "Ref", next := 122112, root := ~, start := 122082, stop := 122111 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 122180, root := ~, start := 122169, stop := 122179 ), rec( content := " is ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "Ref", next := 122220, root := ~, start := 122184, stop := 122219 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 122305, root := ~, start := 122294, stop := 122304 ), rec( content := " is not \ present, then ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "Ref", next := 122365, root := ~, start := 122327, stop := 122364 ), rec( content := " is used\ by default.", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "P", next := 122389, root := ~, start := 122385, stop := 122388 ), rec( content := "\n\n \ The ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ ohnson graph", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "E", next := 122419, root := ~, start := 122399, stop := 122418 ), rec( content := " ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ (n, k)", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "M", next := 122434, root := ~, start := 122420, stop := 122433 ), rec( content := " has ver\ tices given by all the\n ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "A", next := 122477, root := ~, start := 122469, stop := 122476 ), rec( content := "-subsets\ of the range ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 5, 7, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := \ "Text", name := "A", next := 122516, root := ~, start := 122508, stop := 12251\ 5 ), rec( content := "]", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "C", next := 122521, root := ~, start := 122499, stop := 122520 ), rec( content := ", and tw\ o vertices are\n connected by an edge iff their intersection has size ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "k", count := [ 3, 5, 7, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := \ "Text", name := "A", next := 122614, root := ~, start := 122606, stop := 12261\ 3 ), rec( content := " - 1", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "M", next := 122622, root := ~, start := 122603, stop := 122621 ), rec( content := ".\n\n \ ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := JohnsonDigraph(3, 1);\n\ngap> OutNeighbours(gr);\n[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ]\ngap> g\ r := JohnsonDigraph(4, 2);\n\ngap> OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 1, 3, 4, 6 ], [ 1, 2, 5, 6\ ], [ 1, 2, 5, 6 ], \n [ 1, 3, 4, 6 ], [ 2, 3, 4, 5 ] ]\ngap> JohnsonDigraph(\ 1, 0);\n\ngap> JohnsonDigraph(IsMutable\ Digraph, 1, 0);\n\n", count := [ 3, 5, 7, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 6 ], mathmode := "Text", name := "Example", next := 123171, root := ~, start := 122629, stop := 123170 ), rec( content := "\n ", count := [ 3, 5, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "Description", next := 123188, root := ~, start := 121681, stop := 123187 ) ], count := [ 3, 5, 7, 1 ], mathmode := "Text", name := "ManSection", next := 123202, root := ~, start := 121583, stop := 123201 ), rec( content := "\n\n ", count := [ 3, 5, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt]", Name := "PetersenGraph" ), content := 0, count := [ 3, 5, 8, 2 ], mathmode := "Text", name := "Oper", next := 123264, root := ~, start := 123223, stop := 123263 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "Returns", next := 123296, root := ~, start := 123267, stop := 123295 ), rec( attributes := rec( ), content := [ rec( content := "\n\n F\ rom ", count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://en.wikipedia.org/wiki/Petersen_graph", count := [ 3, 5, 8, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "URL", next := 123378, root := ~, start := 123323, stop := 123377 ), rec( content := ":", count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 4 ], mathmode := "Text", name := "P", next := 123383, root := ~, start := 123379, stop := 123382 ), rec( content := "\n\n \ ", count := [ 3, 5, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ he Petersen graph is an undirected graph with 10 vertices and 15 edges.\n I\ t is a small graph that serves as a useful example and counterexample for\n \ many problems in graph theory. The Petersen graph is named after Julius\n \ Petersen, who in 1898 constructed it to be the smallest bridgeless cubic\n \ graph with no three-edge-coloring.", count := [ 3, 5, 8, 4 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 4 ], mathmode := "Text", name := "Q", next := 123739, root := ~, start := 123389, stop := 123738 ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "P", next := 123743, root := ~, start := 123739, stop := 123742 ), rec( content := "\n \n\ If the optional first argument ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 123795, root := ~, start := 123784, stop := 123794 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 123941, root := ~, start := 123930, stop := 123940 ), rec( content := " is ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "Ref", next := 123975, root := ~, start := 123945, stop := 123974 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 124043, root := ~, start := 124032, stop := 124042 ), rec( content := " is ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "Ref", next := 124083, root := ~, start := 124047, stop := 124082 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 124168, root := ~, start := 124157, stop := 124167 ), rec( content := " is not \ present, then ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "Ref", next := 124228, root := ~, start := 124190, stop := 124227 ), rec( content := " is used\ by default.", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "P", next := 124252, root := ~, start := 124248, stop := 124251 ), rec( content := "\n\n \ See also ", count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "General\ isedPetersenGraph" ), content := 0, count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "Ref", next := 124305, root := ~, start := 124267, stop := 124304 ), rec( content := ".\n\n", count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ChromaticNumber(PetersenGraph());\n3\ngap> PetersenGraph(IsMutableDigrap\ h);\n\n", count := [ 3, 5, 8, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 8, 7 ], mathmode := "Text", name := "Example", next := 124464, root := ~, start := 124308, stop := 124463 ), rec( content := "\n ", count := [ 3, 5, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "Description", next := 124481, root := ~, start := 123299, stop := 124480 ) ], count := [ 3, 5, 8, 1 ], mathmode := "Text", name := "ManSection", next := 124495, root := ~, start := 123208, stop := 124494 ), rec( content := "\n\n ", count := [ 3, 5, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n, k", Name := "GeneralisedPete\ rsenGraph" ), content := 0, count := [ 3, 5, 9, 2 ], mathmode := "Text", name := "Oper", next := 124574, root := ~, start := 124516, stop := 124573 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "Returns", next := 124606, root := ~, start := 124577, stop := 124605 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "A", next := 124638, root := ~, start := 124630, stop := 124637 ), rec( content := " is a po\ sitive integer and ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "A", next := 124673, root := ~, start := 124665, stop := 124672 ), rec( content := " is a no\ n-negative\n integer less than ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "n", count := [ 3, 5, 9, 3 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := \ "Text", name := "A", next := 124725, root := ~, start := 124717, stop := 12472\ 4 ), rec( content := " / 2", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "C", next := 124733, root := ~, start := 124714, stop := 124732 ), rec( content := ", then t\ his operation returns\n the ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ eneralised Petersen graph", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "E", next := 124804, root := ~, start := 124771, stop := 124803 ), rec( content := " ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ PG(", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 5, 9, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := \ "Text", name := "A", next := 124820, root := ~, start := 124812, stop := 12481\ 9 ), rec( content := ", k)", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "M", next := 124828, root := ~, start := 124805, stop := 124827 ), rec( content := ". ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "P", next := 124834, root := ~, start := 124830, stop := 124833 ), rec( content := "\n\n \ From\n ", count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://en.wikipedia.org/wiki/Generalized_Petersen_graph", count := [ 3, 5, 9, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "URL", next := 124916, root := ~, start := 124849, stop := 124915 ), rec( content := ":\n " , count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "P", next := 124926, root := ~, start := 124922, stop := 124925 ), rec( content := "\n\n \ ", count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ he generalized Petersen graphs are a family of cubic\n graphs formed by con\ necting the vertices of a regular polygon to the\n corresponding vertices o\ f a star polygon. They include the Petersen graph\n and generalize one of t\ he ways of constructing the Petersen graph. The\n generalized Petersen grap\ h family was introduced in 1950 by H. S. M.\n Coxeter and was given its nam\ e in 1969 by Mark Watkins.", count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "Q", next := 125351, root := ~, start := 124932, stop := 125350 ), rec( content := "\n ", count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "P", next := 125360, root := ~, start := 125356, stop := 125359 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 125408, root := ~, start := 125397, stop := 125407 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 125554, root := ~, start := 125543, stop := 125553 ), rec( content := " is ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "Ref", next := 125588, root := ~, start := 125558, stop := 125587 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 125656, root := ~, start := 125645, stop := 125655 ), rec( content := " is ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "Ref", next := 125696, root := ~, start := 125660, stop := 125695 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 125781, root := ~, start := 125770, stop := 125780 ), rec( content := " is not \ present, then ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "Ref", next := 125841, root := ~, start := 125803, stop := 125840 ), rec( content := " is used\ by default.", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "P", next := 125865, root := ~, start := 125861, stop := 125864 ), rec( content := "\n\n \ See also ", count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Peterse\ nGraph" ), content := 0, count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "Ref", next := 125907, root := ~, start := 125880, stop := 125906 ), rec( content := ".\n\n", count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> GeneralisedPetersenGraph(7, 2);\n\ngap> GeneralisedPetersenGraph(40, 1);\n\ngap> D := GeneralisedPetersenGraph(5, 2\ );\n\ngap> IsIsomorphi\ cDigraph(D, PetersenGraph());\ntrue\ngap> GeneralisedPetersenGraph(IsMutableDi\ graph, 9, 4);\n\n", count := [ 3, 5, 9, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 8 ], mathmode := "Text", name := "Example", next := 126382, root := ~, start := 125910, stop := 126381 ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "Description", next := 126399, root := ~, start := 124609, stop := 126398 ) ], count := [ 3, 5, 9, 1 ], mathmode := "Text", name := "ManSection", next := 126413, root := ~, start := 124501, stop := 126412 ), rec( content := "\n\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 126427, root := ~, start := 114157, stop := 126426 ) , rec( content := "\n\n", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 126439, root := ~, start := 19180, stop := 126438 ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " creating ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Operators" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Operators", count := [ 4, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 126516, root := ~, start := 126488, stop := 126515 ) , rec( content := "\n\n ", count := [ 4, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Operators for dig\ raphs", count := [ 4, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 126570, root := ~, start := 126529, stop := 126569 ), rec( content := "\n ", count := [ 4, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "digraph1", count := [ 4, 1, 0, 4 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 4 ], mathmode := \ "Text", name := "A", next := 126611, root := ~, start := 126596, stop := 12661\ 0 ), rec( content := " = ", count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "A", next := 1266\ 29, root := ~, start := 126614, stop := 126628 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "C", next := 126633, root := ~, start := 126593, stop := 126632 ) ], count := [ 4, 1, 0, 4 ], name := "Mark", next := 126640, root := ~, start := 126587, stop := 126639 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "=" ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "=", count := [ 4, 1, 0, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "C", next := 126683, root := ~, start := 126675, stop := 12668\ 2 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 5 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "Index", next := 126706, root := ~, start := 126660, stop := 126705 ), rec( content := "\n \ returns ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "K", next := 126733, root := ~, start := 126722, stop := 126732 ), rec( content := " if ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "A", next := 126752, root := ~, start := 126737, stop := 126751 ), rec( content := " and ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "A", next := 126772, root := ~, start := 126757, stop := 126771 ), rec( content := " have th\ e same\n vertices, and ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph1", count := [ 4, 1, 0, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "A", next := 126839, root := ~, start := 126824, stop := 12683\ 8 ), rec( content := ") =\n DigraphEdges(", count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph2", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "A", next := 1268\ 80, root := ~, start := 126865, stop := 126879 ), rec( content := ")", count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "C", next := 126885, root := ~, start := 126808, stop := 126884 ), rec( content := ", up to \ some re-ordering of the edge\n lists. ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "P", next := 126940, root := ~, start := 126936, stop := 126939 ), rec( content := "\n\n \ Note that this operator does not compare the vertex labels\n of ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "A", next := 127033, root := ~, start := 127018, stop := 127032 ), rec( content := " and ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "A", next := 127053, root := ~, start := 127038, stop := 127052 ), rec( content := ".\n \ ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], name := "Item", next := 127067, root := ~, start := 126646, stop := 127066 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "digraph1", count := [ 4, 1, 0, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 7 ], mathmode := \ "Text", name := "A", next := 127098, root := ~, start := 127083, stop := 12709\ 7 ), rec( content := " < ", count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "A", next := 1271\ 19, root := ~, start := 127104, stop := 127118 ) ], count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "C", next := 127123, root := ~, start := 127080, stop := 127122 ) ], count := [ 4, 1, 0, 7 ], name := "Mark", next := 127130, root := ~, start := 127074, stop := 127129 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "<" ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "<", count := [ 4, 1, 0, 8 ], mathmode \ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 4, 1, 0, 8 ] , mathmode := \ "Text", name := "C", next := 127179, root := ~, start := 127168, stop := 12717\ 8 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 8 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "Index", next := 127202, root := ~, start := 127150, stop := 127201 ), rec( content := "\n\n \ This operator returns ", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "K", next := 127244, root := ~, start := 127233, stop := 127243 ), rec( content := " if one \ of the following holds:\n\n ", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n The number ", count := [ \ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "n_1", count := [ 4, 1, 0, 10 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 10 ], math\ mode := "Text", name := "M", next := 127339, root := ~, start := 127329, stop \ := 127338 ), rec( content := " of vertices in ", count := [ 4, 1, 0, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "digraph1", count := [ 4, 1, 0, 10 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 10 ], mathmode := "T\ ext", name := "A", next := 127370, root := ~, start := 127355, stop := 127369 ), rec( cont\ ent := " is less than\n the number ", count := [ 4, 1, 0, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "n_2", count := [ 4, 1, 0, 10 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 10 ], mathmode := "Text", name\ := "M", next := 127416, root := ~, start := 127406, stop := 127415 ), rec( cont\ ent := " of vertices in ", count := [ 4, 1, 0, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "digraph2", count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "A", next\ := 127447, root := ~, start := 127432, stop := 127446 ), rec( content := ";\n\ ", count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 0, 10 ], name := "Item", next := 127465, root := ~, start := 127\ 300, stop := 127464 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n_\ 1 = n_2", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "M", next := \ 127509, root := ~, start := 127493, stop := 127508 ), rec( content := ", and t\ he number ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "m_1", count\ := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "M", next := 127536, root :=\ ~, start := 127526, stop := 127535 ), rec( content := " of edges in\n \ ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph1", coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "A", next := 127575, root :=\ ~, start := 127560, stop := 127574 ), rec( content := " is less than the numb\ er ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "m_2", count := \ [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 4, 1, 0, 12 ], mathmode := "Text", name := "M", next := 127610, root := ~, star\ t := 127600, stop := 127609 ), rec( content := " of edges in\n ", coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph2", count := [ 4, 1, 0\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, \ 0, 12 ], mathmode := "Text", name := "A", next := 127649, root := ~, start := \ 127634, stop := 127648 ), rec( content := ";\n ", count := [ 4, 1, 0, \ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 12 ], name := "Ite\ m", next := 127667, root := ~, start := 127475, stop := 127666 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 4, 1, 0, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "n_1 = n_2", count := [ 4, 1, 0, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 14 ], math\ mode := "Text", name := "M", next := 127711, root := ~, start := 127695, stop \ := 127710 ), rec( content := ", ", count := [ 4, 1, 0, 14 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "m_1 = m_2", count := [ 4, 1, 0, 14 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 14 ], mathmode := "Text", name \ := "M", next := 127729, root := ~, start := 127713, stop := 127728 ), rec( cont\ ent := ", and\n ", count := [ 4, 1, 0, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "DigraphEdges(", count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "digraph1", count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "A", next \ := 127777, root := ~, start := 127762, stop := 127776 ), rec( content := ")", \ count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 14 ], mathmode := "Text", name := "C", next := 127782, root :=\ ~, start := 127746, stop := 127781 ), rec( content := " is less than\n \ ", count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphEdges("\ , count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ),\ rec( attributes := rec( ), content := [ rec( content := "digraph2", count :=\ [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 4, 1, 0, 14 ], mathmode := "Text", name := "A", next := 127838, root := ~,\ start := 127823, stop := 127837 ), rec( content := ")", count := [ 4, 1, 0, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, \ 14 ], mathmode := "Text", name := "C", next := 127843, root := ~, start := 127\ 807, stop := 127842 ), rec( content := " after having both of these sets\n \ have been sorted with respect to the lexicographical order.\n " , coun\ t := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 14 ], name := "Ite\ m", next := 127963, root := ~, start := 127677, stop := 127962 ) ], count := [ 4, 1, 0, 9 ], mathmode := "Text", name := "List", next := 127978, root := ~, start := 127284, stop := 127977 ), rec( content := "\n " , count := [ 4, 1, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], name := "Item", next := 127991, root := ~, start := 127136, stop := 127990 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "List", next := 128002, root := ~, start := 126575, stop := 128001 ), rec( content := "\n\n ", count := [ 4, 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "super, sub", Name := "IsSubdigraph" ) , content := 0, count := [ 4, 1, 1, 2 ], mathmode := "Text", name := "Oper", next := 128066, root := ~, start := 128022, stop := 128065 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 128089, root := ~, start := 128078, stop := 128088 ), rec( content := " or ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 128105, root := ~, start := 128093, stop := 128104 ), rec( content := ".", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 128116, root := ~, start := 128069, stop := 128115 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 128152, root := ~, start := 128140, stop := 128151 ), rec( content := " and ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 128167, root := ~, start := 128157, stop := 128166 ), rec( content := " are dig\ raphs, then this operation returns\n ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 128225, root := ~, start := 128214, stop := 128224 ), rec( content := " if ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 128239, root := ~, start := 128229, stop := 128238 ), rec( content := " is a su\ bdigraph of ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 128271, root := ~, start := 128259, stop := 128270 ), rec( content := ", and ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 128289, root := ~, start := 128277, stop := 128288 ), rec( content := "\n if\ it is not. ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "P", next := 128312, root := ~, start := 128308, stop := 128311 ), rec( content := "\n\n \ A digraph ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128338, root := ~, start := 128328, stop := 128337 ), rec( content := " is a ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ubdigraph", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "E", next := 128361, root := ~, start := 128344, stop := 128360 ), rec( content := " of a di\ graph ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128387, root := ~, start := 128375, stop := 128386 ), rec( content := " if\n \ ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128405, root := ~, start := 128395, stop := 128404 ), rec( content := " and ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128422, root := ~, start := 128410, stop := 128421 ), rec( content := " share t\ he same number of vertices, and the\n collection of edges of ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128505, root := ~, start := 128493, stop := 128504 ), rec( content := " (includ\ ing repeats) contains the\n collection of edges of ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128576, root := ~, start := 128566, stop := 128575 ), rec( content := " (includ\ ing repeats). ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "P", next := 128603, root := ~, start := 128599, stop := 128602 ), rec( content := "\n\n \ In other words, ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 128635, root := ~, start := 128625, stop := 128634 ), rec( content := " is a su\ bdigraph of ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 128667, root := ~, start := 128655, stop := 128666 ), rec( content := " if and \ only if\n ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "su\ b", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "A", next := 128718, root := ~, start := 128708, stop := 12871\ 7 ), rec( content := ") = DigraphNrVertices(", count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "super", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 1287\ 52, root := ~, start := 128740, stop := 128751 ), rec( content := ")", count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 128757, root := ~, start := 128687, stop := 128756 ), rec( content := ", and\n \ for each pair of vertices ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 128801, root := ~, start := 128793, stop := 128800 ), rec( content := " and ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 128814, root := ~, start := 128806, stop := 128813 ), rec( content := ", there \ are at least as many\n edges of the form ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 128878, root := ~, start := 128865, stop := 128877 ), rec( content := " in ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 128894, root := ~, start := 128882, stop := 128893 ), rec( content := " as ther\ e are in ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 128921, root := ~, start := 128911, stop := 128920 ), rec( content := ".\n " , count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 1, 6 ], mathmode := "Text", name := "P", next := 128931, root := ~, start := 128927, stop := 128930 ), rec( content := "\n\n \ ", count := [ 4, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> g := Digraph([[2, 3], [1], [2, 3]]);\n\ngap> h := Digraph([[2, 3], [], [2]]);\n\ngap> IsSubdigraph(g, h);\ntrue\ngap> IsSubdigraph(h, g);\nf\ alse\ngap> IsSubdigraph(CompleteDigraph(4), CycleDigraph(4));\ntrue\ngap> IsSu\ bdigraph(CycleDigraph(4), ChainDigraph(4));\ntrue\ngap> g := Digraph([[2, 2], \ [1]]);\n\ngap> h := Digraph([\ [2], [1]]);\n\ngap> IsSubdigraph(g\ , h);\ntrue\ngap> IsSubdigraph(h, g);\nfalse", count := [ 4, 1, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 1, 7 ], mathmode := "Text", name := "Example", next := 129539, root := ~, start := 128937, stop := 129538 ), rec( content := "\n ", count := [ 4, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 129556, root := ~, start := 128119, stop := 129555 ) ], count := [ 4, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 129570, root := ~, start := 128007, stop := 129569 ), rec( content := "\n\n ", count := [ 4, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "super, sub", Name := "IsUndirectedSpa\ nningTree" ), content := 0, count := [ 4, 1, 2, 2 ], mathmode := "Text", name := "Oper", next := 129646, root := ~, start := 129590, stop := 129645 ), rec( attributes := rec( Arg := "super, sub", Name := "IsUndirectedSpa\ nningForest" ), content := 0, count := [ 4, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 129707, root := ~, start := 129649, stop := 129706 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 129730, root := ~, start := 129719, stop := 129729 ), rec( content := " or ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 129746, root := ~, start := 129734, stop := 129745 ), rec( content := ".", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 129757, root := ~, start := 129710, stop := 129756 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ operation ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedSpanningTree", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "C", next := 129823, root := ~, start := 129792, stop := 129822 ), rec( content := " returns\ ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 129843, root := ~, start := 129832, stop := 129842 ), rec( content := " if the\\ n digraph ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 129873, root := ~, start := 129863, stop := 129872 ), rec( content := " is an u\ ndirected spanning tree of the digraph\n ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 129936, root := ~, start := 129924, stop := 129935 ), rec( content := ", and th\ e operation ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedSpanningForest", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "C", next := 129989, root := ~, start := 129956, stop := 129988 ), rec( content := " returns\ \n ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 130013, root := ~, start := 130002, stop := 130012 ), rec( content := " if the \ digraph ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 130039, root := ~, start := 130029, stop := 130038 ), rec( content := " is an u\ ndirected spanning forest of\n the digraph ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 130104, root := ~, start := 130092, stop := 130103 ), rec( content := ". ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "P", next := 130110, root := ~, start := 130106, stop := 130109 ), rec( content := "\n\n \ An ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected spanning tree", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "E", next := 130150, root := ~, start := 130119, stop := 130149 ), rec( content := " of a di\ graph ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "A", next := 130176, root := ~, start := 130164, stop := 130175 ), rec( content := " is a su\ bdigraph\n of ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "A", next := 130212, root := ~, start := 130200, stop := 130211 ), rec( content := " that is\ an undirected tree (see ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsSubdi\ graph" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 130272, root := ~, start := 130245, stop := 130271 ), rec( content := "\n an\ d ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsUndir\ ectedTree" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 130312, root := ~, start := 130281, stop := 130311 ), rec( content := "). Note \ that a digraph whose ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraph" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 130387, root := ~, start := 130341, stop := 130386 ), rec( content := " is not \ connected has no undirected\n spanning trees (see ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 130479, root := ~, start := 130447, stop := 130478 ), rec( content := "). ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "P", next := 130486, root := ~, start := 130482, stop := 130485 ), rec( content := "\n\n \ An ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected spanning forest", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "E", next := 130528, root := ~, start := 130495, stop := 130527 ), rec( content := " of a di\ graph ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 130554, root := ~, start := 130542, stop := 130553 ), rec( content := " is a\n \ subdigraph of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 130590, root := ~, start := 130578, stop := 130589 ), rec( content := " that is\ an undirected forest (see ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsSubdi\ graph" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 130657, root := ~, start := 130625, stop := 130656 ), rec( content := " and ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsUndir\ ectedForest" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 130694, root := ~, start := 130662, stop := 130693 ), rec( content := "), and i\ s not\n contained in any larger such subdigraph of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 130767, root := ~, start := 130755, stop := 130766 ), rec( content := ". Equiv\ alently, an\n undirected spanning forest is a subdigraph of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 130849, root := ~, start := 130837, stop := 130848 ), rec( content := " whose c\ onnected\n components coincide with those of the ", count := [ 4, 1, 2, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraph" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 130954, root := ~, start := 130908, stop := 130953 ), rec( content := " of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 130970, root := ~, start := 130958, stop := 130969 ), rec( content := " (see ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 131022, root := ~, start := 130976, stop := 131021 ), rec( content := "). ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 2, 8 ], mathmode := "Text", name := "P", next := 131029, root := ~, start := 131025, stop := 131028 ), rec( content := "\n\n \ Note that an undirected spanning tree is an undirected spanning forest that\n \ is connected.\n\n ", count := [ 4, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);\n\n\ gap> tree := Digraph([[3], [4], [1, 4], [2, 3]]);\n\ngap> IsSubdigraph(D, tree) and IsUndirectedTree(tree);\ntru\ e\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap> forest := EmptyDigraph(\ 4);\n\ngap> IsSubdigraph(D, forest) a\ nd IsUndirectedForest(forest);\ntrue\ngap> IsUndirectedSpanningForest(D, fores\ t);\nfalse\ngap> IsSubdigraph(tree, forest);\ntrue\ngap> D := DigraphDisjointU\ nion(CycleDigraph(2), CycleDigraph(2));\n\ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedForest(D) and IsUn\ directedSpanningForest(D, D);\ntrue", count := [ 4, 1, 2, 9 ], name := "PCDATA\ ", root := ~ ) ], count := [ 4, 1, 2, 9 ], mathmode := "Text", name := "Example", next := 131881, root := ~, start := 131134, stop := 131880 ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 131898, root := ~, start := 129760, stop := 131897 ) ], count := [ 4, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 131912, root := ~, start := 129575, stop := 131911 ), rec( content := "\n\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 131926, root := ~, start := 126520, stop := 131925 ) , rec( content := "\n", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 131937, root := ~, start := 126461, stop := 131936 ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " operators/hashs ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Attributes and operations" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Attributes and operations\ ", count := [ 5, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 132053, root := ~, start := 132009, stop := 132052 ), rec( content := "\n ", count := [ 5, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Vertices and edge\ s", count := [ 5, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 132102, root := ~, start := 132065, stop := 132101 ), rec( content := "\n ", count := [ 5, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphVertices\ " ), content := 0, count := [ 5, 1, 1, 2 ], mathmode := "Text", name := "Attr", next := 132166, root := ~, start := 132122, stop := 132165 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ integers.", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 132207, root := ~, start := 132169, stop := 132206 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the vertices of the digraph ", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "A", next := 132278, root := ~, start := 132264, stop := 132277 ), rec( content := ". ", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "P", next := 132284, root := ~, start := 132280, stop := 132283 ), rec( content := "\n\n \ Note that the vertices of a digraph are always a range of\n positive intege\ rs from ", count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "C", next := 132383, root := ~, start := 132375, stop := 132382 ), rec( content := " to the \ number of vertices of the graph.\n ", count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphVertices(gr);\n[ 1 .. 3 ]\ngap> gr := Digraph\ ([1, 2, 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> \ [2, 7, 5, 3, 7]);\n\ngap> DigraphV\ ertices(gr);\n[ 1 .. 6 ]\ngap> DigraphVertices(RandomDigraph(100));\n[ 1 .. 10\ 0 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphVertices(D);\n[ 1 .. 3 ]\n", count := [ 5, 1, 1, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 1, 5 ], mathmode := "Text", name := "Example", next := 133024, root := ~, start := 132428, stop := 133023 ), rec( content := "\n ", count := [ 5, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 133041, root := ~, start := 132210, stop := 133040 ) ], count := [ 5, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 133055, root := ~, start := 132107, stop := 133054 ), rec( content := "\n\n ", count := [ 5, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphNrVertic\ es" ), content := 0, count := [ 5, 1, 2, 2 ], mathmode := "Text", name := "Attr", next := 133122, root := ~, start := 133076, stop := 133121 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "Returns", next := 133155, root := ~, start := 133125, stop := 133154 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the number of vertices of the digraph ", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "A", next := 133236, root := ~, start := 133222, stop := 133235 ), rec( content := ". ", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 2, 4 ], mathmode := "Text", name := "P", next := 133242, root := ~, start := 133238, stop := 133241 ), rec( content := "\n\n \ ", count := [ 5, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphNrVertices(gr);\n3\ngap> gr := Digraph([1, 2,\ 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> [2, 7, \ 5, 3, 7]);\n\ngap> DigraphNrVertic\ es(gr);\n6\ngap> DigraphNrVertices(RandomDigraph(100));\n100\ngap> D := CycleD\ igraph(IsMutableDigraph, 3);\n\ngap>\ DigraphNrVertices(D);\n3\n", count := [ 5, 1, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 5 ], mathmode := "Text", name := "Example", next := 133816, root := ~, start := 133248, stop := 133815 ), rec( content := "\n ", count := [ 5, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 133833, root := ~, start := 133158, stop := 133832 ) ], count := [ 5, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 133847, root := ~, start := 133061, stop := 133846 ), rec( content := "\n\n ", count := [ 5, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphEdges" ) , content := 0, count := [ 5, 1, 3, 2 ], mathmode := "Text", name := "Attr", next := 133909, root := ~, start := 133868, stop := 133908 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists.", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 133947, root := ~, start := 133912, stop := 133946 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "C", next := 133987, root := ~, start := 133968, stop := 133986 ), rec( content := " returns\ a list of edges of the digraph ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "A", next := 134041, root := ~, start := 134027, stop := 134040 ), rec( content := ",\n w\ here each edge is a pair of elements of ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 134117, root := ~, start := 134088, stop := 134116 ), rec( content := " of\n \ the form ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ source,range]", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "C", next := 134155, root := ~, start := 134134, stop := 134154 ), rec( content := ".\n " , count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "P", next := 134165, root := ~, start := 134161, stop := 134164 ), rec( content := "\n\n \ The entries of ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 134206, root := ~, start := 134186, stop := 134205 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 134220, root := ~, start := 134206, stop := 134219 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 134228, root := ~, start := 134220, stop := 134227 ), rec( content := " are in \ one-to-one\n corresponence with the edges of ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 134297, root := ~, start := 134283, stop := 134296 ), rec( content := ". Hence\ \n ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 134330, root := ~, start := 134310, stop := 134329 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 134344, root := ~, start := 134330, stop := 134343 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 134352, root := ~, start := 134344, stop := 134351 ), rec( content := " is dupl\ icate-free if and only if\n ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 134404, root := ~, start := 134390, stop := 134403 ), rec( content := " contain\ s no multiple edges. ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "P", next := 134437, root := ~, start := 134433, stop := 134436 ), rec( content := "\n\n \ The entries of ", count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges", count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "C", next := 134477, root := ~, start := 134458, stop := 134476 ), rec( content := " are gua\ ranteed to be sorted by their\n first component (i.e. by the source of each\ edge), but they are not\n necessarily then sorted by the second component.\ \n ", count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := DigraphFromDiSparse6String(\".DaXbOe?EAM@G~\");\n\ngap> edges := ShallowCopy(DigraphEdges(gr))\ ;; Sort(edges);\ngap> edges;\n[ [ 1, 1 ], [ 1, 3 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5\ ], [ 2, 1 ], \n [ 2, 2 ], [ 2, 3 ], [ 2, 5 ], [ 3, 2 ], [ 3, 4 ], [ 3, 5 ], \ \n [ 4, 2 ], [ 4, 4 ], [ 4, 5 ], [ 5, 1 ] ]\ngap> D := CycleDigraph(IsMutable\ Digraph, 3);\n\ngap> DigraphEdges(D)\ ;\n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\n", count := [ 5, 1, 3, 6 ], name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 3, 6 ], mathmode := "Text", name := "Example", next := 135167, root := ~, start := 134644, stop := 135166 ), rec( content := "\n ", count := [ 5, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 135184, root := ~, start := 133950, stop := 135183 ) ], count := [ 5, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 135198, root := ~, start := 133853, stop := 135197 ), rec( content := "\n\n ", count := [ 5, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphNrEdges"\ ), content := 0, count := [ 5, 1, 4, 2 ], mathmode := "Text", name := "Attr", next := 135262, root := ~, start := 135219, stop := 135261 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 135295, root := ~, start := 135265, stop := 135294 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the number of edges of the digraph ", count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "A", next := 135387, root := ~, start := 135373, stop := 135386 ), rec( content := ".\n " , count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]\ ]);;\ngap> DigraphNrEdges(gr);\n15\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\ \n> [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\",\ \"a\"]);\n\ngap> DigraphNrEd\ ges(gr);\n3\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphNrEdges(D);\n3\n", count := [ 5, 1, 4, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 4, 4 ], mathmode := "Text", name := "Example", next := 135834, root := ~, start := 135393, stop := 135833 ), rec( content := "\n ", count := [ 5, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 135851, root := ~, start := 135298, stop := 135850 ) ], count := [ 5, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 135865, root := ~, start := 135204, stop := 135864 ), rec( content := "\n\n ", count := [ 5, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSinks" ) , content := 0, count := [ 5, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 135927, root := ~, start := 135886, stop := 135926 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 135968, root := ~, start := 135930, stop := 135967 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a list of the sinks of the digraph\n ", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "A", next := 136064, root := ~, start := 136050, stop := 136063 ), rec( content := ".\n A\ sink of a digraph is a vertex with out-degree zero.\n See ", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Ref", next := 136163, root := ~, start := 136132, stop := 136162 ), rec( content := ".\n " , count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);\n\ngap> DigraphSinks(gr);\n[ 3, 5 ]\ngap> D\ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSinks(D);\n[ ]\n", count := [ 5, 1, 5, 4 ], name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 5, 4 ], mathmode := "Text", name := "Example", next := 136461, root := ~, start := 136169, stop := 136460 ), rec( content := "\n ", count := [ 5, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 136478, root := ~, start := 135971, stop := 136477 ) ], count := [ 5, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 136492, root := ~, start := 135871, stop := 136491 ), rec( content := "\n\n ", count := [ 5, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSources"\ ), content := 0, count := [ 5, 1, 6, 2 ], mathmode := "Text", name := "Attr", next := 136556, root := ~, start := 136513, stop := 136555 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 136597, root := ~, start := 136559, stop := 136596 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns an immutable list of the sources of the digraph\n ", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "A", next := 136706, root := ~, start := 136692, stop := 136705 ), rec( content := ".\n A\ source of a digraph is a vertex with in-degree zero.\n See ", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "InDegre\ eOfVertex" ), content := 0, count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Ref", next := 136805, root := ~, start := 136775, stop := 136804 ), rec( content := ".\n " , count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);\n\ngap> DigraphSources(gr);\n[ 1, 4 ]\ngap>\ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSources(D);\n[ ]\n", count := [ 5, 1, 6, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 6, 4 ], mathmode := "Text", name := "Example", next := 137107, root := ~, start := 136811, stop := 137106 ), rec( content := "\n ", count := [ 5, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 137124, root := ~, start := 136600, stop := 137123 ) ], count := [ 5, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 137138, root := ~, start := 136498, stop := 137137 ), rec( content := "\n\n ", count := [ 5, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTopologi\ calSort" ), content := 0, count := [ 5, 1, 7, 2 ], mathmode := "Text", name := "Attr", next := 137210, root := ~, start := 137159, stop := 137209 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers, or ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "K", next := 137265, root := ~, start := 137254, stop := 137264 ), rec( content := ".", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Returns", next := 137276, root := ~, start := 137213, stop := 137275 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "A", next := 137314, root := ~, start := 137300, stop := 137313 ), rec( content := " is a di\ graph whose only directed cycles are loops, then\n ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphTopologicalSort", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "C", next := 137404, root := ~, start := 137375, stop := 137403 ), rec( content := " returns\ the vertices of ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "A", next := 137443, root := ~, start := 137429, stop := 137442 ), rec( content := " ordered\ \n so that every edge's source appears no earlier in the list than its rang\ e.\n If the digraph ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "A", next := 137564, root := ~, start := 137550, stop := 137563 ), rec( content := " contain\ s directed cycles of length greater\n than ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "M", next := 137625, root := ~, start := 137617, stop := 137624 ), rec( content := ", then t\ his operation returns ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "K", next := 137666, root := ~, start := 137655, stop := 137665 ), rec( content := ".\n " , count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "P", next := 137676, root := ~, start := 137672, stop := 137675 ), rec( content := "\n\n \ See section ", count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "Ref", next := 137738, root := ~, start := 137694, stop := 137737 ), rec( content := " for the\ definition\n of a directed cycle, and the definition of a loop.\n\n ", count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "P", next := 137822, root := ~, start := 137818, stop := 137821 ), rec( content := "\n\n \ The method used for this attribute has complexity ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "M", next := 137891, root := ~, start := 137878, stop := 137890 ), rec( content := " where\n\ ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "M", next := 137910, root := ~, start := 137902, stop := 137909 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "M", next := 137983, root := ~, start := 137975, stop := 137982 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 7, 6 ], mathmode := "Text", name := "P", next := 138030, root := ~, start := 138026, stop := 138029 ), rec( content := "\n ", count := [ 5, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 3], [], [4, 6], [5], [], [7, 8, 9], [], [], []]);\ \n\ngap> DigraphTopologicalSort(D)\ ;\n[ 2, 5, 4, 7, 8, 9, 6, 3, 1 ]\ngap> D := Digraph(IsMutableDigraph, [[2, 3],\ [3], [4], []]);\n\ngap> DigraphTopo\ logicalSort(D);\n[ 4, 3, 2, 1 ]\n", count := [ 5, 1, 7, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 7 ], mathmode := "Text", name := "Example", next := 138401, root := ~, start := 138035, stop := 138400 ), rec( content := "\n ", count := [ 5, 1, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Description", next := 138418, root := ~, start := 137279, stop := 138417 ) ], count := [ 5, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 138432, root := ~, start := 137144, stop := 138431 ), rec( content := "\n\n ", count := [ 5, 1, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i", Name := "DigraphVertexLa\ bel" ), content := 0, count := [ 5, 1, 8, 2 ], mathmode := "Text", name := "Oper", next := 138505, root := ~, start := 138453, stop := 138504 ), rec( attributes := rec( Arg := "digraph, i, obj" , Name := "SetDigraphVerte\ xLabel" ), content := 0, count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "Oper", next := 138567, root := ~, start := 138508, stop := 138566 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 138605, root := ~, start := 138591, stop := 138604 ), rec( content := " is a di\ graph, then the first operation returns the label\n of the vertex ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 138689, root := ~, start := 138681, stop := 138688 ), rec( content := ". The se\ cond operation can be used to set the label\n of the vertex ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 138767, root := ~, start := 138759, stop := 138766 ), rec( content := " in ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 138785, root := ~, start := 138771, stop := 138784 ), rec( content := " to the \ arbitrary ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object\\ n ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ bj", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "A", next := 138830, root := ~, start := 138820, stop := 138829 ), rec( content := ". ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "P", next := 138837, root := ~, start := 138833, stop := 138836 ), rec( content := "\n\n \ The label of a vertex can be changed an arbitrary number of times. If no\n \ label has been set for the vertex ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "A", next := 138962, root := ~, start := 138954, stop := 138961 ), rec( content := ", then t\ he default value is\n ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "A", next := 139002, root := ~, start := 138994, stop := 139001 ), rec( content := ". ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "P", next := 139008, root := ~, start := 139004, stop := 139007 ), rec( content := "\n\n \ If ", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "A", next := 139031, root := ~, start := 139017, stop := 139030 ), rec( content := " is a di\ graph created from a record with a component\n ", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertices", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "C", next := 139103, root := ~, start := 139088, stop := 139102 ), rec( content := ", then t\ he labels of the vertices are set to the value of\n this component.", count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 8 ], mathmode := "Text", name := "P", next := 139184, root := ~, start := 139180, stop := 139183 ), rec( content := "\n\n \ Induced subdigraphs, and some other operations which create new digraphs from\ \n old ones, inherit their labels from their parents.\n\n ", count := [ 5, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphVertexLabel(D, 3);\n3\ngap> D := Digraph([\"a\ \", \"b\", \"c\"], [], []);\n\ngap> D\ igraphVertexLabel(D, 2);\n\"b\"\ngap> SetDigraphVertexLabel(D, 2, \"d\");\ngap\ > DigraphVertexLabel(D, 2);\n\"d\"\ngap> D := InducedSubdigraph(D, [1, 2]);\n<\ immutable empty digraph with 2 vertices>\ngap> DigraphVertexLabel(D, 2);\n\"d\ \"\ngap> D := Digraph(IsMutableDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> DigraphVertexLabel(D, 1);\n\"e\"\ngap\ > SetDigraphVertexLabel(D, 1, \"h\");\ngap> DigraphVertexLabel(D, 1);\n\"h\"\n\ gap> InducedSubdigraph(D, [1, 2]);\n\ng\ ap> DigraphVertexLabel(D, 1);\n\"h\"\n", count := [ 5, 1, 8, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 8, 9 ], mathmode := "Text", name := "Example", next := 140120, root := ~, start := 139328, stop := 140119 ), rec( content := "\n ", count := [ 5, 1, 8, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "Description", next := 140137, root := ~, start := 138570, stop := 140136 ) ], count := [ 5, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 140151, root := ~, start := 138438, stop := 140150 ), rec( content := "\n\n ", count := [ 5, 1, 8, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphVertexLa\ bels" ), content := 0, count := [ 5, 1, 9, 2 ], mathmode := "Text", name := "Oper", next := 140220, root := ~, start := 140172, stop := 140219 ), rec( attributes := rec( Arg := "digraph, list", Name := "SetDigraphVerte\ xLabels" ), content := 0, count := [ 5, 1, 9, 4 ], mathmode := "Text", name := "Oper", next := 140280, root := ~, start := 140223, stop := 140279 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 140318, root := ~, start := 140304, stop := 140317 ), rec( content := " is a di\ graph, then ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabels", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "C", next := 140364, root := ~, start := 140338, stop := 140363 ), rec( content := " returns\ a\n copy of the labels of the vertices in ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 140431, root := ~, start := 140417, stop := 140430 ), rec( content := ".\n " , count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphVertexLabels", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "C", next := 140466, root := ~, start := 140437, stop := 140465 ), rec( content := " can be \ used to set the labels of the vertices\n in ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 140534, root := ~, start := 140520, stop := 140533 ), rec( content := " to the \ list of\n arbitrary ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 140589, root := ~, start := 140578, stop := 140588 ), rec( content := ". ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "P", next := 140595, root := ~, start := 140591, stop := 140594 ), rec( content := "\n\n \ The label of a vertex can be changed an arbitrary number of times. If no\n \ label has been set for the vertex ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "A", next := 140720, root := ~, start := 140712, stop := 140719 ), rec( content := ", then t\ he default value is\n ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "A", next := 140760, root := ~, start := 140752, stop := 140759 ), rec( content := ". ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "P", next := 140766, root := ~, start := 140762, stop := 140765 ), rec( content := "\n\n \ If ", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "A", next := 140789, root := ~, start := 140775, stop := 140788 ), rec( content := " is a di\ graph created from a record with a component\n ", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertices", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "C", next := 140861, root := ~, start := 140846, stop := 140860 ), rec( content := ", then t\ he labels of the vertices are set to the value of\n this component.", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 9, 8 ], mathmode := "Text", name := "P", next := 140942, root := ~, start := 140938, stop := 140941 ), rec( content := "\n\n \ Induced subdigraphs, and other operations which create new digraphs from\n \ old ones, inherit their labels from their parents.\n\n ", count := [ 5, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphVertexLabels(D);\n[ 1 .. 5 ]\ngap> D := Digra\ ph([\"a\", \"b\", \"c\"], [], []);\n\ \ngap> DigraphVertexLabels(D);\n[ \"a\", \"b\", \"c\" ]\ngap> SetDigraphVertex\ Label(D, 2, \"d\");\ngap> DigraphVertexLabels(D);\n[ \"a\", \"d\", \"c\" ]\nga\ p> D := InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"a\", \"c\" ]\ngap> D := Digraph(IsMutabl\ eDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> SetDigraphVertexLabels(D, [\"h\", \"i\", \"j\"]);\ngap> DigraphVert\ exLabels(D);\n[ \"h\", \"i\", \"j\" ]\ngap> InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"h\", \ \"j\" ]\n", count := [ 5, 1, 9, 9 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 9 ], mathmode := "Text", name := "Example", next := 141905, root := ~, start := 141081, stop := 141904 ), rec( content := "\n ", count := [ 5, 1, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "Description", next := 141922, root := ~, start := 140283, stop := 141921 ) ], count := [ 5, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 141936, root := ~, start := 140157, stop := 141935 ), rec( content := "\n\n ", count := [ 5, 1, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i, j", Name := "DigraphEdgeLabe\ l" ), content := 0, count := [ 5, 1, 10, 2 ], mathmode := "Text", name := "Oper", next := 142010, root := ~, start := 141957, stop := 142009 ), rec( attributes := rec( Arg := "digraph, i, j, o\ bj", Name := "SetDigraphEdgeLabel" ), content := 0, count := [ 5, 1, 10, 4 ], mathmode := "Text", name := "Oper", next := 142074, root := ~, start := 142013, stop := 142073 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142112, root := ~, start := 142098, stop := 142111 ), rec( content := " is a di\ graph without multiple edges, then the first\n operation returns the label \ of the edge from vertex ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142229, root := ~, start := 142221, stop := 142228 ), rec( content := " to vert\ ex\n ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142252, root := ~, start := 142244, stop := 142251 ), rec( content := ". The s\ econd operation can be used to set the label of the edge\n between vertex " , count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142344, root := ~, start := 142336, stop := 142343 ), rec( content := " and ver\ tex ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142364, root := ~, start := 142356, stop := 142363 ), rec( content := " to the \ arbitrary ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object\\ n ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ bj", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 142409, root := ~, start := 142399, stop := 142408 ), rec( content := ". ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "P", next := 142415, root := ~, start := 142411, stop := 142414 ), rec( content := "\n\n \ The label of an edge can be changed an arbitrary number of times. If no\n l\ abel has been set for the edge, then the default value is ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "A", next := 142564, root := ~, start := 142556, stop := 142563 ), rec( content := ". ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "P", next := 142570, root := ~, start := 142566, stop := 142569 ), rec( content := "\n\n \ Induced subdigraphs, and some other operations which create new digraphs from\ \n old ones, inherit their edge labels from their parents.\n\n See also \ ", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Oper := "Digraph\ EdgeLabels" ), content := 0, count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "Ref", next := 142759, root := ~, start := 142728, stop := 142758 ), rec( content := ".\n\n \ ", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphEdgeLabel(D, 3, 1);\n1\ngap> SetDigraphEdgeLa\ bel(D, 2, 5, [42]);\ngap> DigraphEdgeLabel(D, 2, 5);\n[ 42 ]\ngap> D := Induce\ dSubdigraph(D, [2, 5]);\n\ngap> Di\ graphEdgeLabel(D, 1, 2);\n[ 42 ]\ngap> D := ChainDigraph(IsMutableDigraph, 5);\ \n\ngap> DigraphEdgeLabel(D, 2, 3);\ \n1\ngap> SetDigraphEdgeLabel(D, 4, 5, [1729]);\ngap> DigraphEdgeLabel(D, 4, 5\ );\n[ 1729 ]\ngap> InducedSubdigraph(D, [4, 5]);\n\ngap> DigraphEdgeLabel(D, 1, 2);\n[ 1729 ]\n", count := [ 5, 1, 10, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "Example", next := 143454, root := ~, start := 142766, stop := 143453 ), rec( content := "\n ", count := [ 5, 1, 10, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "Description", next := 143471, root := ~, start := 142077, stop := 143470 ) ], count := [ 5, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 143485, root := ~, start := 141942, stop := 143484 ), rec( content := "\n\n ", count := [ 5, 1, 10, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphEdgeLabe\ ls" ), content := 0, count := [ 5, 1, 11, 2 ], mathmode := "Text", name := "Oper", next := 143552, root := ~, start := 143506, stop := 143551 ), rec( attributes := rec( Arg := "digraph, labels" , Label := "for a digraph \ and a list of lists", Name := "SetDigraphEdgeLabels" ), content := 0, count := [ 5, 1, 11, 4 ], mathmode := "Text", name := "Oper", next := 143658, root := ~, start := 143555, stop := 143657 ), rec( attributes := rec( Arg := "digraph, func", Label := "for a digraph \ and a function", Name := "SetDigraphEdgeLabels" ), content := 0, count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "Oper", next := 143757, root := ~, start := 143661, stop := 143756 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 143795, root := ~, start := 143781, stop := 143794 ), rec( content := " is a di\ graph without multiple edges, then\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeLabels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143866, root := ~, start := 143842, stop := 143865 ), rec( content := " returns\ a copy of the labels of the edges\n in ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 143930, root := ~, start := 143916, stop := 143929 ), rec( content := " as a li\ st of lists ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143963, root := ~, start := 143950, stop := 143962 ), rec( content := " such th\ at\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels[i][j]", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 143997, root := ~, start := 143978, stop := 143996 ), rec( content := " is the \ label on the edge from vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144043, root := ~, start := 144035, stop := 144042 ), rec( content := "\n to\ vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours(digraph)[i][j]", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144093, root := ~, start := 144058, stop := 144092 ), rec( content := ".\n\n \ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphEdgeLabels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144127, root := ~, start := 144100, stop := 144126 ), rec( content := " can be \ used to set the labels of the edges in\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 144192, root := ~, start := 144178, stop := 144191 ), rec( content := " without\ multiple edges to the list ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 144241, root := ~, start := 144228, stop := 144240 ), rec( content := " of list\ s of\n arbitrary ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ such that ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[i][j]", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144309, root := ~, start := 144292, stop := 144308 ), rec( content := " is the \ label on the edge\n from vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144359, root := ~, start := 144351, stop := 144358 ), rec( content := " to the \ vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours(digraph>[i][j]", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144409, root := ~, start := 144374, stop := 144408 ), rec( content := ".\n\n \ Alternatively ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphEdgeLabels", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144457, root := ~, start := 144430, stop := 144456 ), rec( content := " can be \ called with binary function\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "A", next := 144508, root := ~, start := 144497, stop := 144507 ), rec( content := " that as\ its second argument that when passed two vertices ", count := [ 5, 1, 11, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144575, root := ~, start := 144567, stop := 144574 ), rec( content := "\n an\ d ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144592, root := ~, start := 144584, stop := 144591 ), rec( content := " returns\ the label for the edge between vertex ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144647, root := ~, start := 144639, stop := 144646 ), rec( content := " and ver\ tex\n ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "C", next := 144671, root := ~, start := 144663, stop := 144670 ), rec( content := ". ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "P", next := 144677, root := ~, start := 144673, stop := 144676 ), rec( content := "\n\n \ The label of an edge can be changed an arbitrary number of times. If no\n l\ abel has been set for an edge, then the default value is ", count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "C", next := 144825, root := ~, start := 144817, stop := 144824 ), rec( content := ". ", count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 11, 9 ], mathmode := "Text", name := "P", next := 144831, root := ~, start := 144827, stop := 144830 ), rec( content := "\n\n \ Induced subdigraphs, and some other operations which create new digraphs\n \ from old ones, inherit their labels from their parents.\n\n ", count := [ 5, 1, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphEdgeLabels(D);\n[ [ 1 ], [ 1, 1, 1 ], [ 1 ], \ [ 1, 1, 1 ], [ 1, 1, 1 ] ]\ngap> SetDigraphEdgeLabel(D, 2, 1, \"d\");\ngap> Di\ graphEdgeLabels(D);\n[ [ 1 ], [ \"d\", 1, 1 ], [ 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ]\ ]\ngap> D := InducedSubdigraph(D, [1, 2, 3]);\n\ngap> DigraphEdgeLabels(D);\n[ [ 1 ], [ \"d\", 1 ], [ 1 ] ]\nga\ p> OutNeighbours(D);\n[ [ 3 ], [ 1, 3 ], [ 1 ] ]\ngap> D := CompleteBipartiteD\ igraph(IsMutableDigraph, 2, 3);\n\n\ gap> DigraphEdgeLabels(D);\n[ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ \ 1, 1 ] ]\ngap> SetDigraphEdgeLabel(D, 2, 4, \"a\");\ngap> DigraphEdgeLabels(D)\ ;\n[ [ 1, 1, 1 ], [ 1, \"a\", 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ]\ngap> Induce\ dSubdigraph(D, [1, 2, 3, 4]);\n\ngap\ > DigraphEdgeLabels(D);\n[ [ 1, 1 ], [ 1, \"a\" ], [ 1, 1 ], [ 1, 1 ] ]\ngap> \ OutNeighbors(D);\n[ [ 3, 4 ], [ 3, 4 ], [ 1, 2 ], [ 1, 2 ] ]\n", count := [ 5, 1, 11, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 10 ], mathmode := "Text", name := "Example", next := 146042, root := ~, start := 144975, stop := 146041 ), rec( content := "\n ", count := [ 5, 1, 11, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "Description", next := 146059, root := ~, start := 143760, stop := 146058 ) ], count := [ 5, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 146073, root := ~, start := 143491, stop := 146072 ), rec( content := "\n\n ", count := [ 5, 1, 11, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphInEdges"\ ), content := 0, count := [ 5, 1, 12, 2 ], mathmode := "Text", name := "Oper", next := 146145, root := ~, start := 146094, stop := 146144 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ edges.", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 146183, root := ~, start := 146148, stop := 146182 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphInEdges", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "C", next := 146225, root := ~, start := 146204, stop := 146224 ), rec( content := " returns\ the list of all edges of ", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "A", next := 146273, root := ~, start := 146259, stop := 146272 ), rec( content := "\n wh\ ich have ", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "A", next := 146302, root := ~, start := 146289, stop := 146301 ), rec( content := " as thei\ r range.\n ", count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n\ngap> DigraphInEdges(D, 2);\n[ [ 1, 2 ], [ 1, 2 ] ]\ \n", count := [ 5, 1, 12, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 4 ], mathmode := "Text", name := "Example", next := 146508, root := ~, start := 146323, stop := 146507 ), rec( content := "\n ", count := [ 5, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 146525, root := ~, start := 146186, stop := 146524 ) ], count := [ 5, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 146539, root := ~, start := 146079, stop := 146538 ), rec( content := "\n\n ", count := [ 5, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphOutEdges\ " ), content := 0, count := [ 5, 1, 13, 2 ], mathmode := "Text", name := "Oper", next := 146612, root := ~, start := 146560, stop := 146611 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ edges.", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "Returns", next := 146650, root := ~, start := 146615, stop := 146649 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOutEdges", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "C", next := 146693, root := ~, start := 146671, stop := 146692 ), rec( content := " returns\ the list of all edges of ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "A", next := 146741, root := ~, start := 146727, stop := 146740 ), rec( content := "\n wh\ ich have ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "A", next := 146770, root := ~, start := 146757, stop := 146769 ), rec( content := " as thei\ r source.\n ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n\ngap> DigraphOutEdges(D, 2);\n[ [ 2, 3 ], [ 2, 3 ] \ ]\n", count := [ 5, 1, 13, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 4 ], mathmode := "Text", name := "Example", next := 146978, root := ~, start := 146792, stop := 146977 ), rec( content := "\n ", count := [ 5, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 146995, root := ~, start := 146653, stop := 146994 ) ], count := [ 5, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 147009, root := ~, start := 146545, stop := 147008 ), rec( content := "\n\n ", count := [ 5, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Label := "for digraph an\ d list", Name := "IsDigraphEdge" ), content := 0, count := [ 5, 1, 14, 2 ], mathmode := "Text", name := "Oper", next := 147115, root := ~, start := 147030, stop := 147114 ), rec( attributes := rec( Arg := "digraph, u, v", Label := "for digraph an\ d two pos ints", Name := "IsDigraphEdge" ), content := 0, count := [ 5, 1, 14, 4 ], mathmode := "Text", name := "Oper", next := 147211, root := ~, start := 147118, stop := 147210 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 147234, root := ~, start := 147223, stop := 147233 ), rec( content := " or ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 147250, root := ~, start := 147238, stop := 147249 ), rec( content := ".", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "Returns", next := 147261, root := ~, start := 147214, stop := 147260 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, this function returns ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 147334, root := ~, start := 147323, stop := 147333 ), rec( content := " if and \ only if the list\n ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 147374, root := ~, start := 147363, stop := 147373 ), rec( content := " specifi\ es an edge in the digraph ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 147422, root := ~, start := 147408, stop := 147421 ), rec( content := ". Speci\ fically,\n this operation returns ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 147477, root := ~, start := 147466, stop := 147476 ), rec( content := " if ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 147492, root := ~, start := 147481, stop := 147491 ), rec( content := " is a pa\ ir of positive\n integers where ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 147545, root := ~, start := 147534, stop := 147544 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "C", next := 147555, root := ~, start := 147545, stop := 147554 ), rec( content := " is the \ source and ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 147585, root := ~, start := 147574, stop := 147584 ), rec( attributes := rec( ), content := [ rec( content := "[\ 2]", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "C", next := 147595, root := ~, start := 147585, stop := 147594 ), rec( content := "\n is\ the range of an edge in ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "A", next := 147641, root := ~, start := 147627, stop := 147640 ), rec( content := ", and ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "K", next := 147659, root := ~, start := 147647, stop := 147658 ), rec( content := " otherwi\ se. ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "P", next := 147675, root := ~, start := 147671, stop := 147674 ), rec( content := "\n\n \ The second form simply returns ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "K", next := 147723, root := ~, start := 147712, stop := 147722 ), rec( content := " if ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "u", count := [ 5, 1, 14, 6 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := \ "Text", name := "A", next := 147739, root := ~, start := 147731, stop := 14773\ 8 ), rec( content := ", ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := \ "Text", name := "A", next := 147749, root := ~, start := 147741, stop := 14774\ 8 ), rec( content := "]", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "C", next := 147754, root := ~, start := 147727, stop := 147753 ), rec( content := " is\n \ an edge in ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "A", next := 147787, root := ~, start := 147773, stop := 147786 ), rec( content := ", and ", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "K", next := 147805, root := ~, start := 147793, stop := 147804 ), rec( content := " otherwi\ se.\n\n", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [6], [], [3], [], [1]]);\n\ngap> IsDigraphEdge(D, [1, 1]);\nfalse\ngap> IsDigra\ phEdge(D, [1, 2]);\ntrue\ngap> IsDigraphEdge(D, [1, 8]);\nfalse\n", count := [ 5, 1, 14, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 14, 7 ], mathmode := "Text", name := "Example", next := 148062, root := ~, start := 147818, stop := 148061 ), rec( content := "\n ", count := [ 5, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "Description", next := 148079, root := ~, start := 147264, stop := 148078 ) ], count := [ 5, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 148093, root := ~, start := 147015, stop := 148092 ), rec( content := "\n\n ", count := [ 5, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Name := "IsMatching" ), content := 0, count := [ 5, 1, 15, 2 ], mathmode := "Text", name := "Oper", next := 148159, root := ~, start := 148114, stop := 148158 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsMaximalMatchi\ ng" ), content := 0, count := [ 5, 1, 15, 4 ], mathmode := "Text", name := "Oper", next := 148214, root := ~, start := 148162, stop := 148213 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsMaximumMatchi\ ng" ), content := 0, count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "Oper", next := 148269, root := ~, start := 148217, stop := 148268 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsPerfectMatchi\ ng" ), content := 0, count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "Oper", next := 148324, root := ~, start := 148272, stop := 148323 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "K", next := 148347, root := ~, start := 148336, stop := 148346 ), rec( content := " or ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "K", next := 148363, root := ~, start := 148351, stop := 148362 ), rec( content := ".", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "Returns", next := 148374, root := ~, start := 148327, stop := 148373 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "A", next := 148412, root := ~, start := 148398, stop := 148411 ), rec( content := " is a di\ graph and ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "A", next := 148441, root := ~, start := 148430, stop := 148440 ), rec( content := " is a li\ st of pairs of\n vertices of ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "A", next := 148494, root := ~, start := 148480, stop := 148493 ), rec( content := ", then " , count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMatching", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "C", next := 148518, root := ~, start := 148501, stop := 148517 ), rec( content := " returns\ ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "K", next := 148538, root := ~, start := 148527, stop := 148537 ), rec( content := " if\n \ ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "A", next := 148557, root := ~, start := 148546, stop := 148556 ), rec( content := " is a ma\ tching of ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "A", next := 148589, root := ~, start := 148575, stop := 148588 ), rec( content := ". The o\ peration\n ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalMatching", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "C", next := 148634, root := ~, start := 148610, stop := 148633 ), rec( content := " returns\ ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "K", next := 148654, root := ~, start := 148643, stop := 148653 ), rec( content := " if ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "A", next := 148669, root := ~, start := 148658, stop := 148668 ), rec( content := " is a ma\ ximal \n matching, ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximumMatching", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "C", next := 148722, root := ~, start := 148698, stop := 148721 ), rec( content := " returns\ ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "K", next := 148742, root := ~, start := 148731, stop := 148741 ), rec( content := " if ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "A", next := 148757, root := ~, start := 148746, stop := 148756 ), rec( content := " is a\n \ maximum matching and ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPerfectMatching", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "C", next := 148812, root := ~, start := 148788, stop := 148811 ), rec( content := " returns\ ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "K", next := 148832, root := ~, start := 148821, stop := 148831 ), rec( content := " if \n \ ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "A", next := 148852, root := ~, start := 148841, stop := 148851 ), rec( content := " is a pe\ rfect, matching of ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "A", next := 148893, root := ~, start := 148879, stop := 148892 ), rec( content := ", respec\ tively. \n Otherwise, each of these operations return ", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "K", next := 148970, root := ~, start := 148958, stop := 148969 ), rec( content := ".\n " , count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "P", next := 148980, root := ~, start := 148976, stop := 148979 ), rec( content := "\n\n \ A ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ atching", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "E", next := 149003, root := ~, start := 148988, stop := 149002 ), rec( content := " ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "C", next := 149012, root := ~, start := 149004, stop := 149011 ), rec( content := " of a di\ graph ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "A", next := 149040, root := ~, start := 149026, stop := 149039 ), rec( content := " is a su\ bset of the\n edges of ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "A", next := 149087, root := ~, start := 149073, stop := 149086 ), rec( content := ", i.e. " , count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 5, 1, 15, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := \ "Text", name := "A", next := 149124, root := ~, start := 149110, stop := 14912\ 3 ), rec( content := ")", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "C", next := 149129, root := ~, start := 149094, stop := 149128 ), rec( content := ", such\n\ that no pair of distinct edges in ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "C", next := 149182, root := ~, start := 149174, stop := 149181 ), rec( content := " are inc\ ident to the same vertex\n of ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "A", next := 149236, root := ~, start := 149222, stop := 149235 ), rec( content := ". Note \ that this definition allows a matching to contain\n loops. See ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "Ref", next := 149340, root := ~, start := 149310, stop := 149339 ), rec( content := ". The m\ atching ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "C", next := 149364, root := ~, start := 149356, stop := 149363 ), rec( content := " is\n \ ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "E", next := 149386, root := ~, start := 149372, stop := 149385 ), rec( content := " if it i\ s contained in no larger matching of the digraph,\n is ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximum", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "E", next := 149465, root := ~, start := 149451, stop := 149464 ), rec( content := " if it h\ as the greatest cardinality among all matchings and\n is ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erfect", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "E", next := 149546, root := ~, start := 149532, stop := 149545 ), rec( content := " if ever\ y vertex of the digraph is incident to an edge in\n the matching. Every ma\ ximum or perfect matching is maximal. Note, however,\n that not every perfe\ ct matching of digraphs with loops is maximum.\n\n ", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [1, 2], [2, 3, 4], [3, 5], [1]]);\n\ngap> IsMatching(D, [[2, 1], [3, 2]]);\nfalse\n\ gap> edges := [[3, 2]];;\ngap> IsMatching(D, edges);\ntrue\ngap> IsMaximalMatc\ hing(D, edges);\nfalse\ngap> edges := [[2, 1], [3, 4]];;\ngap> IsMaximalMatchi\ ng(D, edges);\ntrue\ngap> IsPerfectMatching(D, edges);\nfalse\ngap> edges := [\ [1, 2], [3, 3], [4, 5]];;\ngap> IsPerfectMatching(D, edges);\ntrue\ngap> IsMax\ imumMatching(D, edges);\nfalse\ngap> edges := [[1, 1], [2, 2], [3, 3], [4, 5]]\ ;;\ngap> IsMaximumMatching(D, edges);\ntrue\n", count := [ 5, 1, 15, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 15, 11 ], mathmode := "Text", name := "Example", next := 150360, root := ~, start := 149760, stop := 150359 ), rec( content := "\n ", count := [ 5, 1, 15, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "Description", next := 150377, root := ~, start := 148377, stop := 150376 ) ], count := [ 5, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 150391, root := ~, start := 148099, stop := 150390 ), rec( content := "\n\n ", count := [ 5, 1, 15, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalM\ atching" ), content := 0, count := [ 5, 1, 16, 2 ], mathmode := "Text", name := "Attr", next := 150463, root := ~, start := 150412, stop := 150462 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ pairs of vertices.", count := [ 5, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 3 ], mathmode := "Text", name := "Returns", next := 150513, root := ~, start := 150466, stop := 150512 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a maximal matching of the digraph ", count := [ 5, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 3 ], mathmode := "Text", name := "A", next := 150604, root := ~, start := 150590, stop := 150603 ), rec( content := ".\n " , count := [ 5, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 16, 4 ], mathmode := "Text", name := "P", next := 150614, root := ~, start := 150610, stop := 150613 ), rec( content := "\n\n \ For the definition of a maximal matching, see ", count := [ 5, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alMatching" ), content := 0, count := [ 5, 1, 16, 4 ], mathmode := "Text", name := "Ref", next := 150697, root := ~, start := 150666, stop := 150696 ), rec( content := ". \n\n \ ", count := [ 5, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDiSparse6String(\".IeAoXCJU@|SHAe?d\");\n\ngap> M := DigraphMaximalMatching(D);; IsMaxim\ alMatching(D, M);\ntrue\ngap> D := RandomDigraph(100);;\ngap> IsMaximalMatchin\ g(D, DigraphMaximalMatching(D));\ntrue\ngap> D := GeneralisedPetersenGraph(IsM\ utableDigraph, 9, 2);\n\ngap> IsMa\ ximalMatching(D, DigraphMaximalMatching(D));\ntrue\n", count := [ 5, 1, 16, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 16, 5 ], mathmode := "Text", name := "Example", next := 151165, root := ~, start := 150705, stop := 151164 ), rec( content := "\n ", count := [ 5, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 3 ], mathmode := "Text", name := "Description", next := 151182, root := ~, start := 150516, stop := 151181 ) ], count := [ 5, 1, 16, 1 ], mathmode := "Text", name := "ManSection", next := 151196, root := ~, start := 150397, stop := 151195 ), rec( content := "\n\n ", count := [ 5, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximumM\ atching" ), content := 0, count := [ 5, 1, 17, 2 ], mathmode := "Text", name := "Attr", next := 151268, root := ~, start := 151217, stop := 151267 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ pairs of vertices.", count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "Returns", next := 151318, root := ~, start := 151271, stop := 151317 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a maximum matching of the digraph ", count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "A", next := 151409, root := ~, start := 151395, stop := 151408 ), rec( content := ".\n " , count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "P", next := 151419, root := ~, start := 151415, stop := 151418 ), rec( content := "\n\n \ For the definition of a maximum matching, see ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ umMatching" ), content := 0, count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "Ref", next := 151502, root := ~, start := 151471, stop := 151501 ), rec( content := ".\n I\ f ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "A", next := 151525, root := ~, start := 151511, stop := 151524 ), rec( content := " is bipa\ rtite (see ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsBipar\ titeDigraph" ), content := 0, count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "Ref", next := 151576, root := ~, start := 151544, stop := 151575 ), rec( content := "), then \ \n the algorithm used has complexity ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m*sqrt(n))", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "C", next := 151642, root := ~, start := 151623, stop := 151641 ), rec( content := ". Otherw\ ise for general \n graphs the complexity is ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m*n*log(n))", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "C", next := 151716, root := ~, start := 151696, stop := 151715 ), rec( content := ". Here " , count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "C", next := 151731, root := ~, start := 151723, stop := 151730 ), rec( content := " is the \ number of vertices \n and ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "C", next := 151775, root := ~, start := 151767, stop := 151774 ), rec( content := " is the \ number of edges.\n\n ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&I@EA_A?AdDp[_c??OO\");\n\ngap> M := DigraphMaximumMatching(D);; IsMaxi\ malMatching(D, M);\ntrue\ngap> Length(M);\n5\ngap> D := Digraph([[5, 6, 7, 8],\ [6, 7, 8], [7, 8], [8], \n> [], [], [], []]);;\ngap> M := Di\ graphMaximumMatching(D);\n[ [ 1, 5 ], [ 2, 6 ], [ 3, 7 ], [ 4, 8 ] ]\ngap> D :\ = GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> M := DigraphMaximumMatching(D);;\ngap> IsMaximalMatc\ hing(D, M);\ntrue\ngap> Length(M);\n9\n", count := [ 5, 1, 17, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 17, 5 ], mathmode := "Text", name := "Example", next := 152402, root := ~, start := 151805, stop := 152401 ), rec( content := "\n ", count := [ 5, 1, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "Description", next := 152419, root := ~, start := 151321, stop := 152418 ) ], count := [ 5, 1, 17, 1 ], mathmode := "Text", name := "ManSection", next := 152433, root := ~, start := 151202, stop := 152432 ), rec( content := "\n\n ", count := [ 5, 1, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 152447, root := ~, start := 132056, stop := 152446 ) , rec( content := "\n\n ", count := [ 5, 1, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Neighbours and de\ gree", count := [ 5, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 152500, root := ~, start := 152460, stop := 152499 ), rec( content := "\n ", count := [ 5, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AdjacencyMatrix\ " ), content := 0, count := [ 5, 2, 1, 2 ], mathmode := "Text", name := "Attr", next := 152564, root := ~, start := 152520, stop := 152563 ), rec( attributes := rec( Arg := "digraph", Name := "AdjacencyMatrix\ MutableCopy" ), content := 0, count := [ 5, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 152622, root := ~, start := 152567, stop := 152621 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of non-negative integers.", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 152685, root := ~, start := 152625, stop := 152684 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the adjacency matrix ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 152759, root := ~, start := 152749, stop := 152758 ), rec( content := " of the \ digraph\n ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "A", next := 152793, root := ~, start := 152779, stop := 152792 ), rec( content := ".\n T\ he value of the matrix entry ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 152845, root := ~, start := 152829, stop := 152844 ), rec( content := " is the \ number of edges\n in ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "A", next := 152890, root := ~, start := 152876, stop := 152889 ), rec( content := " with so\ urce ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 152911, root := ~, start := 152903, stop := 152910 ), rec( content := " and ran\ ge ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 152930, root := ~, start := 152922, stop := 152929 ), rec( content := ". If ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "A", next := 152949, root := ~, start := 152935, stop := 152948 ), rec( content := "\n ha\ s no vertices, then the empty list is returned. ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "P", next := 153008, root := ~, start := 153004, stop := 153007 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacencyMatrix", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "C", next := 153049, root := ~, start := 153027, stop := 153048 ), rec( content := " returns\ an immutable list of\n lists, whereas the function ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacencyMatrixMutableCopy", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "C", next := 153144, root := ~, start := 153111, stop := 153143 ), rec( content := " returns\ a copy\n of ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacencyMatrix", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "C", next := 153189, root := ~, start := 153167, stop := 153188 ), rec( content := " that is\ a mutable list of mutable lists. ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 1, 7 ], mathmode := "Text", name := "P", next := 153235, root := ~, start := 153231, stop := 153234 ), rec( content := "\n\n \ ", count := [ 5, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [2, 2, 2], [1, 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, \ 3, 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],\n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4,\ 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\n\ngap> mat := AdjacencyMatrix(gr);;\ngap> Display(mat);\n[ [ 0, \ 3, 0, 0, 0, 0, 0, 0, 0, 0 ],\n [ 1, 0, 1, 0, 0, 1, 0, 1, 1, \ 1 ],\n [ 0, 0, 0, 1, 0, 1, 0, 1, 0, 0 ],\n [ 1, 1, 1, 0, 0, \ 0, 0, 0, 1, 0 ],\n [ 0, 0, 2, 0, 0, 0, 0, 0, 0, 0 ],\n [ 0, \ 0, 1, 0, 1, 1, 0, 0, 0, 1 ],\n [ 1, 1, 0, 0, 0, 0, 1, 0, 0,\ 0 ],\n [ 1, 1, 1, 0, 1, 1, 0, 0, 0, 2 ],\n [ 1, 0, 1, 1, 1,\ 0, 0, 1, 0, 1 ],\n [ 0, 1, 1, 1, 0, 1, 1, 0, 0, 1 ] ]\ngap> D\ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> Display(AdjacencyMatrix(D));\n[ [ 0, 1, 0 ],\n [ 0, 0, 1 ],\ \n [ 1, 0, 0 ] ]\n", count := [ 5, 2, 1, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 8 ], mathmode := "Text", name := "Example", next := 154205, root := ~, start := 153241, stop := 154204 ), rec( content := "\n ", count := [ 5, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 154222, root := ~, start := 152688, stop := 154221 ) ], count := [ 5, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 154236, root := ~, start := 152505, stop := 154235 ), rec( content := "\n\n ", count := [ 5, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "CharacteristicP\ olynomial" ), content := 0, count := [ 5, 2, 2, 2 ], mathmode := "Text", name := "Attr", next := 154310, root := ~, start := 154257, stop := 154309 ), rec( attributes := rec( ), content := [ rec( content := "A polynom\ ial with integer coefficients.", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 154371, root := ~, start := 154313, stop := 154370 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the characteristic polynomial of the digraph\n ", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "A", next := 154478, root := ~, start := 154464, stop := 154477 ), rec( content := ". That i\ s it returns the characteristic polynomial\n of the adjacency matrix of t\ he digraph ", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "A", next := 154588, root := ~, start := 154574, stop := 154587 ), rec( content := "\n ", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 2], [1, 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, 3\ , 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],\n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4, \ 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\n\ngap> CharacteristicPolynomial(D);\nx_1^10-3*x_1^9-7*x_1^8-x_1^7+1\ 4*x_1^6+x_1^5-26*x_1^4+51*x_1^3-10*x_1^2\\\n+18*x_1-30\ngap> D := CompleteDigr\ aph(5);\n\ngap> CharacteristicPoly\ nomial(D);\nx_1^5-10*x_1^3-20*x_1^2-15*x_1-4\ngap> D := CycleDigraph(IsMutable\ Digraph, 3);\n\ngap> CharacteristicP\ olynomial(D);\nx_1^3-1\n", count := [ 5, 2, 2, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 4 ], mathmode := "Text", name := "Example", next := 155254, root := ~, start := 154593, stop := 155253 ), rec( content := "\n ", count := [ 5, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 155271, root := ~, start := 154374, stop := 155270 ) ], count := [ 5, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 155285, root := ~, start := 154242, stop := 155284 ), rec( content := "\n\n ", count := [ 5, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "BooleanAdjacenc\ yMatrix" ), content := 0, count := [ 5, 2, 3, 2 ], mathmode := "Text", name := "Attr", next := 155357, root := ~, start := 155306, stop := 155356 ), rec( attributes := rec( Arg := "digraph", Name := "BooleanAdjacenc\ yMatrixMutableCopy" ), content := 0, count := [ 5, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 155422, root := ~, start := 155360, stop := 155421 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of booleans.", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "Returns", next := 155472, root := ~, start := 155425, stop := 155471 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 155510, root := ~, start := 155496, stop := 155509 ), rec( content := " is a di\ graph with a positive number of vertices\n ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 155571, root := ~, start := 155563, stop := 155570 ), rec( content := ", then " , count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrix(", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 155608, root := ~, start := 155578, stop := 155607 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 155622, root := ~, start := 155608, stop := 155621 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 155630, root := ~, start := 155622, stop := 155629 ), rec( content := "\n re\ turns the boolean adjacency matrix ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 155682, root := ~, start := 155672, stop := 155681 ), rec( content := " of ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 155700, root := ~, start := 155686, stop := 155699 ), rec( content := ". The\n\ value of the matrix entry ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[j][i]", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 155753, root := ~, start := 155737, stop := 155752 ), rec( content := " is ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "K", next := 155768, root := ~, start := 155757, stop := 155767 ), rec( content := " if and \ only if\n there exists an edge in ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 155826, root := ~, start := 155812, stop := 155825 ), rec( content := " with so\ urce ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 155847, root := ~, start := 155839, stop := 155846 ), rec( content := " and ran\ ge\n ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 155870, root := ~, start := 155862, stop := 155869 ), rec( content := ". If ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 155890, root := ~, start := 155876, stop := 155889 ), rec( content := " has no \ vertices, then the empty list is\n returned. ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 3, 6 ], mathmode := "Text", name := "P", next := 155949, root := ~, start := 155945, stop := 155948 ), rec( content := "\n\n \ Note that the boolean adjacency matrix loses information about multiple\n e\ dges. ", count := [ 5, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "P", next := 156043, root := ~, start := 156039, stop := 156042 ), rec( content := "\n\n \ The attribute ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrix", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "C", next := 156092, root := ~, start := 156063, stop := 156091 ), rec( content := " returns\ an immutable list of\n immutable lists, whereas the function\n ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrixMutableCopy", count := [ 5, 2, 3, 7 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "C", next := 156208, root := ~, start := 156168, stop := 156207 ), rec( content := " returns\ a copy of the\n ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrix", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "C", next := 156264, root := ~, start := 156235, stop := 156263 ), rec( content := " that is\ a mutable list of mutable lists. ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 3, 8 ], mathmode := "Text", name := "P", next := 156310, root := ~, start := 156306, stop := 156309 ), rec( content := "\n ", count := [ 5, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[3, 4], [2, 3], [1, 2, 4], [4]]);\n\ngap> PrintArray(BooleanAdjacencyMatrix(gr));\n[ [ fal\ se, false, true, true ],\n [ false, true, true, false ],\n [ tr\ ue, true, false, true ],\n [ false, false, false, true ] ]\ngap> gr\ := CycleDigraph(4);;\ngap> PrintArray(BooleanAdjacencyMatrix(gr));\n[ [ fals\ e, true, false, false ],\n [ false, false, true, false ],\n [ fals\ e, false, false, true ],\n [ true, false, false, false ] ]\ngap> Boo\ leanAdjacencyMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMutableD\ igraph, 3);\n\ngap> PrintArray(Boole\ anAdjacencyMatrix(D));\n[ [ false, true, false ],\n [ false, false, t\ rue ],\n [ true, false, false ] ]\n", count := [ 5, 2, 3, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 9 ], mathmode := "Text", name := "Example", next := 157145, root := ~, start := 156315, stop := 157144 ), rec( content := "\n ", count := [ 5, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "Description", next := 157162, root := ~, start := 155475, stop := 157161 ) ], count := [ 5, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 157176, root := ~, start := 155291, stop := 157175 ), rec( content := "\n\n ", count := [ 5, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAdjacenc\ yFunction" ), content := 0, count := [ 5, 2, 4, 2 ], mathmode := "Text", name := "Attr", next := 157250, root := ~, start := 157197, stop := 157249 ), rec( attributes := rec( ), content := [ rec( content := "A functio\ n.", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 157283, root := ~, start := 157253, stop := 157282 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "A", next := 157321, root := ~, start := 157307, stop := 157320 ), rec( content := " is a di\ graph, then ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAdjacencyFunction", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 157372, root := ~, start := 157341, stop := 157371 ), rec( content := " returns\ \n a function which takes two integer parameters ", count := [ 5, 2, 4, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ , y", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 157442, root := ~, start := 157431, stop := 157441 ), rec( content := " and ret\ urns\n ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "K", next := 157470, root := ~, start := 157459, stop := 157469 ), rec( content := " if ther\ e exists an edge from vertex ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 157515, root := ~, start := 157507, stop := 157514 ), rec( content := " to vert\ ex ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "y\ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 157534, root := ~, start := 157526, stop := 157533 ), rec( content := "\n in\ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "A", next := 157556, root := ~, start := 157542, stop := 157555 ), rec( content := " and ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "K", next := 157573, root := ~, start := 157561, stop := 157572 ), rec( content := " if not.\ \n\n ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := Digraph([[1, 2], [3], []]);\n\ngap> foo := DigraphAdjacencyFunction(digraph);\nfunction( u, v )\ ... end\ngap> foo(1, 1);\ntrue\ngap> foo(1, 2);\ntrue\ngap> foo(1, 3);\nfalse\ \ngap> foo(3, 1);\nfalse\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"a\"]);\n\ \ngap> foo := DigraphAdjacenc\ yFunction(gr);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(3, 2\ );\nfalse\ngap> foo(3, 1);\nfalse\ngap> D := CycleDigraph(IsMutableDigraph, 3)\ ;\n\ngap> foo := DigraphAdjacencyFun\ ction(D);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(2, 1);\nf\ alse\n", count := [ 5, 2, 4, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 4 ], mathmode := "Text", name := "Example", next := 158354, root := ~, start := 157587, stop := 158353 ), rec( content := "\n ", count := [ 5, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 158371, root := ~, start := 157286, stop := 158370 ) ], count := [ 5, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 158385, root := ~, start := 157182, stop := 158384 ), rec( content := "\n\n ", count := [ 5, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphRange" ) , content := 0, count := [ 5, 2, 5, 2 ], mathmode := "Text", name := "Attr", next := 158447, root := ~, start := 158406, stop := 158446 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSource" ), content := 0, count := [ 5, 2, 5, 4 ], mathmode := "Text", name := "Attr", next := 158492, root := ~, start := 158450, stop := 158491 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers.", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "Returns", next := 158542, root := ~, start := 158495, stop := 158541 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphRange", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 158582, root := ~, start := 158563, stop := 158581 ), rec( content := " and ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphSource", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 158607, root := ~, start := 158587, stop := 158606 ), rec( content := " return \ the range and source of\n the digraph ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "A", next := 158669, root := ~, start := 158655, stop := 158668 ), rec( content := ". More p\ recisely, position ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 158704, root := ~, start := 158696, stop := 158703 ), rec( content := " in\n \ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphRange(", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 5, 2, 5, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := \ "Text", name := "A", next := 158742, root := ~, start := 158728, stop := 15874\ 1 ), rec( content := ")", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 158747, root := ~, start := 158712, stop := 158746 ), rec( content := " is the \ range of the ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 158776, root := ~, start := 158768, stop := 158775 ), rec( content := "th edge \ of\n ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "A", next := 158805, root := ~, start := 158791, stop := 158804 ), rec( content := ".\n\n \ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [2, 1, 3, 5], [1, 3, 4], [2, 3], [2], [1, 2, 3, 4]]);\ \n\ngap> DigraphRange(gr);\n[ 2, \ 1, 3, 5, 1, 3, 4, 2, 3, 2, 1, 2, 3, 4 ]\ngap> DigraphSource(gr);\n[ 1, 1, 1, 1\ , 2, 2, 2, 3, 3, 4, 5, 5, 5, 5 ]\ngap> DigraphEdges(gr);\n[ [ 1, 2 ], [ 1, 1 ]\ , [ 1, 3 ], [ 1, 5 ], [ 2, 1 ], [ 2, 3 ], \n [ 2, 4 ], [ 3, 2 ], [ 3, 3 ], [ \ 4, 2 ], [ 5, 1 ], [ 5, 2 ], \n [ 5, 3 ], [ 5, 4 ] ]\ngap> D := CycleDigraph(I\ sMutableDigraph, 3);\n\ngap> Digraph\ Range(D);\n[ 2, 3, 1 ]\ngap> DigraphSource(D);\n[ 1, 2, 3 ]\n", count := [ 5, 2, 5, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 5, 6 ], mathmode := "Text", name := "Example", next := 159433, root := ~, start := 158812, stop := 159432 ), rec( content := "\n ", count := [ 5, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "Description", next := 159450, root := ~, start := 158545, stop := 159449 ) ], count := [ 5, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 159464, root := ~, start := 158391, stop := 159463 ), rec( content := "\n\n ", count := [ 5, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "OutNeighbours" ), content := 0, count := [ 5, 2, 6, 2 ], mathmode := "Text", name := "Attr", next := 159527, root := ~, start := 159485, stop := 159526 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighbors" ) , content := 0, count := [ 5, 2, 6, 4 ], mathmode := "Text", name := "Attr", next := 159571, root := ~, start := 159530, stop := 159570 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighboursMu\ tableCopy" ), content := 0, count := [ 5, 2, 6, 6 ], mathmode := "Text", name := "Oper", next := 159627, root := ~, start := 159574, stop := 159626 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighborsMut\ ableCopy" ), content := 0, count := [ 5, 2, 6, 8 ], mathmode := "Text", name := "Oper", next := 159682, root := ~, start := 159630, stop := 159681 ), rec( attributes := rec( ), content := [ rec( content := "The adjac\ encies of a digraph.", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "Returns", next := 159733, root := ~, start := 159685, stop := 159732 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the list ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 159795, root := ~, start := 159785, stop := 159794 ), rec( content := " of out-\ neighbours of each vertex\n of the digraph ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "A", next := 159862, root := ~, start := 159848, stop := 159861 ), rec( content := ".\n M\ ore specifically, a vertex ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 159904, root := ~, start := 159896, stop := 159903 ), rec( content := " appears\ in ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut[i]", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 159929, root := ~, start := 159916, stop := 159928 ), rec( content := " each ti\ me\n there exists an edge with source ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 159985, root := ~, start := 159977, stop := 159984 ), rec( content := " and ran\ ge ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 160004, root := ~, start := 159996, stop := 160003 ), rec( content := " in\n \ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "A", next := 160026, root := ~, start := 160012, stop := 160025 ), rec( content := ". ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "P", next := 160032, root := ~, start := 160028, stop := 160031 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "C", next := 160071, root := ~, start := 160051, stop := 160070 ), rec( content := " returns\ an immutable list of\n lists, whereas the function ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighboursMutableCopy", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "C", next := 160164, root := ~, start := 160133, stop := 160163 ), rec( content := " returns\ a copy\n of ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "C", next := 160207, root := ~, start := 160187, stop := 160206 ), rec( content := " which i\ s a mutable list of mutable lists. ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 6, 11 ], mathmode := "Text", name := "P", next := 160254, root := ~, start := 160250, stop := 160253 ), rec( content := "\n\n \ ", count := [ 5, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"a\", \"c\"]);\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ 1, 3 ], [ ] ]\ngap> \ gr := Digraph([[1, 2, 3], [2, 1], [3]]);\n\ngap> OutNeighbours(gr);\n[ [ 1, 2, 3 ], [ 2, 1 ], [ 3 ] ]\ngap> gr :\ = DigraphByAdjacencyMatrix([\n> [1, 2, 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n<\ immutable multidigraph with 3 vertices, 7 edges>\ngap> OutNeighbours(gr);\n[ [\ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> OutNeighboursMutableCopy(gr);\n[ [ 1, 2\ , 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 \ ], [ 1 ] ]\n", count := [ 5, 2, 6, 12 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 12 ], mathmode := "Text", name := "Example", next := 161042, root := ~, start := 160260, stop := 161041 ), rec( content := "\n ", count := [ 5, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "Description", next := 161059, root := ~, start := 159736, stop := 161058 ) ], count := [ 5, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 161073, root := ~, start := 159470, stop := 161072 ), rec( content := "\n\n ", count := [ 5, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "InNeighbours" ) , content := 0, count := [ 5, 2, 7, 2 ], mathmode := "Text", name := "Attr", next := 161135, root := ~, start := 161094, stop := 161134 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighbors" ), content := 0, count := [ 5, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 161178, root := ~, start := 161138, stop := 161177 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighboursMut\ ableCopy" ), content := 0, count := [ 5, 2, 7, 6 ], mathmode := "Text", name := "Oper", next := 161233, root := ~, start := 161181, stop := 161232 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighborsMuta\ bleCopy" ), content := 0, count := [ 5, 2, 7, 8 ], mathmode := "Text", name := "Oper", next := 161287, root := ~, start := 161236, stop := 161286 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists of vertices.", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "Returns", next := 161337, root := ~, start := 161290, stop := 161336 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the list ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 161399, root := ~, start := 161389, stop := 161398 ), rec( content := " of in-n\ eighbours of each vertex\n of the digraph ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "A", next := 161465, root := ~, start := 161451, stop := 161464 ), rec( content := ".\n M\ ore specifically, a vertex ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 161507, root := ~, start := 161499, stop := 161506 ), rec( content := " appears\ in ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn[i]", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 161532, root := ~, start := 161519, stop := 161531 ), rec( content := " each ti\ me\n there exists an edge with source ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 161588, root := ~, start := 161580, stop := 161587 ), rec( content := " and ran\ ge ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 161607, root := ~, start := 161599, stop := 161606 ), rec( content := " in\n \ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "A", next := 161629, root := ~, start := 161615, stop := 161628 ), rec( content := ". ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "P", next := 161635, root := ~, start := 161631, stop := 161634 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nNeighbours", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "C", next := 161673, root := ~, start := 161654, stop := 161672 ), rec( content := " returns\ an immutable list of\n lists, whereas the function ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nNeighboursMutableCopy", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "C", next := 161765, root := ~, start := 161735, stop := 161764 ), rec( content := " returns\ a copy\n of ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nNeighbours", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "C", next := 161807, root := ~, start := 161788, stop := 161806 ), rec( content := " which i\ s a mutable list of mutable lists. ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "P", next := 161854, root := ~, start := 161850, stop := 161853 ), rec( content := "\n\n \ Note that each entry of ", count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "C", next := 161894, root := ~, start := 161884, stop := 161893 ), rec( content := " is sort\ ed into ascending order. ", count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 7, 12 ], mathmode := "Text", name := "P", next := 161931, root := ~, start := 161927, stop := 161930 ), rec( content := "\n\n \ ", count := [ 5, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"a\", \"c\"]);\n\ngap> InNeighbours(gr);\n[ [ 2 ], [ 1 ], [ 2 ] ]\ngap> gr \ := Digraph([[1, 2, 3], [2, 1], [3]]);\n\ngap> InNeighbours(gr);\n[ [ 1, 2 ], [ 1, 2 ], [ 1, 3 ] ]\ngap> gr := Di\ graphByAdjacencyMatrix([\n> [1, 2, 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n\ngap> InNeighbours(gr);\n[ [ 1, 2\ ], [ 1, 1, 2 ], [ 1, 3 ] ]\ngap> InNeighboursMutableCopy(gr);\n[ [ 1, 2 ], [ \ 1, 1, 2 ], [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> InNeighbours(D);\n[ [ 3 ], [ 1 ], [ 2 \ ] ]\n", count := [ 5, 2, 7, 13 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 13 ], mathmode := "Text", name := "Example", next := 162712, root := ~, start := 161937, stop := 162711 ), rec( content := "\n ", count := [ 5, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "Description", next := 162729, root := ~, start := 161340, stop := 162728 ) ], count := [ 5, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 162743, root := ~, start := 161079, stop := 162742 ), rec( content := "\n\n ", count := [ 5, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "OutDegrees" ), content := 0, count := [ 5, 2, 8, 2 ], mathmode := "Text", name := "Attr", next := 162803, root := ~, start := 162764, stop := 162802 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSequen\ ce" ), content := 0, count := [ 5, 2, 8, 4 ], mathmode := "Text", name := "Attr", next := 162852, root := ~, start := 162806, stop := 162851 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSet" ) , content := 0, count := [ 5, 2, 8, 6 ], mathmode := "Text", name := "Attr", next := 162896, root := ~, start := 162855, stop := 162895 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ non-negative integers.", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Returns", next := 162950, root := ~, start := 162899, stop := 162949 ), rec( attributes := rec( ), content := [ rec( content := "\n\n G\ iven a digraph ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "A", next := 163002, root := ~, start := 162988, stop := 163001 ), rec( content := " with ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "M", next := 163016, root := ~, start := 163008, stop := 163015 ), rec( content := " vertice\ s, the function\n ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utDegrees", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163061, root := ~, start := 163044, stop := 163060 ), rec( content := " returns\ an immutable list ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163098, root := ~, start := 163088, stop := 163097 ), rec( content := " of leng\ th ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "M", next := 163117, root := ~, start := 163109, stop := 163116 ), rec( content := ", such t\ hat\n for a vertex ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163154, root := ~, start := 163146, stop := 163153 ), rec( content := " in ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "A", next := 163172, root := ~, start := 163158, stop := 163171 ), rec( content := ", the va\ lue of ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut[i]", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163200, root := ~, start := 163187, stop := 163199 ), rec( content := " is the\\ n out-degree of vertex ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163241, root := ~, start := 163233, stop := 163240 ), rec( content := ".\n S\ ee ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Ref", next := 163282, root := ~, start := 163251, stop := 163281 ), rec( content := ". ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "P", next := 163288, root := ~, start := 163284, stop := 163287 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utDegreeSequence", count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "C", next := 163331, root := ~, start := 163307, stop := 163330 ), rec( content := " returns\ the same list,\n after it has been sorted into non-increasing order. ", count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "P", next := 163415, root := ~, start := 163411, stop := 163414 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utDegreeSet", count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "C", next := 163453, root := ~, start := 163434, stop := 163452 ), rec( content := " returns\ the same list, sorted into\n increasing order with duplicate entries remov\ ed. ", count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 8, 10 ], mathmode := "Text", name := "P", next := 163546, root := ~, start := 163542, stop := 163545 ), rec( content := "\n\n \ ", count := [ 5, 2, 8, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3, 2, 2], [], [2, 1], []]);\n\ngap> OutDegrees(D);\n[ 4, 0, 2, 0 ]\ngap> OutDegreeS\ equence(D);\n[ 4, 2, 0, 0 ]\ngap> OutDegreeSet(D);\n[ 0, 2, 4 ]\ngap> D := Cyc\ leDigraph(IsMutableDigraph, 3);\n\ng\ ap> OutDegrees(D);\n[ 1, 1, 1 ]\n", count := [ 5, 2, 8, 11 ], name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 8, 11 ], mathmode := "Text", name := "Example", next := 163916, root := ~, start := 163552, stop := 163915 ), rec( content := "\n ", count := [ 5, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Description", next := 163933, root := ~, start := 162953, stop := 163932 ) ], count := [ 5, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 163947, root := ~, start := 162749, stop := 163946 ), rec( content := "\n\n ", count := [ 5, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "InDegrees" ), content := 0, count := [ 5, 2, 9, 2 ], mathmode := "Text", name := "Attr", next := 164006, root := ~, start := 163968, stop := 164005 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSequenc\ e" ), content := 0, count := [ 5, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 164054, root := ~, start := 164009, stop := 164053 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSet" ), content := 0, count := [ 5, 2, 9, 6 ], mathmode := "Text", name := "Attr", next := 164097, root := ~, start := 164057, stop := 164096 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ non-negative integers.", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Returns", next := 164151, root := ~, start := 164100, stop := 164150 ), rec( attributes := rec( ), content := [ rec( content := "\n\n G\ iven a digraph ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "A", next := 164203, root := ~, start := 164189, stop := 164202 ), rec( content := " with ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "M", next := 164217, root := ~, start := 164209, stop := 164216 ), rec( content := " vertice\ s, the function\n ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nDegrees", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 164261, root := ~, start := 164245, stop := 164260 ), rec( content := " returns\ an immutable list ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 164298, root := ~, start := 164288, stop := 164297 ), rec( content := " of leng\ th ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "M", next := 164317, root := ~, start := 164309, stop := 164316 ), rec( content := ", such t\ hat\n for a vertex ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 164354, root := ~, start := 164346, stop := 164353 ), rec( content := " in ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "A", next := 164372, root := ~, start := 164358, stop := 164371 ), rec( content := ", the va\ lue of ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn[i]", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 164400, root := ~, start := 164387, stop := 164399 ), rec( content := " is\n \ the in-degree of vertex ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 164440, root := ~, start := 164432, stop := 164439 ), rec( content := ".\n S\ ee ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "InDegre\ eOfVertex" ), content := 0, count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Ref", next := 164480, root := ~, start := 164450, stop := 164479 ), rec( content := ". ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "P", next := 164486, root := ~, start := 164482, stop := 164485 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nDegreeSequence", count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "C", next := 164528, root := ~, start := 164505, stop := 164527 ), rec( content := " returns\ the same list,\n after it has been sorted into non-increasing order. ", count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "P", next := 164612, root := ~, start := 164608, stop := 164611 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nDegreeSet", count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "C", next := 164649, root := ~, start := 164631, stop := 164648 ), rec( content := " returns\ the same list, sorted into\n increasing order with duplicate entries remov\ ed. ", count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 9, 10 ], mathmode := "Text", name := "P", next := 164742, root := ~, start := 164738, stop := 164741 ), rec( content := "\n\n \ ", count := [ 5, 2, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3, 2, 2, 4], [], [2, 1, 4], []]);\n\ngap> InDegrees(D);\n[ 2, 3, 1, 2 ]\ngap> InDeg\ reeSequence(D);\n[ 3, 2, 2, 1 ]\ngap> InDegreeSet(D);\n[ 1, 2, 3 ]\ngap> D := \ CycleDigraph(IsMutableDigraph, 3);\n\ \ngap> InDegrees(D);\n[ 1, 1, 1 ]\n", count := [ 5, 2, 9, 11 ], name := "PCDAT\ A", root := ~ ) ], count := [ 5, 2, 9, 11 ], mathmode := "Text", name := "Example", next := 165114, root := ~, start := 164748, stop := 165113 ), rec( content := "\n ", count := [ 5, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Description", next := 165131, root := ~, start := 164154, stop := 165130 ) ], count := [ 5, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 165145, root := ~, start := 163953, stop := 165144 ), rec( content := "\n\n ", count := [ 5, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutDegreeOfVert\ ex" ), content := 0, count := [ 5, 2, 10, 2 ], mathmode := "Text", name := "Oper", next := 165220, root := ~, start := 165166, stop := 165219 ), rec( attributes := rec( ), content := [ rec( content := "The non-n\ egative integer.", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "Returns", next := 165267, root := ~, start := 165223, stop := 165266 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns the out-degree of the vertex ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 165353, root := ~, start := 165340, stop := 165352 ), rec( content := " in the\\ n digraph ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 165387, root := ~, start := 165373, stop := 165386 ), rec( content := ".\n T\ he out-degree of ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 165424, root := ~, start := 165411, stop := 165423 ), rec( content := " is the \ number of edges in ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 165465, root := ~, start := 165451, stop := 165464 ), rec( content := "\n wh\ ose source is ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 165499, root := ~, start := 165486, stop := 165498 ), rec( content := ".\n " , count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 10, 4 ], mathmode := "Text", name := "P", next := 165509, root := ~, start := 165505, stop := 165508 ), rec( content := "\n ", count := [ 5, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> OutDegreeOfVerte\ x(D, 1);\n3\ngap> OutDegreeOfVertex(D, 2);\n2\ngap> OutDegreeOfVertex(D, 3);\n\ 4\ngap> OutDegreeOfVertex(D, 4);\n7\n", count := [ 5, 2, 10, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 10, 5 ], mathmode := "Text", name := "Example", next := 165805, root := ~, start := 165514, stop := 165804 ), rec( content := "\n ", count := [ 5, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 165822, root := ~, start := 165270, stop := 165821 ) ], count := [ 5, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 165836, root := ~, start := 165151, stop := 165835 ), rec( content := "\n\n ", count := [ 5, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutNeighboursOf\ Vertex" ), content := 0, count := [ 5, 2, 11, 2 ], mathmode := "Text", name := "Oper", next := 165915, root := ~, start := 165857, stop := 165914 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutNeighborsOfV\ ertex" ), content := 0, count := [ 5, 2, 11, 4 ], mathmode := "Text", name := "Oper", next := 165975, root := ~, start := 165918, stop := 165974 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "Returns", next := 166016, root := ~, start := 165978, stop := 166015 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation returns the list ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 166080, root := ~, start := 166070, stop := 166079 ), rec( content := " of vert\ ices of the digraph\n ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "A", next := 166126, root := ~, start := 166112, stop := 166125 ), rec( content := ".\n A\ vertex ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 166149, root := ~, start := 166141, stop := 166148 ), rec( content := " appears\ in the list ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 166180, root := ~, start := 166170, stop := 166179 ), rec( content := " each ti\ me there exists an\n edge with source ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "A", next := 166242, root := ~, start := 166229, stop := 166241 ), rec( content := " and ran\ ge ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 166261, root := ~, start := 166253, stop := 166260 ), rec( content := " in ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "A", next := 166279, root := ~, start := 166265, stop := 166278 ), rec( content := "; in\n \ particular, this means that ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 166326, root := ~, start := 166316, stop := 166325 ), rec( content := " may con\ tain duplicates.", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 11, 6 ], mathmode := "Text", name := "P", next := 166354, root := ~, start := 166350, stop := 166353 ), rec( content := "\n ", count := [ 5, 2, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> OutNeighboursOfV\ ertex(D, 1);\n[ 2, 2, 3 ]\ngap> OutNeighboursOfVertex(D, 3);\n[ 2, 2, 3 ]\n", count := [ 5, 2, 11, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 11, 7 ], mathmode := "Text", name := "Example", next := 166614, root := ~, start := 166359, stop := 166613 ), rec( content := "\n ", count := [ 5, 2, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "Description", next := 166631, root := ~, start := 166019, stop := 166630 ) ], count := [ 5, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 166645, root := ~, start := 165842, stop := 166644 ), rec( content := "\n\n ", count := [ 5, 2, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "InDegreeOfVerte\ x" ), content := 0, count := [ 5, 2, 12, 2 ], mathmode := "Text", name := "Oper", next := 166719, root := ~, start := 166666, stop := 166718 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer.", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "Returns", next := 166764, root := ~, start := 166722, stop := 166763 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns the in-degree of the vertex ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 166849, root := ~, start := 166836, stop := 166848 ), rec( content := " in the\\ n digraph ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 166883, root := ~, start := 166869, stop := 166882 ), rec( content := ".\n T\ he in-degree of ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 166919, root := ~, start := 166906, stop := 166918 ), rec( content := " is the \ number of edges in ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 166960, root := ~, start := 166946, stop := 166959 ), rec( content := "\n wh\ ose range is ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 166993, root := ~, start := 166980, stop := 166992 ), rec( content := ".\n " , count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> InDegreeOfVertex\ (D, 1);\n5\ngap> InDegreeOfVertex(D, 2);\n9\ngap> InDegreeOfVertex(D, 3);\n0\n\ gap> InDegreeOfVertex(D, 4);\n2\n", count := [ 5, 2, 12, 4 ], name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 12, 4 ], mathmode := "Text", name := "Example", next := 167286, root := ~, start := 166999, stop := 167285 ), rec( content := "\n ", count := [ 5, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 167303, root := ~, start := 166767, stop := 167302 ) ], count := [ 5, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 167317, root := ~, start := 166651, stop := 167316 ), rec( content := "\n\n ", count := [ 5, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "InNeighboursOfV\ ertex" ), content := 0, count := [ 5, 2, 13, 2 ], mathmode := "Text", name := "Oper", next := 167395, root := ~, start := 167338, stop := 167394 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "InNeighborsOfVe\ rtex" ), content := 0, count := [ 5, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 167454, root := ~, start := 167398, stop := 167453 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ postitive vertices.", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "Returns", next := 167505, root := ~, start := 167457, stop := 167504 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation returns the list ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 167569, root := ~, start := 167559, stop := 167568 ), rec( content := " of vert\ ices of the digraph\n ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "A", next := 167615, root := ~, start := 167601, stop := 167614 ), rec( content := ".\n A\ vertex ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 167638, root := ~, start := 167630, stop := 167637 ), rec( content := " appears\ in the list ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 167669, root := ~, start := 167659, stop := 167668 ), rec( content := " each ti\ me there exists an\n edge with source ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 167726, root := ~, start := 167718, stop := 167725 ), rec( content := " and ran\ ge ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "A", next := 167750, root := ~, start := 167737, stop := 167749 ), rec( content := " in ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "A", next := 167768, root := ~, start := 167754, stop := 167767 ), rec( content := "; in\n \ particular, this means that ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 167815, root := ~, start := 167805, stop := 167814 ), rec( content := " may con\ tain duplicates. ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 13, 6 ], mathmode := "Text", name := "P", next := 167844, root := ~, start := 167840, stop := 167843 ), rec( content := "\n\n \ ", count := [ 5, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> InNeighboursOfVe\ rtex(D, 1);\n[ 2, 4, 4, 4 ]\ngap> InNeighboursOfVertex(D, 2);\n[ 1, 1, 3, 3, 4\ , 4, 4, 4 ]\n", count := [ 5, 2, 13, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 7 ], mathmode := "Text", name := "Example", next := 168121, root := ~, start := 167850, stop := 168120 ), rec( content := "\n ", count := [ 5, 2, 13, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 168138, root := ~, start := 167508, stop := 168137 ) ], count := [ 5, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 168152, root := ~, start := 167323, stop := 168151 ), rec( content := "\n\n ", count := [ 5, 2, 13, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphLoops" ) , content := 0, count := [ 5, 2, 14, 2 ], mathmode := "Text", name := "Attr", next := 168214, root := ~, start := 168173, stop := 168213 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Returns", next := 168255, root := ~, start := 168217, stop := 168254 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "A", next := 168293, root := ~, start := 168279, stop := 168292 ), rec( content := " is a di\ graph, then ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphLoops", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "C", next := 168332, root := ~, start := 168313, stop := 168331 ), rec( content := " returns\ the list\n consisting of the ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Ref", next := 168401, root := ~, start := 168372, stop := 168400 ), rec( content := " of ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "A", next := 168419, root := ~, start := 168405, stop := 168418 ), rec( content := "\n at\ which there is a loop. See ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Ref", next := 168483, root := ~, start := 168454, stop := 168482 ), rec( content := ". ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 14, 4 ], mathmode := "Text", name := "P", next := 168489, root := ~, start := 168485, stop := 168488 ), rec( content := "\n\n \ ", count := [ 5, 2, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], []]);\n\ngap> DigraphHasLoops(D);\nfalse\ngap> DigraphLoops(D);\n[ ]\ngap> D := D\ igraph([[3, 5], [1], [2, 4, 3], [4], [2, 1]]);\n\ngap> DigraphLoops(D);\n[ 3, 4 ]\ngap> D := Digraph(IsMutableDi\ graph, [[1], [1]]);\n\ngap> DigraphL\ oops(D);\n[ 1 ]\n", count := [ 5, 2, 14, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 5 ], mathmode := "Text", name := "Example", next := 168919, root := ~, start := 168495, stop := 168918 ), rec( content := "\n ", count := [ 5, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 168936, root := ~, start := 168258, stop := 168935 ) ], count := [ 5, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 168950, root := ~, start := 168158, stop := 168949 ), rec( content := "\n\n ", count := [ 5, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "PartialOrderDig\ raphMeetOfVertices" ), content := 0, count := [ 5, 2, 15, 2 ], mathmode := "Text", name := "Oper", next := 169043, root := ~, start := 168971, stop := 169042 ), rec( attributes := rec( Arg := "digraph, u, v", Name := "PartialOrderDig\ raphJoinOfVertices" ), content := 0, count := [ 5, 2, 15, 4 ], mathmode := "Text", name := "Oper", next := 169118, root := ~, start := 169046, stop := 169117 ), rec( attributes := rec( ), content := [ rec( content := "A positiv\ e integer or ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "K", next := 169163, root := ~, start := 169152, stop := 169162 ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Returns", next := 169173, root := ~, start := 169121, stop := 169172 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the first argument is a partial order digraph ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Ref", next := 169284, root := ~, start := 169245, stop := 169283 ), rec( content := " then th\ ese operations return the meet, or\n the join, of the two input vertices. I\ f the meet (or join) is does not\n exist then ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "K", next := 169428, root := ~, start := 169417, stop := 169427 ), rec( content := " is retu\ rned. The meet (or join) is guaranteed to\n exist when the first argument s\ atisfies\n ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMeetS\ emilatticeDigraph" ), content := 0, count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Ref", next := 169564, root := ~, start := 169526, stop := 169563 ), rec( content := " (or\n \ ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Ref", next := 169612, root := ~, start := 169574, stop := 169611 ), rec( content := ") - see \ the documentation for these\n properties for the definition of the meet (or\ the join).\n\n ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [1, 2], [1, 3], [1, 2, 3, 4]]);\n\ngap> PartialOrderDigraphMeetOfVertices(D, 2, 3);\n4\ \ngap> PartialOrderDigraphJoinOfVertices(D, 2, 3);\n1\ngap> PartialOrderDigrap\ hMeetOfVertices(D, 1, 2);\n2\ngap> PartialOrderDigraphJoinOfVertices(D, 3, 4);\ \n3\ngap> D := Digraph([[1], [2], [1, 2, 3], [1, 2, 4]]);\n\ngap> PartialOrderDigraphMeetOfVertices(D, 3, 4);\nf\ ail\ngap> PartialOrderDigraphJoinOfVertices(D, 3, 4);\nfail\ngap> PartialOrder\ DigraphMeetOfVertices(D, 1, 2);\nfail\ngap> PartialOrderDigraphJoinOfVertices(\ D, 1, 2);\nfail\n", count := [ 5, 2, 15, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 6 ], mathmode := "Text", name := "Example", next := 170365, root := ~, start := 169714, stop := 170364 ), rec( content := "\n ", count := [ 5, 2, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Description", next := 170382, root := ~, start := 169178, stop := 170381 ) ], count := [ 5, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 170396, root := ~, start := 168956, stop := 170395 ), rec( content := "\n\n ", count := [ 5, 2, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DegreeMatrix" ) , content := 0, count := [ 5, 2, 16, 2 ], mathmode := "Text", name := "Attr", next := 170458, root := ~, start := 170417, stop := 170457 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of non-negative integers.", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "Returns", next := 170521, root := ~, start := 170461, stop := 170520 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the outdegree matrix ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 170595, root := ~, start := 170585, stop := 170594 ), rec( content := " of the \ digraph\n ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "A", next := 170629, root := ~, start := 170615, stop := 170628 ), rec( content := ". The va\ lue of the ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 170656, root := ~, start := 170648, stop := 170655 ), rec( content := "th diago\ nal matrix entry is the\n outdegree of the vertex ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 170724, root := ~, start := 170716, stop := 170723 ), rec( content := " in ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "A", next := 170742, root := ~, start := 170728, stop := 170741 ), rec( content := ". All of\ f-diagonal entries \n are ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 170786, root := ~, start := 170778, stop := 170785 ), rec( content := ".\n I\ f ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "A", next := 170809, root := ~, start := 170795, stop := 170808 ), rec( content := " has no \ vertices, then the empty list is returned. ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "P", next := 170864, root := ~, start := 170860, stop := 170863 ), rec( content := "\n\n \ See ", count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDegr\ ees" ), content := 0, count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "Ref", next := 170898, root := ~, start := 170874, stop := 170897 ), rec( content := " for mor\ e information.\n ", count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 3], [4], [1, 3, 4], []]);\n\ngap> PrintArray(DegreeMatrix(D));\n[ [ 3, 0, 0, 0 ],\ \n [ 0, 1, 0, 0 ],\n [ 0, 0, 3, 0 ],\n [ 0, 0, 0, 0 ] ]\ngap> D\ := CycleDigraph(5);;\ngap> PrintArray(DegreeMatrix(D));\n[ [ 1, 0, 0, 0, \ 0 ],\n [ 0, 1, 0, 0, 0 ],\n [ 0, 0, 1, 0, 0 ],\n [ 0, 0, 0, \ 1, 0 ],\n [ 0, 0, 0, 0, 1 ] ]\ngap> DegreeMatrix(EmptyDigraph(0));\n[ \ ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> Display(DegreeMatrix(D));\n[ [ 1, 0, 0 ],\n [ 0, 1, \ 0 ],\n [ 0, 0, 1 ] ]\n", count := [ 5, 2, 16, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 5 ], mathmode := "Text", name := "Example", next := 171585, root := ~, start := 170925, stop := 171584 ), rec( content := "\n ", count := [ 5, 2, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 171602, root := ~, start := 170524, stop := 171601 ) ], count := [ 5, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 171616, root := ~, start := 170402, stop := 171615 ), rec( content := "\n\n ", count := [ 5, 2, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "LaplacianMatrix\ " ), content := 0, count := [ 5, 2, 17, 2 ], mathmode := "Text", name := "Attr", next := 171681, root := ~, start := 171637, stop := 171680 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of integers.", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "Returns", next := 171731, root := ~, start := 171684, stop := 171730 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the outdegree Laplacian matrix ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "C", next := 171815, root := ~, start := 171805, stop := 171814 ), rec( content := " of the\\ n digraph ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "A", next := 171849, root := ~, start := 171835, stop := 171848 ), rec( content := ". The ou\ tdegree Laplacian matrix is defined as\n ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ egreeMatrix(digraph) - AdjacencyMatrix(digraph)", count := [ 5, 2, 17, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "C", next := 171955, root := ~, start := 171900, stop := 171954 ), rec( content := ". If\n \ ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "A", next := 171979, root := ~, start := 171965, stop := 171978 ), rec( content := " has no \ vertices, then the empty list is returned. ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "P", next := 172034, root := ~, start := 172030, stop := 172033 ), rec( content := "\n\n \ See ", count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DegreeM\ atrix" ), content := 0, count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "Ref", next := 172070, root := ~, start := 172044, stop := 172069 ), rec( content := " and ", count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "Ref", next := 172104, root := ~, start := 172075, stop := 172103 ), rec( content := " for mor\ e\n information.\n ", count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[1, 2, 3], [4], [1, 3, 4], []]);\n\ngap> PrintArray(LaplacianMatrix(gr));\n[ [ 2, -1, -\ 1, 0 ],\n [ 0, 1, 0, -1 ],\n [ -1, 0, 2, -1 ],\n [ 0, 0,\ 0, 0 ] ]\ngap> LaplacianMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDi\ graph(IsMutableDigraph, 3);\n\ngap> \ Display(LaplacianMatrix(D));\n[ [ 1, -1, 0 ],\n [ 0, 1, -1 ],\n [ \ -1, 0, 1 ] ]\n", count := [ 5, 2, 17, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 5 ], mathmode := "Text", name := "Example", next := 172638, root := ~, start := 172135, stop := 172637 ), rec( content := "\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 172655, root := ~, start := 171734, stop := 172654 ) ], count := [ 5, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 172669, root := ~, start := 171622, stop := 172668 ), rec( content := "\n\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 172683, root := ~, start := 152451, stop := 172682 ) , rec( content := "\n\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Reachability and \ connectivity", count := [ 5, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 172744, root := ~, start := 172696, stop := 172743 ), rec( content := "\n ", count := [ 5, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDiameter\ " ), content := 0, count := [ 5, 3, 1, 2 ], mathmode := "Text", name := "Attr", next := 172808, root := ~, start := 172764, stop := 172807 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r or ", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "C", next := 172845, root := ~, start := 172834, stop := 172844 ), rec( content := ".", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 172856, root := ~, start := 172811, stop := 172855 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the diameter of the digraph ", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "A", next := 172941, root := ~, start := 172927, stop := 172940 ), rec( content := ".\n " , count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "P", next := 172951, root := ~, start := 172947, stop := 172950 ), rec( content := "\n\n \ If a digraph ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "A", next := 172984, root := ~, start := 172970, stop := 172983 ), rec( content := " is stro\ ngly connected and has at least 1\n vertex, then the ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ iameter", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "E", next := 173062, root := ~, start := 173047, stop := 173061 ), rec( content := " is the \ maximum shortest distance between\n any pair of distinct vertices. Otherwis\ e then the diameter of\n ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "A", next := 173188, root := ~, start := 173174, stop := 173187 ), rec( content := " is unde\ fined, and this function returns the value\n ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "C", next := 173254, root := ~, start := 173243, stop := 173253 ), rec( content := ". ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "P", next := 173260, root := ~, start := 173256, stop := 173259 ), rec( content := "\n\n \ See ", count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ShortestDistances" ), content := 0, count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "Ref", next := 173308, root := ~, start := 173270, stop := 173307 ), rec( content := ". ", count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 1, 6 ], mathmode := "Text", name := "P", next := 173314, root := ~, start := 173310, stop := 173313 ), rec( content := "\n ", count := [ 5, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [4, 5], [5], [1, 2, 3, 4, 5]]);\n\ngap> DigraphDiameter(D);\n3\ngap> D := ChainDi\ graph(2);\n\ngap> DigraphDiameter(D);\ \nfail\ngap> IsStronglyConnectedDigraph(D);\nfalse\ngap> D := GeneralisedPeter\ senGraph(IsMutableDigraph, 6, 2);\n\ngap> DigraphDiameter(D);\n4\n", count := [ 5, 3, 1, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 7 ], mathmode := "Text", name := "Example", next := 173758, root := ~, start := 173319, stop := 173757 ), rec( content := "\n ", count := [ 5, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 173775, root := ~, start := 172859, stop := 173774 ) ], count := [ 5, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 173789, root := ~, start := 172749, stop := 173788 ), rec( content := "\n\n ", count := [ 5, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Label := "for a digraph \ and two vertices", Name := "DigraphShortestDistance" ), content := 0, count := [ 5, 3, 2, 2 ], mathmode := "Text", name := "Oper", next := 173915, root := ~, start := 173810, stop := 173914 ), rec( attributes := rec( Arg := "digraph, list", Label := "for a digraph \ and a list", Name := "DigraphShortestDistance" ), content := 0, count := [ 5, 3, 2, 4 ], mathmode := "Text", name := "Oper", next := 174017, root := ~, start := 173918, stop := 174016 ), rec( attributes := rec( Arg := "digraph, list1, \ list2", Label := "for a digraph, a list, and a list", Name := "DigraphShortest\ Distance" ), content := 0, count := [ 5, 3, 2, 6 ], mathmode := "Text", name := "Oper", next := 174136, root := ~, start := 174020, stop := 174135 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r or ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "K", next := 174173, root := ~, start := 174162, stop := 174172 ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Returns", next := 174183, root := ~, start := 174139, stop := 174182 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ there is a directed path in the digraph ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "A", next := 174261, root := ~, start := 174247, stop := 174260 ), rec( content := " between\ vertex\n ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "A", next := 174289, root := ~, start := 174281, stop := 174288 ), rec( content := " and ver\ tex ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "A", next := 174309, root := ~, start := 174301, stop := 174308 ), rec( content := ", then t\ his operation returns the length of the\n shortest such directed path. If \ no such directed path exists, then this\n operation returns ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "K", next := 174467, root := ~, start := 174456, stop := 174466 ), rec( content := ". See se\ ction ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Ref", next := 174531, root := ~, start := 174481, stop := 174530 ), rec( content := " for the\ definition of a directed path. ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "P", next := 174575, root := ~, start := 174571, stop := 174574 ), rec( content := "\n\n \ If the second form is used, then ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "A", next := 174625, root := ~, start := 174614, stop := 174624 ), rec( content := " should \ be a list of length two,\n containing two positive integers which correspon\ d to the vertices ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "A", next := 174736, root := ~, start := 174728, stop := 174735 ), rec( content := "\n an\ d ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "A", next := 174753, root := ~, start := 174745, stop := 174752 ), rec( content := ". ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 9 ], mathmode := "Text", name := "P", next := 174759, root := ~, start := 174755, stop := 174758 ), rec( content := "\n\n \ Note that as usual, a vertex is considered to be at distance 0 from\n itsel\ f .", count := [ 5, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "P", next := 174849, root := ~, start := 174845, stop := 174848 ), rec( content := "\n\n \ If the third form is used, then ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 174899, root := ~, start := 174887, stop := 174898 ), rec( content := " and ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 174916, root := ~, start := 174904, stop := 174915 ), rec( content := " are bot\ h lists\n of vertices. The shortest directed path between ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 174996, root := ~, start := 174984, stop := 174995 ), rec( content := " and\n \ ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175017, root := ~, start := 175005, stop := 175016 ), rec( content := " is then\ the length of the shortest directed path which starts\n with a vertex in " , count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175113, root := ~, start := 175101, stop := 175112 ), rec( content := " and ter\ minates at a vertex in ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175156, root := ~, start := 175144, stop := 175155 ), rec( content := ", if\n \ such directed path exists. If ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175207, root := ~, start := 175195, stop := 175206 ), rec( content := " and ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175224, root := ~, start := 175212, stop := 175223 ), rec( content := " have no\ n-empty\n intersection, the operation returns ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "C", next := 175288, root := ~, start := 175280, stop := 175287 ), rec( content := ".", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 11 ], mathmode := "Text", name := "P", next := 175293, root := ~, start := 175289, stop := 175292 ), rec( content := "\n ", count := [ 5, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [1, 4], [1, 3], [5]]);\n\ngap> DigraphShortestDistance(D, 1, 3);\n2\ngap> DigraphS\ hortestDistance(D, [3, 3]);\n0\ngap> DigraphShortestDistance(D, 5, 2);\nfail\n\ gap> DigraphShortestDistance(D, [1, 2], [4, 5]);\n2\ngap> DigraphShortestDista\ nce(D, [1, 3], [3, 5]);\n0\n", count := [ 5, 3, 2, 12 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 12 ], mathmode := "Text", name := "Example", next := 175657, root := ~, start := 175298, stop := 175656 ), rec( content := "\n ", count := [ 5, 3, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Description", next := 175674, root := ~, start := 174186, stop := 175673 ) ], count := [ 5, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 175688, root := ~, start := 173795, stop := 175687 ), rec( content := "\n\n ", count := [ 5, 3, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphShortest\ Distances" ), content := 0, count := [ 5, 3, 3, 2 ], mathmode := "Text", name := "Attr", next := 175762, root := ~, start := 175709, stop := 175761 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix.", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 175800, root := ~, start := 175765, stop := 175799 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "A", next := 175838, root := ~, start := 175824, stop := 175837 ), rec( content := " is a di\ graph with ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "M", next := 175865, root := ~, start := 175857, stop := 175864 ), rec( content := " vertice\ s, then this\n function returns an ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ \\times n", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "M", next := 175927, root := ~, start := 175910, stop := 175926 ), rec( content := " matrix \ ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "C", next := 175945, root := ~, start := 175935, stop := 175944 ), rec( content := ", where \ each entry is\n either a non-negative integer, or ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "K", next := 176016, root := ~, start := 176005, stop := 176015 ), rec( content := ". If ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ = 0", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "M", next := 176034, root := ~, start := 176022, stop := 176033 ), rec( content := ", then a\ n\n empty list is returned. ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "P", next := 176076, root := ~, start := 176072, stop := 176075 ), rec( content := "\n\n \ If there is a directed path from vertex ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176130, root := ~, start := 176122, stop := 176129 ), rec( content := " to vert\ ex ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176149, root := ~, start := 176141, stop := 176148 ), rec( content := ", then\n\ the value of ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176189, root := ~, start := 176173, stop := 176188 ), rec( content := " is the \ length of the shortest such directed\n path. If no such directed path exist\ s, then the value of ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176311, root := ~, start := 176295, stop := 176310 ), rec( content := " is\n \ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176330, root := ~, start := 176319, stop := 176329 ), rec( content := ". We us\ e the convention that the distance from every vertex to\n itself is ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176416, root := ~, start := 176408, stop := 176415 ), rec( content := ", i.e. " , count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][i] = 0", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176443, root := ~, start := 176423, stop := 176442 ), rec( content := " for all\ vertices ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176469, root := ~, start := 176461, stop := 176468 ), rec( content := ".\n " , count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "P", next := 176479, root := ~, start := 176475, stop := 176478 ), rec( content := "\n\n \ The method used in this function is a version of the Floyd-Warshall\n algor\ ithm, and has complexity ", count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (n^3)", count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "M", next := 176600, root := ~, start := 176587, stop := 176599 ), rec( content := ".\n\n \ ", count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], [1, 2], [4]]);\n\ngap> mat := DigraphShortestDistances(D);;\ngap> PrintArray(ma\ t);\n[ [ 0, 1, 2, fail ],\n [ 2, 0, 1, fail ],\n [\ 1, 1, 0, fail ],\n [ fail, fail, fail, 0 ] ]\ngap> D := \ CycleDigraph(IsMutableDigraph, 3);\n\ \ngap> DigraphShortestDistances(D);\n[ [ 0, 1, 2 ], [ 2, 0, 1 ], [ 1, 2, 0 ] ]\ \n", count := [ 5, 3, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 6 ], mathmode := "Text", name := "Example", next := 177096, root := ~, start := 176607, stop := 177095 ), rec( content := "\n ", count := [ 5, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 177113, root := ~, start := 175803, stop := 177112 ) ], count := [ 5, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 177127, root := ~, start := 175694, stop := 177126 ), rec( content := "\n\n ", count := [ 5, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, v", Name := "DigraphLongestD\ istanceFromVertex" ), content := 0, count := [ 5, 3, 4, 2 ], mathmode := "Text", name := "Oper", next := 177212, root := ~, start := 177148, stop := 177211 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r, or ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "K", next := 177254, root := ~, start := 177239, stop := 177253 ), rec( content := ".", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 177265, root := ~, start := 177215, stop := 177264 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 177303, root := ~, start := 177289, stop := 177302 ), rec( content := " is a di\ graph and ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 177329, root := ~, start := 177321, stop := 177328 ), rec( content := " is a ve\ rtex in ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 177359, root := ~, start := 177345, stop := 177358 ), rec( content := ",\n t\ hen this operation returns the length of the longest directed walk in\n ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 177454, root := ~, start := 177440, stop := 177453 ), rec( content := " which b\ egins at vertex ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 177486, root := ~, start := 177478, stop := 177485 ), rec( content := ". See s\ ection ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 177551, root := ~, start := 177501, stop := 177550 ), rec( content := " for the\ definitions of directed\n walk, directed cycle, and loop.\n ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 4, 4 ], mathmode := "Text", name := "P", next := 177628, root := ~, start := 177624, stop := 177627 ), rec( content := "\n\n \ ", count := [ 5, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n If there exists a directed w\ alk starting at vertex ", count := [ 5, 3, 4, 6 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "v", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "A", next := 1777\ 21, root := ~, start := 177713, stop := 177720 ), rec( content := "\n w\ hich traverses a loop or a directed cycle,\n then we consider there to \ be a walk of infinite length from ", count := [ 5, 3, 4, 6 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "v", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "A", next\ := 177850, root := ~, start := 177842, stop := 177849 ), rec( content := "\n \ (realised by repeatedly traversing the loop/directed cycle),\n a\ nd so the result is ", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "infinity", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "K", next := \ 177964, root := ~, start := 177949, stop := 177963 ), rec( content := ".\n \ To disallow walks using loops, try using\n ", count := [ 5, 3, 4, 6\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ Oper := "DigraphRemoveLoops" ), content := 0, count := [ 5, 3, 4, 6 ], mathmo\ de := "Text", name := "Ref", next := 178055, root := ~, start := 178023, stop \ := 178054 ), rec( content := ":", count := [ 5, 3, 4, 6 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 5, 3, 4, 7 ], mathmode := "Text", name := "P", next := 178060, root := ~, star\ t := 178056, stop := 178059 ), rec( content := "\n\n ", count := [ 5, 3\ , 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "DigraphLongestDistanceFromVertex(Digr\ aphRemoveLoops(", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digr\ aph", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "A", next := 178139\ , root := ~, start := 178125, stop := 178138 ), rec( content := ",", count := \ [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "v", count := [ 5, 3, 4, 7 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 7 ], ma\ thmode := "Text", name := "A", next := 178148, root := ~, start := 178140, sto\ p := 178147 ), rec( content := "))", count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "C", next := 178154, root := ~, start := 178070, stop := 178153 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 3, 4, 6 ], name := "Item", next := 178169, root := ~, start := 177\ 647, stop := 178168 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n Otherwise, if all directed walks\n starting at vertex ", coun\ t := [ 5, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "v", count := [ 5, 3, 4, 9 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 9 ], math\ mode := "Text", name := "A", next := 178260, root := ~, start := 178252, stop \ := 178259 ), rec( content := " have finite length,\n then the length of\ the longest such walk is returned.\n ", count := [ 5, 3, 4, 9 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 9 ], name := "Ite\ m", next := 178356, root := ~, start := 178177, stop := 178355 ) ], count := [ 5, 3, 4, 5 ], mathmode := "Text", name := "List", next := 178368, root := ~, start := 177634, stop := 178367 ), rec( content := "\n\n \ Note that the result is ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "C", next := 178406, root := ~, start := 178398, stop := 178405 ), rec( content := " if and \ only if ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "A", next := 178430, root := ~, start := 178422, stop := 178429 ), rec( content := " is a si\ nk of\n ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "A", next := 178462, root := ~, start := 178448, stop := 178461 ), rec( content := ". See " , count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Sinks" ), content := 0, count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "Ref", next := 178495, root := ~, start := 178469, stop := 178494 ), rec( content := ".\n\n", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3, 4], [], [5], [], [6]]);\n\ngap> DigraphLongestDistanceFromVertex(D, 1);\n3\ngap> Di\ graphLongestDistanceFromVertex(D, 3);\n0\ngap> 3 in DigraphSinks(D);\ntrue\nga\ p> DigraphLongestDistanceFromVertex(D, 6);\ninfinity\n", count := [ 5, 3, 4, 12 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 4, 12 ], mathmode := "Text", name := "Example", next := 178807, root := ~, start := 178498, stop := 178806 ), rec( content := "\n ", count := [ 5, 3, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 178824, root := ~, start := 177268, stop := 178823 ) ], count := [ 5, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 178838, root := ~, start := 177133, stop := 178837 ), rec( content := "\n\n ", count := [ 5, 3, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex,\ distance", Label := "for a digraph, a pos int, and an int", Name := "DigraphDistance\ Set" ), content := 0, count := [ 5, 3, 5, 2 ], mathmode := "Text", name := "Oper", next := 178969, root := ~, start := 178859, stop := 178968 ), rec( attributes := rec( Arg := "digraph, vertex,\ distances", Label := "for a digraph, a pos int, and a list", Name := "DigraphDistance\ Set" ), content := 0, count := [ 5, 3, 5, 4 ], mathmode := "Text", name := "Oper", next := 179083, root := ~, start := 178972, stop := 179082 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "Returns", next := 179123, root := ~, start := 179086, stop := 179122 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns the list of all vertices in digraph ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 179217, root := ~, start := 179203, stop := 179216 ), rec( content := "\n su\ ch that the shortest distance to a vertex ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 179279, root := ~, start := 179266, stop := 179278 ), rec( content := " is\n \ ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ istance", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 179302, root := ~, start := 179287, stop := 179301 ), rec( content := " or is i\ n the list ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istances", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 179337, root := ~, start := 179321, stop := 179336 ), rec( content := ". ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "P", next := 179343, root := ~, start := 179339, stop := 179342 ), rec( content := "\n\n \ ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 179363, root := ~, start := 179349, stop := 179362 ), rec( content := " should \ be a digraph, ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 179398, root := ~, start := 179385, stop := 179397 ), rec( content := " should \ be a positive\n integer, ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istance", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 179448, root := ~, start := 179433, stop := 179447 ), rec( content := " should \ be a non-negative integer, and\n ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istances", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 179507, root := ~, start := 179491, stop := 179506 ), rec( content := " should \ be a list of non-negative integers.\n\n ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [1, 4], [1, 3]]);\n\ngap> DigraphDistanceSet(D, 2, [1, 2]);\n[ 3, 1, 4 ]\ngap> Dig\ raphDistanceSet(D, 3, 1);\n[ 1, 4 ]\ngap> DigraphDistanceSet(D, 2, 0);\n[ 2 ]\ \n", count := [ 5, 3, 5, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 7 ], mathmode := "Text", name := "Example", next := 179814, root := ~, start := 179556, stop := 179813 ), rec( content := "\n ", count := [ 5, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 179831, root := ~, start := 179126, stop := 179830 ) ], count := [ 5, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 179845, root := ~, start := 178844, stop := 179844 ), rec( content := "\n\n ", count := [ 5, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphGirth" ) , content := 0, count := [ 5, 3, 6, 2 ], mathmode := "Text", name := "Attr", next := 179907, root := ~, start := 179866, stop := 179906 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r, or ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "K", next := 179949, root := ~, start := 179934, stop := 179948 ), rec( content := ".", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 179960, root := ~, start := 179910, stop := 179959 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s attribute returns the ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ irth", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "E", next := 180020, root := ~, start := 180008, stop := 180019 ), rec( content := " of the \ digraph ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "A", next := 180050, root := ~, start := 180036, stop := 180049 ), rec( content := ".\n T\ he ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ irth", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "E", next := 180072, root := ~, start := 180060, stop := 180071 ), rec( content := " of a di\ graph is the length of its shortest simple circuit.\n See section ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Ref", next := 180192, root := ~, start := 180148, stop := 180191 ), rec( content := " for the\ definitions\n of simple circuit, directed cycle, and loop.\n ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "P", next := 180270, root := ~, start := 180266, stop := 180269 ), rec( content := "\n\n \ If ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "A", next := 180293, root := ~, start := 180279, stop := 180292 ), rec( content := " has no \ directed cycles, then this function will return\n ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "K", next := 180368, root := ~, start := 180353, stop := 180367 ), rec( content := ". If ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "A", next := 180388, root := ~, start := 180374, stop := 180387 ), rec( content := " contain\ s a loop, then this function will\n return ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "C", next := 180449, root := ~, start := 180441, stop := 180448 ), rec( content := ".\n " , count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "P", next := 180459, root := ~, start := 180455, stop := 180458 ), rec( content := "\n\n \ In the worst case, the method used in this function is a version of the\n F\ loyd-Warshall algorithm, and has complexity ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 5, 3, 6, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := \ "Text", name := "A", next := 180599, root := ~, start := 180591, stop := 18059\ 8 ), rec( content := " ^ 3)", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "C", next := 180608, root := ~, start := 180586, stop := 180607 ), rec( content := ", where\\ n ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "A", next := 180628, root := ~, start := 180620, stop := 180627 ), rec( content := " is the \ number of vertices in ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "A", next := 180672, root := ~, start := 180658, stop := 180671 ), rec( content := ". If th\ e digraph has\n known automorphisms [see ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "Ref", next := 180749, root := ~, start := 180723, stop := 180748 ), rec( content := "], then \ the performance\n is likely to be better.\n ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "P", next := 180809, root := ~, start := 180805, stop := 180808 ), rec( content := "\n\n \ For symmetric digraphs, see also ", count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ UndirectedGirth" ), content := 0, count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "Ref", next := 180884, root := ~, start := 180848, stop := 180883 ), rec( content := ". ", count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 7 ], mathmode := "Text", name := "P", next := 180890, root := ~, start := 180886, stop := 180889 ), rec( content := "\n ", count := [ 5, 3, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [1]]);\n\ \ngap> DigraphGirth(D);\n1\ngap> D := Digraph([[2, 3], [3], [4], []]);\n\ngap> DigraphGirth(D);\ninfinity\ngap> \ D := Digraph([[2, 3], [3], [4], [1]]);\n\ngap> DigraphGirth(D);\n3\ngap> D := GeneralisedPetersenGraph(IsMutable\ Digraph, 6, 2);\n\ngap> DigraphGir\ th(D);\n2\n", count := [ 5, 3, 6, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 8 ], mathmode := "Text", name := "Example", next := 181388, root := ~, start := 180895, stop := 181387 ), rec( content := "\n ", count := [ 5, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 181405, root := ~, start := 179963, stop := 181404 ) ], count := [ 5, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 181419, root := ~, start := 179851, stop := 181418 ), rec( content := "\n\n ", count := [ 5, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphOddGirth\ " ), content := 0, count := [ 5, 3, 7, 2 ], mathmode := "Text", name := "Attr", next := 181484, root := ~, start := 181440, stop := 181483 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r, or ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "K", next := 181526, root := ~, start := 181511, stop := 181525 ), rec( content := ".", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 181537, root := ~, start := 181487, stop := 181536 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s attribute returns the ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ dd girth", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "E", next := 181601, root := ~, start := 181585, stop := 181600 ), rec( content := " of the \ digraph ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "A", next := 181631, root := ~, start := 181617, stop := 181630 ), rec( content := ".\n T\ he ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ dd girth", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "E", next := 181657, root := ~, start := 181641, stop := 181656 ), rec( content := " of a di\ graph is the length of its shortest simple\n circuit of odd length. See Se\ ction ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Ref", next := 181796, root := ~, start := 181748, stop := 181795 ), rec( content := " for the\ definitions of simple circuit, directed\n cycle, and loop.\n ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "P", next := 181874, root := ~, start := 181870, stop := 181873 ), rec( content := "\n\n \ If ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "A", next := 181897, root := ~, start := 181883, stop := 181896 ), rec( content := " has no \ directed cycles of odd length, then this function\n will return ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "K", next := 181986, root := ~, start := 181971, stop := 181985 ), rec( content := ", even i\ f ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "A", next := 182010, root := ~, start := 181996, stop := 182009 ), rec( content := " has a d\ irected cycle\n of even length. If ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "A", next := 182070, root := ~, start := 182056, stop := 182069 ), rec( content := " contain\ s a loop, then this function\n will return ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "C", next := 182131, root := ~, start := 182123, stop := 182130 ), rec( content := ".\n " , count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "P", next := 182141, root := ~, start := 182137, stop := 182140 ), rec( content := "\n\n \ See also ", count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Girth" ), content := 0, count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 182182, root := ~, start := 182156, stop := 182181 ), rec( content := ". ", count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 7, 6 ], mathmode := "Text", name := "P", next := 182188, root := ~, start := 182184, stop := 182187 ), rec( content := "\n ", count := [ 5, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3, 1], [1]]);\n\ngap> DigraphOddGirth(D);\n3\ngap> D := CycleDigraph(4);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := D\ igraph([[2], [3], [], [3], [4]]);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := CycleDigraph(IsMutableDigraph\ , 4);\n\ngap> DigraphDisjointUnion(D\ , CycleDigraph(5));\n\ngap> DigraphO\ ddGirth(D);\n5\n", count := [ 5, 3, 7, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 7 ], mathmode := "Text", name := "Example", next := 182768, root := ~, start := 182193, stop := 182767 ), rec( content := "\n ", count := [ 5, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 182785, root := ~, start := 181540, stop := 182784 ) ], count := [ 5, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 182799, root := ~, start := 181425, stop := 182798 ), rec( content := "\n\n ", count := [ 5, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphUndirect\ edGirth" ), content := 0, count := [ 5, 3, 8, 2 ], mathmode := "Text", name := "Attr", next := 182871, root := ~, start := 182820, stop := 182870 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r or ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "K", next := 182912, root := ~, start := 182897, stop := 182911 ), rec( content := ".", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 182923, root := ~, start := 182874, stop := 182922 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "A", next := 182961, root := ~, start := 182947, stop := 182960 ), rec( content := " is a sy\ mmetric digraph, then this function returns the\n girth of ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "A", next := 183044, root := ~, start := 183030, stop := 183043 ), rec( content := " when tr\ eated as an undirected graph (i.e. each pair\n of edges ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 183123, root := ~, start := 183110, stop := 183122 ), rec( content := " and ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j, i]", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 183141, root := ~, start := 183128, stop := 183140 ), rec( content := " is trea\ ted as a single edge between\n ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 183190, root := ~, start := 183182, stop := 183189 ), rec( content := " and ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 183203, root := ~, start := 183195, stop := 183202 ), rec( content := "). ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "P", next := 183210, root := ~, start := 183206, stop := 183209 ), rec( content := "\n\n \ The ", count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ irth", count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "E", next := 183232, root := ~, start := 183220, stop := 183231 ), rec( content := " of an u\ ndirected graph is the length of its shortest simple\n cycle, i.e. the shor\ test non-trivial path starting and ending at the same\n vertex and passing \ through no vertex or edge more than once. ", count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "P", next := 183440, root := ~, start := 183436, stop := 183439 ), rec( content := "\n\n \ If ", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "A", next := 183463, root := ~, start := 183449, stop := 183462 ), rec( content := " has no \ cycles, then this function will return\n ", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "K", next := 183529, root := ~, start := 183514, stop := 183528 ), rec( content := ". ", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 8, 6 ], mathmode := "Text", name := "P", next := 183535, root := ~, start := 183531, stop := 183534 ), rec( content := "\n ", count := [ 5, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 4], [1, 3], [2, 4], [1, 3]]);\n\ngap> DigraphUndirectedGirth(D);\n4\ngap> D := Digraph([\ [2], [1, 3], [2]]);\n\ngap> Digrap\ hUndirectedGirth(D);\ninfinity\ngap> D := Digraph([[1], [], [4], [3]]);\n\ngap> DigraphUndirectedGirth(D);\n1\ng\ ap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> DigraphUndirectedGirth(D);\n5\n", count := [ 5, 3, 8, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 8, 7 ], mathmode := "Text", name := "Example", next := 184087, root := ~, start := 183540, stop := 184086 ), rec( content := "\n ", count := [ 5, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 184104, root := ~, start := 182926, stop := 184103 ) ], count := [ 5, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 184118, root := ~, start := 182805, stop := 184117 ), rec( content := "\n\n ", count := [ 5, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphConnecte\ dComponents" ), content := 0, count := [ 5, 3, 9, 2 ], mathmode := "Text", name := "Attr", next := 184194, root := ~, start := 184139, stop := 184193 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrConnec\ tedComponents" ), content := 0, count := [ 5, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 184254, root := ~, start := 184197, stop := 184253 ), rec( attributes := rec( ), content := [ rec( content := "A record.\ ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 184285, root := ~, start := 184257, stop := 184284 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the record ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "w\ cc", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "C", next := 184349, root := ~, start := 184339, stop := 184348 ), rec( content := " corresp\ onding to the weakly\n connected components of the digraph ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 184432, root := ~, start := 184418, stop := 184431 ), rec( content := ". Two v\ ertices of\n ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 184469, root := ~, start := 184455, stop := 184468 ), rec( content := " are in \ the same weakly connected component whenever they are\n equal, or there exi\ sts a directed path (ignoring the orientation of edges)\n between them. Mo\ re formally, two vertices are in the same weakly connected\n component of " , count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 184721, root := ~, start := 184707, stop := 184720 ), rec( content := " if and \ only if they are in the same strongly\n connected component (see ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ StronglyConnectedComponents" ), content := 0, count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 184844, root := ~, start := 184796, stop := 184843 ), rec( content := ")\n o\ f the ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ SymmetricClosure" ), content := 0, count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 184894, root := ~, start := 184857, stop := 184893 ), rec( content := " of ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 184912, root := ~, start := 184898, stop := 184911 ), rec( content := ". ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "P", next := 184919, root := ~, start := 184915, stop := 184918 ), rec( content := "\n\n \ The set of weakly connected components is a partition of\n the vertex set o\ f ", count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "A", next := 185018, root := ~, start := 185004, stop := 185017 ), rec( content := ". ", count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "P", next := 185024, root := ~, start := 185020, stop := 185023 ), rec( content := "\n\n \ The record ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "w\ cc", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185051, root := ~, start := 185041, stop := 185050 ), rec( content := " has 2 c\ omponents: ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185082, root := ~, start := 185070, stop := 185081 ), rec( content := " and ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185096, root := ~, start := 185087, stop := 185095 ), rec( content := ".\n T\ he component ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185128, root := ~, start := 185116, stop := 185127 ), rec( content := " is a li\ st of the weakly connected components\n of ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "A", next := 185196, root := ~, start := 185182, stop := 185195 ), rec( content := " (each o\ f which is a list of vertices).\n The component ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185263, root := ~, start := 185254, stop := 185262 ), rec( content := " is a li\ st such that the vertex ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185303, root := ~, start := 185295, stop := 185302 ), rec( content := " is an\n\ element of the weakly connected component ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps[id[i]]", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185375, root := ~, start := 185356, stop := 185374 ), rec( content := ". ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "P", next := 185381, root := ~, start := 185377, stop := 185380 ), rec( content := "\n\n \ The method used in this function has complexity ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "M", next := 185449, root := ~, start := 185436, stop := 185448 ), rec( content := ", where\\ n ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "M", next := 185469, root := ~, start := 185461, stop := 185468 ), rec( content := " is the \ number of edges and\n ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "M", next := 185509, root := ~, start := 185501, stop := 185508 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "P", next := 185556, root := ~, start := 185552, stop := 185555 ), rec( content := "\n\n \ ", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrConnectedComponents(", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "A", next := 1856\ 08, root := ~, start := 185594, stop := 185607 ), rec( content := ")", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 185613, root := ~, start := 185562, stop := 185612 ), rec( content := " is simp\ ly a shortcut\n for ", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ength(DigraphConnectedComponents(", count := [ 5, 3, 9, 9 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "digraph", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "A", next := 1856\ 94, root := ~, start := 185680, stop := 185693 ), rec( content := ").comps)", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 185706, root := ~, start := 185643, stop := 185705 ), rec( content := ",\n a\ nd is no more efficient.\n\n ", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2], [3, 1], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ [ 1, 2, 3 ] ], \ id := [ 1, 1, 1 ] )\ngap> gr := Digraph([[1], [1, 2], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps \ := [ [ 1, 2 ], [ 3 ] ], id := [ 1, 1, 2 ] )\ngap> DigraphNrConnectedComponents\ (gr);\n2\ngap> gr := EmptyDigraph(0);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ ], id := [ ] )\nga\ p> D := CycleDigraph(IsMutableDigraph, 2);\n\ngap> G := CycleDigraph(3);\n\ngap> DigraphDisjointUnion(D, G);\n\ngap> DigraphConnectedComponents(D);\nrec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] \ ], id := [ 1, 1, 2, 2, 2 ] )\n", count := [ 5, 3, 9, 10 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 10 ], mathmode := "Text", name := "Example", next := 186640, root := ~, start := 185743, stop := 186639 ), rec( content := "\n ", count := [ 5, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 186657, root := ~, start := 184288, stop := 186656 ) ], count := [ 5, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 186671, root := ~, start := 184124, stop := 186670 ), rec( content := "\n\n ", count := [ 5, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphConnecte\ dComponent" ), content := 0, count := [ 5, 3, 10, 2 ], mathmode := "Text", name := "Oper", next := 186754, root := ~, start := 186692, stop := 186753 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Returns", next := 186795, root := ~, start := 186757, stop := 186794 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 186832, root := ~, start := 186819, stop := 186831 ), rec( content := " is a ve\ rtex in the digraph ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 186874, root := ~, start := 186860, stop := 186873 ), rec( content := ", then\n\ this operation returns the connected component of ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 186948, root := ~, start := 186935, stop := 186947 ), rec( content := "\n in\ ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 186970, root := ~, start := 186956, stop := 186969 ), rec( content := ".\n S\ ee ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Ref", next := 187020, root := ~, start := 186980, stop := 187019 ), rec( content := " for mor\ e information.\n ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [2], [1, 2], [4]]);\n\ngap> DigraphConnectedComponent(D, 3);\n[ 1, 2, 3 ]\ngap> Digraph\ ConnectedComponent(D, 2);\n[ 1, 2, 3 ]\ngap> DigraphConnectedComponent(D, 4);\ \n[ 4 ]\n", count := [ 5, 3, 10, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 4 ], mathmode := "Text", name := "Example", next := 187312, root := ~, start := 187047, stop := 187311 ), rec( content := "\n ", count := [ 5, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 187329, root := ~, start := 186798, stop := 187328 ) ], count := [ 5, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 187343, root := ~, start := 186677, stop := 187342 ), rec( content := "\n\n ", count := [ 5, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphStrongly\ ConnectedComponents" ), content := 0, count := [ 5, 3, 11, 2 ], mathmode := "Text", name := "Attr", next := 187427, root := ~, start := 187364, stop := 187426 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrStrong\ lyConnectedComponents" ), content := 0, count := [ 5, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 187495, root := ~, start := 187430, stop := 187494 ), rec( attributes := rec( ), content := [ rec( content := "A record.\ ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 187526, root := ~, start := 187498, stop := 187525 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the record ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ cc", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "C", next := 187590, root := ~, start := 187580, stop := 187589 ), rec( content := " corresp\ onding to the strongly\n connected components of the digraph ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "A", next := 187675, root := ~, start := 187661, stop := 187674 ), rec( content := ". Two v\ ertices of\n ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "A", next := 187712, root := ~, start := 187698, stop := 187711 ), rec( content := " are in \ the same strongly connected component whenever they\n are equal, or there i\ s a directed path from each vertex to the other. The\n set of strongly con\ nected components is a partition of the vertex set of\n ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "A", next := 187946, root := ~, start := 187932, stop := 187945 ), rec( content := ".\n " , count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "P", next := 187956, root := ~, start := 187952, stop := 187955 ), rec( content := "\n\n \ The record ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ cc", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 187983, root := ~, start := 187973, stop := 187982 ), rec( content := " has 2 c\ omponents: ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188014, root := ~, start := 188002, stop := 188013 ), rec( content := " and ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188028, root := ~, start := 188019, stop := 188027 ), rec( content := ".\n T\ he component ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188060, root := ~, start := 188048, stop := 188059 ), rec( content := " is a li\ st of the strongly connected components\n of ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "A", next := 188129, root := ~, start := 188115, stop := 188128 ), rec( content := " (each o\ f which is a list of vertices).\n The component ", count := [ 5, 3, 11, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188196, root := ~, start := 188187, stop := 188195 ), rec( content := " is a li\ st such that the vertex ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188236, root := ~, start := 188228, stop := 188235 ), rec( content := " is an\n\ element of the strongly connected component ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps[id[i]]", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188310, root := ~, start := 188291, stop := 188309 ), rec( content := ". ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "P", next := 188316, root := ~, start := 188312, stop := 188315 ), rec( content := "\n\n \ The method used in this function is a non-recursive version of Gabow's\n Al\ gorithm ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Gab00" ) , content := 0, count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "Cite", next := 188426, root := ~, start := 188407, stop := 188425 ), rec( content := " and has\ complexity ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "M", next := 188459, root := ~, start := 188446, stop := 188458 ), rec( content := " where\n\ ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "M", next := 188478, root := ~, start := 188470, stop := 188477 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "M", next := 188551, root := ~, start := 188543, stop := 188550 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "P", next := 188598, root := ~, start := 188594, stop := 188597 ), rec( content := "\n\n \ ", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrStronglyConnectedComponents(", count := [ 5, 3, 11, 8 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "A", next := 1886\ 58, root := ~, start := 188644, stop := 188657 ), rec( content := ")", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 188663, root := ~, start := 188604, stop := 188662 ), rec( content := " is simp\ ly a\n shortcut for\n ", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ength(DigraphStronglyConnectedComponents(", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "A", next := 1887\ 56, root := ~, start := 188742, stop := 188755 ), rec( content := ").comps)", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 188768, root := ~, start := 188697, stop := 188767 ), rec( content := ",\n a\ nd is no more efficient.\n\n ", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2], [3, 1], []]);\n\ngap> DigraphStronglyConnectedComponents(gr);\nrec( comps := [ [ 3 ], \ [ 1, 2 ] ], id := [ 2, 2, 1 ] )\ngap> DigraphNrStronglyConnectedComponents(gr)\ ;\n2\ngap> D := DigraphDisjointUnion(CycleDigraph(4), CycleDigraph(5));\n\ngap> DigraphStronglyConnectedComponen\ ts(D);\nrec( comps := [ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8, 9 ] ], \n id := [ 1, 1,\ 1, 1, 2, 2, 2, 2, 2 ] )\ngap> DigraphNrStronglyConnectedComponents(D);\n2\nga\ p> D := CycleDigraph(IsMutableDigraph, 2);\n\ngap> G := CycleDigraph(3);\n\ngap> DigraphDisjointUnion(D, G);\n\ngap> DigraphStronglyConnectedComponents(D);\nrec( comps := [ [ 1, 2 ], [ 3,\ 4, 5 ] ], id := [ 1, 1, 2, 2, 2 ] )\n", count := [ 5, 3, 11, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 11, 9 ], mathmode := "Text", name := "Example", next := 189711, root := ~, start := 188805, stop := 189710 ), rec( content := "\n ", count := [ 5, 3, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 189728, root := ~, start := 187529, stop := 189727 ) ], count := [ 5, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 189742, root := ~, start := 187349, stop := 189741 ), rec( content := "\n\n ", count := [ 5, 3, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphStrongly\ ConnectedComponent" ), content := 0, count := [ 5, 3, 12, 2 ], mathmode := "Text", name := "Oper", next := 189833, root := ~, start := 189763, stop := 189832 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 189874, root := ~, start := 189836, stop := 189873 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 189911, root := ~, start := 189898, stop := 189910 ), rec( content := " is a ve\ rtex in the digraph ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 189953, root := ~, start := 189939, stop := 189952 ), rec( content := ", then\n\ this operation returns the strongly connected component of ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 190036, root := ~, start := 190023, stop := 190035 ), rec( content := "\n in\ ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 190058, root := ~, start := 190044, stop := 190057 ), rec( content := ".\n S\ ee ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ StronglyConnectedComponents" ), content := 0, count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Ref", next := 190116, root := ~, start := 190068, stop := 190115 ), rec( content := " for mor\ e information.\n ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [2], [1, 2], [3]]);\n\ngap> DigraphStronglyConnectedComponent(D, 3);\n[ 1, 3 ]\ngap> Di\ graphStronglyConnectedComponent(D, 2);\n[ 2 ]\ngap> DigraphStronglyConnectedCo\ mponent(D, 4);\n[ 4 ]\n", count := [ 5, 3, 12, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 4 ], mathmode := "Text", name := "Example", next := 190423, root := ~, start := 190143, stop := 190422 ), rec( content := "\n ", count := [ 5, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 190440, root := ~, start := 189877, stop := 190439 ) ], count := [ 5, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 190454, root := ~, start := 189748, stop := 190453 ), rec( content := "\n\n ", count := [ 5, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphBicompon\ ents" ), content := 0, count := [ 5, 3, 13, 2 ], mathmode := "Text", name := "Attr", next := 190523, root := ~, start := 190475, stop := 190522 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists of vertices, or ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "K", next := 190578, root := ~, start := 190567, stop := 190577 ), rec( content := ".", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Returns", next := 190589, root := ~, start := 190526, stop := 190588 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "A", next := 190627, root := ~, start := 190613, stop := 190626 ), rec( content := " is a bi\ partite digraph, i.e. if it satisfies ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Ref", next := 190711, root := ~, start := 190673, stop := 190710 ), rec( content := ", then " , count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphBicomponents", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "C", next := 190744, root := ~, start := 190718, stop := 190743 ), rec( content := " returns\ a\n pair of bicomponents of ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "A", next := 190797, root := ~, start := 190783, stop := 190796 ), rec( content := ". Otherw\ ise,\n ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphBicomponents", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "C", next := 190840, root := ~, start := 190814, stop := 190839 ), rec( content := " returns\ ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "K", next := 190860, root := ~, start := 190849, stop := 190859 ), rec( content := ".", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "P", next := 190865, root := ~, start := 190861, stop := 190864 ), rec( content := "\n\n \ For a bipartite digraph, the vertices can be partitioned into two non-empty\n \ sets such that the source and range of any edge are in distinct sets. The\n\ parts of this partition are called ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ icomponents", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "E", next := 191083, root := ~, start := 191064, stop := 191082 ), rec( content := " of ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "A", next := 191101, root := ~, start := 191087, stop := 191100 ), rec( content := ".\n E\ quivalently, a pair of bicomponents of ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "A", next := 191161, root := ~, start := 191147, stop := 191160 ), rec( content := " consist\ s of the\n color-classes of a 2-coloring of ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "A", next := 191229, root := ~, start := 191215, stop := 191228 ), rec( content := ". ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "P", next := 191235, root := ~, start := 191231, stop := 191234 ), rec( content := "\n\n \ For a bipartite digraph with at least 3 vertices, there is a unique pair of\n \ bicomponents of bipartite if and only if the digraph is connected. See ", count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "Ref", next := 191430, root := ~, start := 191392, stop := 191429 ), rec( content := " for mor\ e information.\n ", count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(3);\n\ngap> D\ igraphBicomponents(D);\nfail\ngap> D := ChainDigraph(5);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 3, 5 ], [ 2, 4 ] ]\ \ngap> D := Digraph([[5], [1, 4], [5], [5], []]);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 3, 4 ], [ 2, 5 ] ]\nga\ p> D := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 2 ], [ 3, 4, 5 \ ] ]\n", count := [ 5, 3, 13, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 6 ], mathmode := "Text", name := "Example", next := 192023, root := ~, start := 191457, stop := 192022 ), rec( content := "\n ", count := [ 5, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Description", next := 192040, root := ~, start := 190592, stop := 192039 ) ], count := [ 5, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 192054, root := ~, start := 190460, stop := 192053 ), rec( content := "\n\n ", count := [ 5, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "ArticulationPoi\ nts" ), content := 0, count := [ 5, 3, 14, 2 ], mathmode := "Text", name := "Attr", next := 192122, root := ~, start := 192075, stop := 192121 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 192163, root := ~, start := 192125, stop := 192162 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ iconnected", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "E", next := 192225, root := ~, start := 192207, stop := 192224 ), rec( content := " if it i\ s still connected (in the\n sense of ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 192304, root := ~, start := 192272, stop := 192303 ), rec( content := ") when a\ ny vertex is removed. \n If the digraph ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "A", next := 192368, root := ~, start := 192354, stop := 192367 ), rec( content := " is not \ biconnected but is connected, then any\n vertex ", count := [ 5, 3, 14, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "C", next := 192434, root := ~, start := 192426, stop := 192433 ), rec( content := " of ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "A", next := 192452, root := ~, start := 192438, stop := 192451 ), rec( content := " whose r\ emoval makes the resulting digraph\n disconnected is called an ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ rticulation point", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "E", next := 192550, root := ~, start := 192525, stop := 192549 ), rec( content := ".", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "P", next := 192555, root := ~, start := 192551, stop := 192554 ), rec( content := "\n\n \ ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "A\ rticulationPoints", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "C", next := 192586, root := ~, start := 192561, stop := 192585 ), rec( content := " returns\ a list of the articulation points of\n ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "A", next := 192650, root := ~, start := 192636, stop := 192649 ), rec( content := ", if any\ , and, in particular, returns the empty list if\n ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "A", next := 192724, root := ~, start := 192710, stop := 192723 ), rec( content := " is not \ connected. ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 5 ], mathmode := "Text", name := "P", next := 192747, root := ~, start := 192743, stop := 192746 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 5, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "P", next := 192800, root := ~, start := 192796, stop := 192799 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "M", next := 192868, root := ~, start := 192855, stop := 192867 ), rec( content := " where\n\ ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "M", next := 192887, root := ~, start := 192879, stop := 192886 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "M", next := 192923, root := ~, start := 192915, stop := 192922 ), rec( content := " is the \ number of vertices in\n the digraph.", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "P", next := 192973, root := ~, start := 192969, stop := 192972 ), rec( content := "\n \ \n If ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "A", next := 192996, root := ~, start := 192988, stop := 192995 ), rec( content := " has a b\ ridge (see ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "Ref", next := 193036, root := ~, start := 193015, stop := 193035 ), rec( content := "), then \ a node incident\n to the bridge is an articulation point if and only if it \ has degree at\n least ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "M", next := 193153, root := ~, start := 193145, stop := 193152 ), rec( content := ". It fo\ llows that if ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "A", next := 193183, root := ~, start := 193175, stop := 193182 ), rec( content := " has a b\ ridge and at least\n ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "M", next := 193222, root := ~, start := 193214, stop := 193221 ), rec( content := " nodes, \ then at least one of the nodes incident to the bridge is\n an articulation \ point. The converse does not hold, there are digraphs\n with articulation \ points, but no bridges.\n ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "P", next := 193416, root := ~, start := 193412, stop := 193415 ), rec( content := "\n\n \ See also ", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "Ref", next := 193465, root := ~, start := 193431, stop := 193464 ), rec( content := " and \n \ ", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "Ref", next := 193508, root := ~, start := 193475, stop := 193507 ), rec( content := ".\n", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ArticulationPoints(CycleDigraph(5));\n[ ]\ngap> D := Digraph([[2, 7], [\ 3, 5], [4], [2], [6], [1], []]);;\ngap> ArticulationPoints(D);\n[ 2, 1 ]\ngap>\ ArticulationPoints(ChainDigraph(5));\n[ 4, 3, 2 ]\ngap> ArticulationPoints(Nu\ llDigraph(5));\n[ ]\ngap> D := ChainDigraph(IsMutableDigraph, 4);\n\ngap> ArticulationPoints(D);\n[ 3, 2 ]\n", count := [ 5, 3, 14, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 9 ], mathmode := "Text", name := "Example", next := 193913, root := ~, start := 193510, stop := 193912 ), rec( content := "\n ", count := [ 5, 3, 14, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 193930, root := ~, start := 192166, stop := 193929 ) ], count := [ 5, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 193944, root := ~, start := 192060, stop := 193943 ), rec( content := "\n\n ", count := [ 5, 3, 14, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "Bridges" ), content := 0, count := [ 5, 3, 15, 2 ], mathmode := "Text", name := "Attr", next := 193995, root := ~, start := 193965, stop := 193994 ), rec( attributes := rec( ), content := [ rec( content := "A (possib\ ly empty) list of edges.", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Returns", next := 194050, root := ~, start := 193998, stop := 194049 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -edge-connected", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "E", next := 194117, root := ~, start := 194094, stop := 194116 ), rec( content := " if it i\ s still connected (in\n the sense of ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Ref", next := 194196, root := ~, start := 194164, stop := 194195 ), rec( content := ") when a\ ny edge is\n removed. If the digraph ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "A", next := 194252, root := ~, start := 194244, stop := 194251 ), rec( content := " is not \ 2-edge-connected but is\n connected, then any edge ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "C", next := 194326, root := ~, start := 194313, stop := 194325 ), rec( content := " of ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "A", next := 194338, root := ~, start := 194330, stop := 194337 ), rec( content := " whose r\ emoval\n makes the resulting digraph disconnected is called a ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ ridge", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "E", next := 194423, root := ~, start := 194410, stop := 194422 ), rec( content := ".", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "P", next := 194428, root := ~, start := 194424, stop := 194427 ), rec( content := "\n\n \ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ ridges", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "C", next := 194448, root := ~, start := 194434, stop := 194447 ), rec( content := " returns\ a list of the bridges of ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "A", next := 194490, root := ~, start := 194482, stop := 194489 ), rec( content := ", if any\ , and, in\n particular, returns the empty list if ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "A", next := 194558, root := ~, start := 194550, stop := 194557 ), rec( content := " is not \ connected. ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "P", next := 194581, root := ~, start := 194577, stop := 194580 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "P", next := 194634, root := ~, start := 194630, stop := 194633 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "M", next := 194702, root := ~, start := 194689, stop := 194701 ), rec( content := " where\n\ ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "M", next := 194721, root := ~, start := 194713, stop := 194720 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "M", next := 194757, root := ~, start := 194749, stop := 194756 ), rec( content := " is the \ number of vertices in\n the digraph.\n ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "P", next := 194812, root := ~, start := 194808, stop := 194811 ), rec( content := "\n\n \ If ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "A", next := 194829, root := ~, start := 194821, stop := 194828 ), rec( content := " has a b\ ridge, then a node incident to the bridge is an\n articulation point (see " , count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 194945, root := ~, start := 194913, stop := 194944 ), rec( content := ") if and\ only if\n it has degree at least ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "M", next := 194997, root := ~, start := 194989, stop := 194996 ), rec( content := ". It fo\ llows that if ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "A", next := 195027, root := ~, start := 195019, stop := 195026 ), rec( content := " has a\n\ bridge and at least ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "M", next := 195066, root := ~, start := 195058, stop := 195065 ), rec( content := " nodes, \ then at least one of the nodes\n incident to the bridge is an articulation \ point. The converse does not\n hold, there are digraphs with articulation p\ oints, but no bridges.\n ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "P", next := 195259, root := ~, start := 195255, stop := 195258 ), rec( content := "\n\n \ See also ", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "Ref", next := 195308, root := ~, start := 195274, stop := 195307 ), rec( content := " and \n \ ", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "Ref", next := 195351, root := ~, start := 195318, stop := 195350 ), rec( content := ".\n", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 5], [1, 3, 4, 5], [2, 4], [2, 3], [1, 2]]);\n\ngap> Bridges(D);\n[ ]\ngap> D := Digrap\ h([[2], [3], [4], [2]]);\n\ngap> B\ ridges(D);\n[ [ 1, 2 ] ]\ngap> Bridges(ChainDigraph(2));\n[ [ 1, 2 ] ]\ngap> A\ rticulationPoints(ChainDigraph(2));\n[ ]\n", count := [ 5, 3, 15, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 15, 9 ], mathmode := "Text", name := "Example", next := 195727, root := ~, start := 195353, stop := 195726 ), rec( content := "\n ", count := [ 5, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Description", next := 195744, root := ~, start := 194053, stop := 195743 ) ], count := [ 5, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 195758, root := ~, start := 193950, stop := 195757 ), rec( content := "\n\n ", count := [ 5, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "StrongOrientati\ on" ), content := 0, count := [ 5, 3, 16, 2 ], mathmode := "Text", name := "Oper", next := 195819, root := ~, start := 195779, stop := 195818 ), rec( attributes := rec( Arg := "D", Name := "StrongOrientati\ onAttr" ), content := 0, count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "Attr", next := 195866, root := ~, start := 195822, stop := 195865 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ or ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "K", next := 195902, root := ~, start := 195891, stop := 195901 ), rec( content := ".", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Returns", next := 195913, root := ~, start := 195869, stop := 195912 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ trong orientation", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "E", next := 195961, root := ~, start := 195936, stop := 195960 ), rec( content := " of a co\ nnected symmetric digraph ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 196003, root := ~, start := 195995, stop := 196002 ), rec( content := " (if\n \ it exists) is a strongly connected subdigraph ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196066, root := ~, start := 196058, stop := 196065 ), rec( content := " of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 196078, root := ~, start := 196070, stop := 196077 ), rec( content := " such\n \ that for every edge ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196121, root := ~, start := 196108, stop := 196120 ), rec( content := " of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 196133, root := ~, start := 196125, stop := 196132 ), rec( content := " either \ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196154, root := ~, start := 196141, stop := 196153 ), rec( content := " or\n \ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v, u]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196175, root := ~, start := 196162, stop := 196174 ), rec( content := " is an e\ dge of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196198, root := ~, start := 196190, stop := 196197 ), rec( content := " but not\ both. Robbin's Theorem states\n that a digraph admits a strong orientatio\ n if and only if it is\n bridgeless (see ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Ref", next := 196359, root := ~, start := 196326, stop := 196358 ), rec( content := "). \n \ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "P", next := 196371, root := ~, start := 196367, stop := 196370 ), rec( content := "\n\n \ This operation returns a strong orientation of the digraph ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 196444, root := ~, start := 196436, stop := 196443 ), rec( content := " if \n \ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 196461, root := ~, start := 196453, stop := 196460 ), rec( content := " is symm\ etric and ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 196487, root := ~, start := 196479, stop := 196486 ), rec( content := " admits \ a strong orientation. If ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 196528, root := ~, start := 196520, stop := 196527 ), rec( content := "\n is\ symmetric but does not admit a strong orientation, then ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "K", next := 196603, root := ~, start := 196592, stop := 196602 ), rec( content := " is\n \ returned. If ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 196632, root := ~, start := 196624, stop := 196631 ), rec( content := " is not \ symmetric, then an error is given. \n ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "P", next := 196684, root := ~, start := 196680, stop := 196683 ), rec( content := "\n\n \ If ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "A", next := 196701, root := ~, start := 196693, stop := 196700 ), rec( content := " is immu\ table, ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ trongOrientation(", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := \ "Text", name := "A", next := 196745, root := ~, start := 196737, stop := 19674\ 4 ), rec( content := ")", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "C", next := 196750, root := ~, start := 196716, stop := 196749 ), rec( content := " returns\ an\n immutable digraph, and if ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "A", next := 196800, root := ~, start := 196792, stop := 196799 ), rec( content := " is muta\ ble, then\n ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ trongOrientation(", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := \ "Text", name := "A", next := 196851, root := ~, start := 196843, stop := 19685\ 0 ), rec( content := ")", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "C", next := 196856, root := ~, start := 196822, stop := 196855 ), rec( content := " returns\ a mutable digraph. ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "P", next := 196888, root := ~, start := 196884, stop := 196887 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "M", next := 196956, root := ~, start := 196943, stop := 196955 ), rec( content := " where\n\ ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "M", next := 196975, root := ~, start := 196967, stop := 196974 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "M", next := 197011, root := ~, start := 197003, stop := 197010 ), rec( content := " is the \ number of vertices in\n the digraph.\n", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> StrongOrientation(DigraphSymmetricClosure(CycleDigraph(5))) \n> = CycleD\ igraph(5);\ntrue\ngap> D := DigraphSymmetricClosure(Digraph(\n> [[2, 7], [3, 5\ ], [4], [2], [6], [1], []]));;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> Stro\ ngOrientation(D);\nfail\ngap> StrongOrientation(NullDigraph(0));\n\ngap> StrongOrientation(DigraphDisjointUnion(Com\ pleteDigraph(3), \n> CompleteDigraph\ (3)));\nfail\n", count := [ 5, 3, 16, 9 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 9 ], mathmode := "Text", name := "Example", next := 197557, root := ~, start := 197058, stop := 197556 ), rec( content := "\n ", count := [ 5, 3, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Description", next := 197574, root := ~, start := 195916, stop := 197573 ) ], count := [ 5, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 197588, root := ~, start := 195764, stop := 197587 ), rec( content := "\n\n ", count := [ 5, 3, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphPeriod" ), content := 0, count := [ 5, 3, 17, 2 ], mathmode := "Text", name := "Attr", next := 197651, root := ~, start := 197609, stop := 197650 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 197684, root := ~, start := 197654, stop := 197683 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the period of the digraph ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "A", next := 197767, root := ~, start := 197753, stop := 197766 ), rec( content := ".", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "P", next := 197772, root := ~, start := 197768, stop := 197771 ), rec( content := "\n\n \ If a digraph ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "A", next := 197805, root := ~, start := 197791, stop := 197804 ), rec( content := " has at \ least one directed cycle, then the period\n is the greatest positive intege\ r which divides the lengths of all directed\n cycles of ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "A", next := 197962, root := ~, start := 197948, stop := 197961 ), rec( content := ". If ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "A", next := 197982, root := ~, start := 197968, stop := 197981 ), rec( content := " has no \ directed cycles, then\n this function returns ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "M", next := 198046, root := ~, start := 198038, stop := 198045 ), rec( content := ". See s\ ection ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "Ref", next := 198111, root := ~, start := 198061, stop := 198110 ), rec( content := " for the\ definition of a directed cycle. ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "P", next := 198156, root := ~, start := 198152, stop := 198155 ), rec( content := "\n\n \ A digraph with a period of ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "M", next := 198197, root := ~, start := 198189, stop := 198196 ), rec( content := " is said\ to be ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ periodic", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "E", next := 198228, root := ~, start := 198212, stop := 198227 ), rec( content := ". See\n\ ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAperi\ odicDigraph" ), content := 0, count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "Ref", next := 198271, root := ~, start := 198239, stop := 198270 ), rec( content := ". ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 17, 6 ], mathmode := "Text", name := "P", next := 198277, root := ~, start := 198273, stop := 198276 ), rec( content := "\n ", count := [ 5, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> DigraphPeriod(D);\n6\ngap> D := Digraph([[2]\ , [3, 5], [4], [5], [1, 2]]);\n\ng\ ap> DigraphPeriod(D);\n1\ngap> D := ChainDigraph(2);\n\ngap> DigraphPeriod(D);\n0\ngap> IsAcyclicDigraph(D);\ntrue\ \ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> DigraphPeriod(D);\n1\n", count := [ 5, 3, 17, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 17, 7 ], mathmode := "Text", name := "Example", next := 198820, root := ~, start := 198282, stop := 198819 ), rec( content := "\n ", count := [ 5, 3, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 198837, root := ~, start := 197687, stop := 198836 ) ], count := [ 5, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 198851, root := ~, start := 197594, stop := 198850 ), rec( content := "\n\n ", count := [ 5, 3, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, func, n\ opath, edge", Name := "DigraphFloydWarshall" ), content := 0, count := [ 5, 3, 18, 2 ], mathmode := "Text", name := "Oper", next := 198941, root := ~, start := 198872, stop := 198940 ), rec( attributes := rec( ), content := [ rec( content := "A matrix.\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 198972, root := ~, start := 198944, stop := 198971 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 199010, root := ~, start := 198996, stop := 199009 ), rec( content := " is a di\ graph with ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "M", next := 199037, root := ~, start := 199029, stop := 199036 ), rec( content := " vertice\ s, then\n this operation returns an ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ \\times n", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "M", next := 199100, root := ~, start := 199083, stop := 199099 ), rec( content := " matrix \ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "C", next := 199118, root := ~, start := 199108, stop := 199117 ), rec( content := " contain\ ing\n the output of a generalised version of the Floyd-Warshall algorithm,\ \n applied to ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 199232, root := ~, start := 199218, stop := 199231 ), rec( content := ". ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "P", next := 199238, root := ~, start := 199234, stop := 199237 ), rec( content := "\n\n \ The operation ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "C", next := 199285, root := ~, start := 199258, stop := 199284 ), rec( content := " is cust\ omised by the arguments\n ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 199332, root := ~, start := 199321, stop := 199331 ), rec( content := ", ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 199347, root := ~, start := 199334, stop := 199346 ), rec( content := ", and ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 199364, root := ~, start := 199353, stop := 199363 ), rec( content := ".\n T\ he arguments ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 199397, root := ~, start := 199384, stop := 199396 ), rec( content := " and ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 199413, root := ~, start := 199402, stop := 199412 ), rec( content := " can be \ arbitrary ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ .\n The argument ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 199474, root := ~, start := 199463, stop := 199473 ), rec( content := " must be\ a function which accepts 4 arguments:\n the matrix ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "C", next := 199546, root := ~, start := 199536, stop := 199545 ), rec( content := ", follow\ ed by 3 postive integers. The function\n ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 199608, root := ~, start := 199597, stop := 199607 ), rec( content := " is wher\ e the work to calculate the desired outcome must be\n performed. ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "P", next := 199687, root := ~, start := 199683, stop := 199686 ), rec( content := "\n\n \ This method initialises the matrix ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 199738, root := ~, start := 199728, stop := 199737 ), rec( content := " by sett\ ing entry\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 199776, root := ~, start := 199760, stop := 199775 ), rec( content := " to equa\ l ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 199797, root := ~, start := 199786, stop := 199796 ), rec( content := " if ther\ e is an edge with source\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 199842, root := ~, start := 199834, stop := 199841 ), rec( content := " and ran\ ge ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 199861, root := ~, start := 199853, stop := 199860 ), rec( content := ", and by\ setting entry ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 199900, root := ~, start := 199884, stop := 199899 ), rec( content := " to equa\ l\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 199927, root := ~, start := 199914, stop := 199926 ), rec( content := " otherwi\ se.\n The final part of ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 199988, root := ~, start := 199961, stop := 199987 ), rec( content := " then ca\ lls the function\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 200028, root := ~, start := 200017, stop := 200027 ), rec( content := " inside \ three nested for loops, over the vertices of\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 200099, root := ~, start := 200085, stop := 200098 ), rec( content := ": ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 6 ], mathmode := "Text", name := "P", next := 200105, root := ~, start := 200101, stop := 200104 ), rec( content := "\n\n", count := [ 5, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ nfor i in DigraphsVertices(digraph) do\n for j in DigraphsVertices(digraph) d\ o\n for k in DigraphsVertices(digraph) do\n func(mat, i, j, k);\n o\ d;\n od;\nod;\n", count := [ 5, 3, 18, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 7 ], mathmode := "Text", name := "Listing", next := 200303, root := ~, start := 200107, stop := 200302 ), rec( content := "\n\n \ The matrix ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "C", next := 200330, root := ~, start := 200320, stop := 200329 ), rec( content := " is then\ returned as the result. An example of using\n ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "C", next := 200414, root := ~, start := 200387, stop := 200413 ), rec( content := " to calc\ ulate the shortest (non-zero) distances\n between the vertices of a digraph\ is shown below: ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 9 ], mathmode := "Text", name := "P", next := 200520, root := ~, start := 200516, stop := 200519 ), rec( content := "\n\n \ ", count := [ 5, 3, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&EAHQeDB\");\n\ngap> func := function(mat, i, j, k)\n> if mat[i][k] <>\ -1 and mat[k][j] <> -1 then\n> if (mat[i][j] = -1) or (mat[i][j] > mat[i]\ [k] + mat[k][j]) then\n> mat[i][j] := mat[i][k] + mat[k][j];\n> fi;\ \n> fi;\n> end;\nfunction( mat, i, j, k ) ... end\ngap> shortest_distances :\ = DigraphFloydWarshall(D, func, -1, 1);;\ngap> Display(shortest_distances);\n[\ [ 3, -1, -1, 2, 1, 2 ],\n [ 4, 2, 1, 3, 2, 1 ],\n [ \ 3, 1, 2, 2, 1, 2 ],\n [ 1, -1, -1, 1, 1, 2 ],\n [ 2, \ -1, -1, 1, 2, 1 ],\n [ 3, -1, -1, 2, 1, 1 ] ]\n", count := [ 5, 3, 18, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "Example", next := 201226, root := ~, start := 200526, stop := 201225 ), rec( content := "\n ", count := [ 5, 3, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 201243, root := ~, start := 198975, stop := 201242 ) ], count := [ 5, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 201257, root := ~, start := 198857, stop := 201256 ), rec( content := "\n\n ", count := [ 5, 3, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "IsReachable" ), content := 0, count := [ 5, 3, 19, 2 ], mathmode := "Text", name := "Oper", next := 201324, root := ~, start := 201278, stop := 201323 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 201347, root := ~, start := 201336, stop := 201346 ), rec( content := " or ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 201363, root := ~, start := 201351, stop := 201362 ), rec( content := ".", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Returns", next := 201374, root := ~, start := 201327, stop := 201373 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 201429, root := ~, start := 201418, stop := 201428 ), rec( content := " if ther\ e exists a non-trivial directed\n walk from vertex ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "A", next := 201498, root := ~, start := 201490, stop := 201497 ), rec( content := " to vert\ ex ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "A", next := 201517, root := ~, start := 201509, stop := 201516 ), rec( content := " in the \ digraph ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "A", next := 201547, root := ~, start := 201533, stop := 201546 ), rec( content := ",\n a\ nd ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 201569, root := ~, start := 201557, stop := 201568 ), rec( content := " if ther\ e does not exist such a directed walk. See section\n ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Ref", next := 201677, root := ~, start := 201633, stop := 201676 ), rec( content := " for the\ definition of a\n non-trivial directed walk.\n ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "P", next := 201741, root := ~, start := 201737, stop := 201740 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sReachable", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "C", next := 201780, root := ~, start := 201762, stop := 201779 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 201831, root := ~, start := 201810, stop := 201830 ), rec( content := " where " , count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 201846, root := ~, start := 201838, stop := 201845 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 201882, root := ~, start := 201874, stop := 201881 ), rec( content := " the num\ ber of\n vertices in ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "A", next := 201927, root := ~, start := 201913, stop := 201926 ), rec( content := ".\n\n", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [2, 3]]);\n\ngap> IsReachable(D, 1, 3);\ntrue\ngap> IsReachable(D, 2, 1);\nfalse\n\ gap> IsReachable(D, 3, 3);\ntrue\ngap> IsReachable(D, 1, 1);\nfalse\n", count := [ 5, 3, 19, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 19, 5 ], mathmode := "Text", name := "Example", next := 202176, root := ~, start := 201930, stop := 202175 ), rec( content := "\n ", count := [ 5, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 202193, root := ~, start := 201377, stop := 202192 ) ], count := [ 5, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 202207, root := ~, start := 201263, stop := 202206 ), rec( content := "\n\n ", count := [ 5, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "DigraphPath" ), content := 0, count := [ 5, 3, 20, 2 ], mathmode := "Text", name := "Oper", next := 202274, root := ~, start := 202228, stop := 202273 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists, or ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "K", next := 202317, root := ~, start := 202306, stop := 202316 ), rec( content := ".", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Returns", next := 202328, root := ~, start := 202277, stop := 202327 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ there exists a non-trivial directed path (or a non-trivial cycle, in the\n \ case that ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 202447, root := ~, start := 202439, stop := 202446 ), rec( content := " ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "=\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "C", next := 202456, root := ~, start := 202448, stop := 202455 ), rec( content := " ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 202465, root := ~, start := 202457, stop := 202464 ), rec( content := ") from v\ ertex ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 202487, root := ~, start := 202479, stop := 202486 ), rec( content := " to vert\ ex\n ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 202510, root := ~, start := 202502, stop := 202509 ), rec( content := " in the \ digraph ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 202540, root := ~, start := 202526, stop := 202539 ), rec( content := ", then t\ his operation returns such a\n directed path (or directed cycle). Otherwise\ , this operation returns\n ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "K", next := 202665, root := ~, start := 202654, stop := 202664 ), rec( content := ". See S\ ection ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Ref", next := 202721, root := ~, start := 202680, stop := 202720 ), rec( content := " for the\ \n definition of a directed path and a directed cycle.\n ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "P", next := 202794, root := ~, start := 202790, stop := 202793 ), rec( content := "\n\n \ A directed path (or directed cycle) of non-zero length ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "C", next := 202865, root := ~, start := 202855, stop := 202864 ), rec( content := ",\n " , count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)", count := [ 5, 3, 20, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "M", next := 202917, root := ~, start := 202871, stop := 202916 ), rec( content := ",\n i\ s represented by a pair of lists ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v,a]", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "C", next := 202969, root := ~, start := 202957, stop := 202968 ), rec( content := " as foll\ ows:\n\n ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 5, 3, 20, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "v", count := [ 5, 3, 20, 6 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 6 ], mathmode := "Text", name\ := "C", next := 203023, root := ~, start := 203015, stop := 203022 ), rec( cont\ ent := " is the list ", count := [ 5, 3, 20, 6 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "[v_1, v_2, ..., v_n]", count := [ 5, 3, 20, 6 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 6 ], mathmode := "Text", name \ := "M", next := 203063, root := ~, start := 203036, stop := 203062 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 20, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 20, 6 ], name := "Item", next := 203078, root := ~, start := 203\ 000, stop := 203077 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "a", count :\ = [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 5, 3, 20, 8 ], mathmode := "Text", name := "C", next := 203108, root := ~ , star\ t := 203100, stop := 203107 ), rec( content := " is the list of positive integ\ ers ", count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "[a_1, a_2, ...\ , a_{n-1}]", count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "M", next \ := 203173, root := ~, start := 203142, stop := 203172 ), rec( content := "\n \ where for each each ", count := [ 5, 3, 20, 8 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i < n", count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "M", next\ := 203217, root := ~, start := 203202, stop := 203216 ), rec( content := ", " , coun\ t := [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "a_i", count := [ 5, 3, 20, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 8\ ], mathmode := "Text", name := "M", next := 203229, root := ~, start := 20321\ 9, stop := 203228 ), rec( content := " is the position of\n ", count :=\ [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "v_{i+1}", count := [ 5, 3, 20\ , 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2\ 0, 8 ], mathmode := "Text", name := "M", next := 203271, root := ~, start := 2\ 03257, stop := 203270 ), rec( content := " in\n ", count := [ 5, 3, 20,\ 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "OutNeighboursOfVertex(", count := [ 5, 3,\ 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3\ , 20, 8 ], mathmode := "Text", name := "C", next := 203312, root := ~, start :\ = 203283, stop := 203311 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "A", next\ := 203326, root := ~, start := 203312, stop := 203325 ), rec( attributes := re\ c( ), content := [ rec( content := ",", count := [ 5, 3, 20, 8 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 8 ], mathmode :\ = "Text", name := "C", next := 203334, root := ~, start := 203326, stop := 203\ 333 ), rec( attributes := rec( ), content := [ rec( content := "v_i", count :\ = [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 5, 3, 20, 8 ], mathmode := "Text", name := "M", next := 203344, root := ~ , star\ t := 203334, stop := 203343 ), rec( attributes := rec( ), content := [ rec( c\ ontent := ")", count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "C", next\ := 203352, root := ~, start := 203344, stop := 203351 ), rec( content := "\n \ corresponding to the edge ", count := [ 5, 3, 20, 8 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "e_i", count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "M\ ", next := 203397, root := ~, start := 203387, stop := 203396 ), rec( content \ := ". This is can be useful if the\n position of a vertex in a list of\ out-neighours is significant, for\n example in orbit digraphs.\n \ ", count := [ 5, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 8 ], name := "Ite\ m", next := 203553, root := ~, start := 203085, stop := 203552 ) ], count := [ 5, 3, 20, 5 ], mathmode := "Text", name := "List", next := 203565, root := ~, start := 202987, stop := 203564 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPath", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "C", next := 203604, root := ~, start := 203586, stop := 203603 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "M", next := 203655, root := ~, start := 203634, stop := 203654 ), rec( content := " where " , count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "M", next := 203670, root := ~, start := 203662, stop := 203669 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "M", next := 203706, root := ~, start := 203698, stop := 203705 ), rec( content := " the num\ ber of\n vertices in ", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "A", next := 203751, root := ~, start := 203737, stop := 203750 ), rec( content := ".\n\n", count := [ 5, 3, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [2, 3]]);\n\ngap> DigraphPath(D, 1, 3);\n[ [ 1, 2, 3 ], [ 1, 1 ] ]\ngap> DigraphPa\ th(D, 2, 1);\nfail\ngap> DigraphPath(D, 3, 3);\n[ [ 3, 3 ], [ 2 ] ]\ngap> Digr\ aphPath(D, 1, 1);\nfail\n", count := [ 5, 3, 20, 11 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 11 ], mathmode := "Text", name := "Example", next := 204034, root := ~, start := 203754, stop := 204033 ), rec( content := "\n ", count := [ 5, 3, 20, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 204051, root := ~, start := 202331, stop := 204050 ) ], count := [ 5, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 204065, root := ~, start := 202213, stop := 204064 ), rec( content := "\n\n ", count := [ 5, 3, 20, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "DigraphShortest\ Path" ), content := 0, count := [ 5, 3, 21, 2 ], mathmode := "Text", name := "Oper", next := 204140, root := ~, start := 204086, stop := 204139 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists, or ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "K", next := 204183, root := ~, start := 204172, stop := 204182 ), rec( content := ".", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Returns", next := 204194, root := ~, start := 204143, stop := 204193 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the shortest directed path in the digraph digraph from the vertex\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 204301, root := ~, start := 204293, stop := 204300 ), rec( content := " to the \ vertex ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 204324, root := ~, start := 204316, stop := 204323 ), rec( content := ", if suc\ h a path exists. If ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "u", count := [ 5, 3, 21, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := \ "Text", name := "A", next := 204363, root := ~, start := 204355, stop := 20436\ 2 ), rec( content := " =\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := \ "Text", name := "A", next := 204378, root := ~, start := 204370, stop := 20437\ 7 ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "C", next := 204382, root := ~, start := 204352, stop := 204381 ), rec( content := ", then t\ he shortest non-trivial cycle is returned, again, if it\n exists. Otherwise\ , this operation returns ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "K", next := 204503, root := ~, start := 204492, stop := 204502 ), rec( content := ". See S\ ection\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Ref", next := 204563, root := ~, start := 204522, stop := 204562 ), rec( content := " for the\ definition of a directed\n path and a directed cycle.\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "P", next := 204636, root := ~, start := 204632, stop := 204635 ), rec( content := "\n\n \ See ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Path" ), content := 0, count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "Ref", next := 204671, root := ~, start := 204646, stop := 204670 ), rec( content := " for det\ ails on the output.\n\n The method for ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphShortestPath", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "C", next := 204745, root := ~, start := 204719, stop := 204744 ), rec( content := " has wor\ st case complexity of\n ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n)", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "M", next := 204794, root := ~, start := 204779, stop := 204793 ), rec( content := " where " , count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "M", next := 204809, root := ~, start := 204801, stop := 204808 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "M", next := 204845, root := ~, start := 204837, stop := 204844 ), rec( content := " the\n \ number of vertices in ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "A", next := 204890, root := ~, start := 204876, stop := 204889 ), rec( content := ".\n\n", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], [2, 4], [1], [2, 4]]);\n\ngap> DigraphShortestPath(D, 5, 1);\n[ [ 5, 4, 1 ], [ \ 2, 1 ] ]\ngap> DigraphShortestPath(D, 3, 3);\n[ [ 3, 2, 3 ], [ 1, 1 ] ]\ngap> \ DigraphShortestPath(D, 5, 5);\nfail\ngap> DigraphShortestPath(D, 1, 1);\n[ [ 1\ , 1 ], [ 1 ] ]\n", count := [ 5, 3, 21, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "Example", next := 205242, root := ~, start := 204893, stop := 205241 ), rec( content := "\n ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Description", next := 205259, root := ~, start := 204197, stop := 205258 ) ], count := [ 5, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 205273, root := ~, start := 204071, stop := 205272 ), rec( content := "\n\n ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "IteratorOfPaths\ " ), content := 0, count := [ 5, 3, 22, 2 ], mathmode := "Text", name := "Oper", next := 205344, root := ~, start := 205294, stop := 205343 ), rec( attributes := rec( ), content := [ rec( content := "An iterat\ or.", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Returns", next := 205378, root := ~, start := 205347, stop := 205377 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 205416, root := ~, start := 205402, stop := 205415 ), rec( content := " is a di\ graph or a list of adjacencies which defines a\n digraph - see ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Ref", next := 205516, root := ~, start := 205489, stop := 205515 ), rec( content := " - then \ this operation returns an\n iterator of the non-trivial directed paths (or \ directed cycles, in the case\n that ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 205647, root := ~, start := 205639, stop := 205646 ), rec( content := " ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "=\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "C", next := 205656, root := ~, start := 205648, stop := 205655 ), rec( content := " ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 205665, root := ~, start := 205657, stop := 205664 ), rec( content := ") in ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 205684, root := ~, start := 205670, stop := 205683 ), rec( content := " from th\ e vertex ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 205709, root := ~, start := 205701, stop := 205708 ), rec( content := "\n to\ the vertex ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 205736, root := ~, start := 205728, stop := 205735 ), rec( content := ".\n " , count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "P", next := 205746, root := ~, start := 205742, stop := 205745 ), rec( content := "\n\n \ See ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Path" ), content := 0, count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "Ref", next := 205781, root := ~, start := 205756, stop := 205780 ), rec( content := " for mor\ e information about the repesentation\n of a directed path or directed cycl\ e which is used, and see ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Sect := "Iterators" ), content := 0, count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "Ref", next := 205935, root := ~, start := 205891, stop := 205934 ), rec( content := " for mor\ e information about iterators.\n See Section ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "Ref", next := 206031, root := ~, start := 205990, stop := 206030 ), rec( content := " for the\ definition of\n a directed path and a directed cycle.\n ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "P", next := 206104, root := ~, start := 206100, stop := 206103 ), rec( content := "\n\n", count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 4, 4, 2], [3, 5], [2, 3], [1, 2], [4]]);\n\ngap> iter := IteratorOfPaths(D, 1, 4);\ \n\ngap> NextIterator(iter);\n[ [ 1, 4 ], [ 2 ] ]\ngap> NextIterator\ (iter);\n[ [ 1, 4 ], [ 3 ] ]\ngap> NextIterator(iter);\n[ [ 1, 2, 5, 4 ], [ 4,\ 2, 1 ] ]\ngap> IsDoneIterator(iter);\ntrue\ngap> iter := IteratorOfPaths(D, 4\ , 3);\n\ngap> NextIterator(iter);\n[ [ 4, 1, 2, 3 ], [ 1, 4, 1 ] ]\n\ ", count := [ 5, 3, 22, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 6 ], mathmode := "Text", name := "Example", next := 206589, root := ~, start := 206106, stop := 206588 ), rec( content := "\n ", count := [ 5, 3, 22, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 206606, root := ~, start := 205381, stop := 206605 ) ], count := [ 5, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 206620, root := ~, start := 205279, stop := 206619 ), rec( content := "\n\n ", count := [ 5, 3, 22, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAllSimpl\ eCircuits" ), content := 0, count := [ 5, 3, 23, 2 ], mathmode := "Text", name := "Attr", next := 206694, root := ~, start := 206641, stop := 206693 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists of vertices.", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Returns", next := 206744, root := ~, start := 206697, stop := 206743 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 206782, root := ~, start := 206768, stop := 206781 ), rec( content := " is a di\ graph, then ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "C", next := 206833, root := ~, start := 206802, stop := 206832 ), rec( content := "\n re\ turns a list of the ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ imple circuits", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "E", next := 206882, root := ~, start := 206860, stop := 206881 ), rec( content := " in ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 206900, root := ~, start := 206886, stop := 206899 ), rec( content := ". ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "P", next := 206906, root := ~, start := 206902, stop := 206905 ), rec( content := "\n\n \ See section ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "Ref", next := 206968, root := ~, start := 206924, stop := 206967 ), rec( content := " for the\ definition\n of a simple circuit, and related notions. Note that a loop is\ a simple\n circuit. ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "P", next := 207080, root := ~, start := 207076, stop := 207079 ), rec( content := "\n\n \ For a digraph without multiple edges, a simple circuit is uniquely\n determ\ ined by its subsequence of vertices. However this is not the case for\n a m\ ultidigraph. The attribute ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "C", next := 207299, root := ~, start := 207268, stop := 207298 ), rec( content := " ignores\ \n multiple edges, and identifies a simple circuit using only its subsequen\ ce\n of vertices. For example, although the simple circuits ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v, e, v)", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "M", next := 207462, root := ~, start := 207446, stop := 207461 ), rec( content := " and\n \ ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v, e', v)", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "M", next := 207488, root := ~, start := 207471, stop := 207487 ), rec( content := " (for di\ stinct edges ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "M", next := 207517, root := ~, start := 207509, stop := 207516 ), rec( content := " and ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ '", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "M", next := 207531, root := ~, start := 207522, stop := 207530 ), rec( content := ") are\n \ mathematically distinct, ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "C", next := 207597, root := ~, start := 207566, stop := 207596 ), rec( content := " conside\ rs them to\n be the same. ", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "P", next := 207637, root := ~, start := 207633, stop := 207636 ), rec( content := "\n\n \ With this approach, a directed circuit of length ", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "C", next := 207700, root := ~, start := 207692, stop := 207699 ), rec( content := " can be \ determined\n by a list of its first ", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "C", next := 207754, root := ~, start := 207746, stop := 207753 ), rec( content := " vertice\ s. Thus a simple circuit ", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v_1,\n e_1, v_2, e_2, ..., e_{n-1}, v_n, e_{n+1}, v_1)", count := [ 5, 3, 23, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "M", next := 207853, root := ~, start := 207787, stop := 207852 ), rec( content := " can be \ represented as\n the list ", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v_1, \\ldots, v_n]", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "M", next := 207914, root := ~, start := 207889, stop := 207913 ), rec( content := ", or any\ cyclic permutation thereof. For\n each simple circuit of ", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "A", next := 207997, root := ~, start := 207983, stop := 207996 ), rec( content := ",\n " , count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits(", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "A", next := 2080\ 45, root := ~, start := 208031, stop := 208044 ), rec( content := ")", count := [ 5, 3, 23, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "C", next := 208050, root := ~, start := 208003, stop := 208049 ), rec( content := " include\ s precisely one such\n list to represent the circuit. ", count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 23, 7 ], mathmode := "Text", name := "P", next := 208119, root := ~, start := 208115, stop := 208118 ), rec( content := "\n\n \ ", count := [ 5, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], [3], [2, 4], [5, 4], [4]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 4 ], [ 4, 5 ], [ 2\ , 3 ] ]\ngap> D := ChainDigraph(10);;\ngap> DigraphAllSimpleCircuits(D);\n[ ]\ \ngap> D := Digraph([[3], [1], [1]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1, 3 ] ]\ngap> D := Digraph([[1,\ 1]]);\n\ngap> DigraphAllSimple\ Circuits(D);\n[ [ 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1,\ 2, 3 ] ]\n", count := [ 5, 3, 23, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 8 ], mathmode := "Text", name := "Example", next := 208772, root := ~, start := 208125, stop := 208771 ), rec( content := "\n ", count := [ 5, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 208789, root := ~, start := 206747, stop := 208788 ) ], count := [ 5, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 208803, root := ~, start := 206626, stop := 208802 ), rec( content := "\n\n ", count := [ 5, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphLongestS\ impleCircuit" ), content := 0, count := [ 5, 3, 24, 2 ], mathmode := "Text", name := "Attr", next := 208880, root := ~, start := 208824, stop := 208879 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices, or ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "K", next := 208926, root := ~, start := 208915, stop := 208925 ), rec( content := ".", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Returns", next := 208937, root := ~, start := 208883, stop := 208936 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "A", next := 208975, root := ~, start := 208961, stop := 208974 ), rec( content := " is a di\ graph, then ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphLongestSimpleCircuit", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "C", next := 209029, root := ~, start := 208995, stop := 209028 ), rec( content := "\n re\ turns the longest ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ imple circuit", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "E", next := 209075, root := ~, start := 209054, stop := 209074 ), rec( content := " in ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "A", next := 209093, root := ~, start := 209079, stop := 209092 ), rec( content := ". See se\ ction\n ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Ref", next := 209155, root := ~, start := 209111, stop := 209154 ), rec( content := " for the\ definition of simple\n circuit, and the definition of length for a simple \ circuit.", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "P", next := 209252, root := ~, start := 209248, stop := 209251 ), rec( content := "\n\n \ This attribute computes\n ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits(", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "C", next := 209318, root := ~, start := 209286, stop := 209317 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "A", next := 209332, root := ~, start := 209318, stop := 209331 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "C", next := 209340, root := ~, start := 209332, stop := 209339 ), rec( content := " to find\ all the\n simple circuits of ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "A", next := 209394, root := ~, start := 209380, stop := 209393 ), rec( content := ", and re\ turns one of maximal length. A\n simple circuit is represented as a list o\ f vertices, in the same way as\n described in ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ AllSimpleCircuits" ), content := 0, count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "Ref", next := 209565, root := ~, start := 209527, stop := 209564 ), rec( content := ".", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "P", next := 209570, root := ~, start := 209566, stop := 209569 ), rec( content := "\n\n \ If ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "A", next := 209593, root := ~, start := 209579, stop := 209592 ), rec( content := " has no \ simple circuits, then this attribute returns\n ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "K", next := 209661, root := ~, start := 209650, stop := 209660 ), rec( content := ". If ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "A", next := 209681, root := ~, start := 209667, stop := 209680 ), rec( content := " has mul\ tiple simple circuits of maximal\n length, then this attribute returns one \ of them.", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "P", next := 209778, root := ~, start := 209774, stop := 209777 ), rec( content := "\n\n \ ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], [3], [2, 4], [5, 4], [4]]);;\ngap> DigraphLongestSimpl\ eCircuit(D);\n[ 4, 5 ]\ngap> D := ChainDigraph(10);;\ngap> DigraphLongestSimpl\ eCircuit(D);\nfail\ngap> D := Digraph([[3], [1], [1, 4], [1, 1]]);;\ngap> Digr\ aphLongestSimpleCircuit(D);\n[ 1, 3, 4 ]\ngap> D := GeneralisedPetersenGraph(I\ sMutableDigraph, 4, 1);\n\ngap> Dig\ raphLongestSimpleCircuit(D);\n[ 1, 2, 3, 4, 8, 7, 6, 5 ]\n", count := [ 5, 3, 24, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 7 ], mathmode := "Text", name := "Example", next := 210250, root := ~, start := 209784, stop := 210249 ), rec( content := "\n ", count := [ 5, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Description", next := 210267, root := ~, start := 208940, stop := 210266 ) ], count := [ 5, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 210281, root := ~, start := 208809, stop := 210280 ), rec( content := "\n\n ", count := [ 5, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphLayers" ), content := 0, count := [ 5, 3, 25, 2 ], mathmode := "Text", name := "Oper", next := 210352, root := ~, start := 210302, stop := 210351 ), rec( attributes := rec( ), content := [ rec( content := "A list.", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Returns", next := 210381, root := ~, start := 210355, stop := 210380 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a list ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "K", next := 210443, root := ~, start := 210432, stop := 210442 ), rec( content := " such th\ at ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[i]", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "K", next := 210468, root := ~, start := 210454, stop := 210467 ), rec( content := " is the\\ n list of vertices whose minimum distance from the vertex ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "A", next := 210549, root := ~, start := 210536, stop := 210548 ), rec( content := " in\n \ ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "A", next := 210571, root := ~, start := 210557, stop := 210570 ), rec( content := " is ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ - 1", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "K", next := 210587, root := ~, start := 210575, stop := 210586 ), rec( content := ". Vertex\ ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "A", next := 210609, root := ~, start := 210596, stop := 210608 ), rec( content := " is\n \ assumed to be at distance ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "K", next := 210651, root := ~, start := 210643, stop := 210650 ), rec( content := " from it\ self.\n ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);;\ngap> DigraphLayers(D, 1);\n[ [ 1 ], [ 2, 3, 4\ ] ]\n", count := [ 5, 3, 25, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "Example", next := 210781, root := ~, start := 210669, stop := 210780 ), rec( content := "\n ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Description", next := 210798, root := ~, start := 210384, stop := 210797 ) ], count := [ 5, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 210812, root := ~, start := 210287, stop := 210811 ), rec( content := "\n\n ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDegenera\ cy" ), content := 0, count := [ 5, 3, 26, 2 ], mathmode := "Text", name := "Attr", next := 210879, root := ~, start := 210833, stop := 210878 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer, or ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 210929, root := ~, start := 210918, stop := 210928 ), rec( content := ".", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Returns", next := 210940, root := ~, start := 210882, stop := 210939 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "A", next := 210978, root := ~, start := 210964, stop := 210977 ), rec( content := " is a sy\ mmetric digraph without multiple edges - see\n ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Ref", next := 211067, root := ~, start := 211035, stop := 211066 ), rec( content := " and ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Ref", next := 211100, root := ~, start := 211072, stop := 211099 ), rec( content := " - then\\ n this attribute returns the degeneracy of ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "A", next := 211167, root := ~, start := 211153, stop := 211166 ), rec( content := ". ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "P", next := 211173, root := ~, start := 211169, stop := 211172 ), rec( content := "\n\n \ The degeneracy of a digraph is the least integer ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "C", next := 211236, root := ~, start := 211228, stop := 211235 ), rec( content := " such\n \ that every induced of ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "A", next := 211282, root := ~, start := 211268, stop := 211281 ), rec( content := " contain\ s a vertex whose number of\n neighbours (excluding itself) is at most ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "C", next := 211370, root := ~, start := 211362, stop := 211369 ), rec( content := ". Note t\ hat this means\n that loops are ignored.", count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "P", next := 211424, root := ~, start := 211420, stop := 211423 ), rec( content := "\n\n \ If ", count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "A", next := 211447, root := ~, start := 211433, stop := 211446 ), rec( content := " is not \ symmetric or has multiple edges then this\n attribute returns ", count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "K", next := 211530, root := ~, start := 211519, stop := 211529 ), rec( content := ".\n " , count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> DigraphDegeneracy(\ D);\n1\ngap> D := CompleteDigraph(5);;\ngap> DigraphDegeneracy(D);\n4\ngap> D \ := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]);\n\ngap> DigraphDegeneracy(D);\n1\ngap> D := Generalis\ edPetersenGraph(IsMutableDigraph, 10, 3);\n\ngap> DigraphDegeneracy(D);\n3\n", count := [ 5, 3, 26, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 26, 6 ], mathmode := "Text", name := "Example", next := 211985, root := ~, start := 211536, stop := 211984 ), rec( content := "\n ", count := [ 5, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Description", next := 212002, root := ~, start := 210943, stop := 212001 ) ], count := [ 5, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 212016, root := ~, start := 210818, stop := 212015 ), rec( content := "\n\n ", count := [ 5, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDegenera\ cyOrdering" ), content := 0, count := [ 5, 3, 27, 2 ], mathmode := "Text", name := "Attr", next := 212091, root := ~, start := 212037, stop := 212090 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ integers, or ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "K", next := 212137, root := ~, start := 212126, stop := 212136 ), rec( content := ".", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Returns", next := 212148, root := ~, start := 212094, stop := 212147 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "A", next := 212186, root := ~, start := 212172, stop := 212185 ), rec( content := " is a di\ graph for which\n ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDegeneracy(", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "C", next := 212239, root := ~, start := 212214, stop := 212238 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "A", next := 212253, root := ~, start := 212239, stop := 212252 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "C", next := 212261, root := ~, start := 212253, stop := 212260 ), rec( content := " is a no\ n-negative integer\n ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "C", next := 212300, root := ~, start := 212292, stop := 212299 ), rec( content := " - see " , count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Degeneracy" ), content := 0, count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Ref", next := 212338, root := ~, start := 212307, stop := 212337 ), rec( content := " - then \ this attribute\n returns a degeneracy ordering of the vertices of the verti\ ces of\n ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "A", next := 212448, root := ~, start := 212434, stop := 212447 ), rec( content := ".", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "P", next := 212453, root := ~, start := 212449, stop := 212452 ), rec( content := "\n\n \ A degeneracy ordering of ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "A", next := 212498, root := ~, start := 212484, stop := 212497 ), rec( content := " is a li\ st ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rdering", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "C", next := 212524, root := ~, start := 212509, stop := 212523 ), rec( content := " of the\\ n vertices of ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "A", next := 212562, root := ~, start := 212548, stop := 212561 ), rec( content := " ordered\ such that for any\n position ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "C", next := 212610, root := ~, start := 212602, stop := 212609 ), rec( content := " of the \ list, the vertex ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rdering[i]", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "C", next := 212653, root := ~, start := 212635, stop := 212652 ), rec( content := " has at \ most\n ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "C", next := 212678, root := ~, start := 212670, stop := 212677 ), rec( content := " neighbo\ urs in later position of the list.", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "P", next := 212724, root := ~, start := 212720, stop := 212723 ), rec( content := "\n\n \ If ", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDegeneracy(", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "C", next := 212758, root := ~, start := 212733, stop := 212757 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "A", next := 212772, root := ~, start := 212758, stop := 212771 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "C", next := 212780, root := ~, start := 212772, stop := 212779 ), rec( content := " returns\ ", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "K", next := 212800, root := ~, start := 212789, stop := 212799 ), rec( content := ",\n t\ hen this attribute returns ", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "K", next := 212845, root := ~, start := 212834, stop := 212844 ), rec( content := ".\n " , count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> DigraphDegeneracyO\ rdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap> D := CompleteDigraph(5);;\ngap> DigraphDe\ generacyOrdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap> D := Digraph([[1], [2, 4, 5], [3\ , 4], [2, 3, 4], [2], []]);\n\nga\ p> DigraphDegeneracyOrdering(D);\n[ 1, 6, 5, 2, 4, 3 ]\ngap> D := GeneralisedP\ etersenGraph(IsMutableDigraph, 3, 1);\n\ngap> DigraphDegeneracyOrdering(D);\n[ 6, 5, 4, 1, 3, 2 ]\n", count := [ 5, 3, 27, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 27, 6 ], mathmode := "Text", name := "Example", next := 213400, root := ~, start := 212851, stop := 213399 ), rec( content := "\n ", count := [ 5, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Description", next := 213417, root := ~, start := 212151, stop := 213416 ) ], count := [ 5, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 213431, root := ~, start := 212022, stop := 213430 ), rec( content := "\n\n ", count := [ 5, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "HamiltonianPath\ " ), content := 0, count := [ 5, 3, 28, 2 ], mathmode := "Text", name := "Attr", next := 213496, root := ~, start := 213452, stop := 213495 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "K", next := 213529, root := ~, start := 213518, stop := 213528 ), rec( content := ".", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Returns", next := 213540, root := ~, start := 213499, stop := 213539 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns a Hamiltonian path if one exists, ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "K", next := 213614, root := ~, start := 213603, stop := 213613 ), rec( content := " if not.\ ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "P", next := 213626, root := ~, start := 213622, stop := 213625 ), rec( content := "\n\n \ A ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonian path", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "E", next := 213657, root := ~, start := 213634, stop := 213656 ), rec( content := " of a di\ graph with ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 213684, root := ~, start := 213676, stop := 213683 ), rec( content := " vertice\ s is directed\n cycle of length ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 213734, root := ~, start := 213726, stop := 213733 ), rec( content := ". If ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "A", next := 213753, root := ~, start := 213739, stop := 213752 ), rec( content := " is a di\ graph that contains a\n Hamiltonian path, then this function returns one, d\ escribed in the form\n used by ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ AllSimpleCircuits" ), content := 0, count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "Ref", next := 213909, root := ~, start := 213871, stop := 213908 ), rec( content := ". Note i\ f ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "A", next := 213933, root := ~, start := 213919, stop := 213932 ), rec( content := " has\n \ 0 or 1 vertices, then ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonianPath", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 213986, root := ~, start := 213964, stop := 213985 ), rec( content := " returns\ ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ]", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 214004, root := ~, start := 213995, stop := 214003 ), rec( content := " or\n \ ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 214022, root := ~, start := 214012, stop := 214021 ), rec( content := ", respec\ tively.", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "P", next := 214041, root := ~, start := 214037, stop := 214040 ), rec( content := "\n\n \ The method used in this attribute has the same worst case complexity as\n " , count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "Ref", next := 214156, root := ~, start := 214123, stop := 214155 ), rec( content := ".\n\n \ ", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[]]);\n\ngap> Hamil\ tonianPath(D);\n[ 1 ]\ngap> D := Digraph([[2], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2 ]\ngap> D := Digraph([\ [3], [], [2]]);\n\ngap> Hamiltonia\ nPath(D);\nfail\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2, 3 ]\ngap> D := General\ isedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> HamiltonianPath(D);\nfail\n", count := [ 5, 3, 28, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 28, 6 ], mathmode := "Text", name := "Example", next := 214764, root := ~, start := 214163, stop := 214763 ), rec( content := "\n ", count := [ 5, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Description", next := 214781, root := ~, start := 213543, stop := 214780 ) ], count := [ 5, 3, 28, 1 ], mathmode := "Text", name := "ManSection", next := 214795, root := ~, start := 213437, stop := 214794 ), rec( content := "\n\n ", count := [ 5, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "NrSpanningTrees\ " ), content := 0, count := [ 5, 3, 29, 2 ], mathmode := "Text", name := "Attr", next := 214860, root := ~, start := 214816, stop := 214859 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "Returns", next := 214893, root := ~, start := 214863, stop := 214892 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the number of spanning trees of the symmetric digraph\n ", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "A", next := 214994, root := ~, start := 214980, stop := 214993 ), rec( content := ".\n " , count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ rSpanningTrees", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "C", next := 215022, root := ~, start := 215000, stop := 215021 ), rec( content := " will re\ turn an error if ", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "A", next := 215061, root := ~, start := 215047, stop := 215060 ), rec( content := " is not \ a\n symmetric digraph. ", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "P", next := 215098, root := ~, start := 215094, stop := 215097 ), rec( content := " \n\n \ See ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "Ref", next := 215141, root := ~, start := 215109, stop := 215140 ), rec( content := " and\n \ ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningTree" ), content := 0, count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "Ref", next := 215189, root := ~, start := 215150, stop := 215188 ), rec( content := " for mor\ e information. \n\n ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(5);\n\n\ gap> NrSpanningTrees(D);\n125\ngap> D := DigraphSymmetricClosure(CycleDigraph(\ 24));;\ngap> NrSpanningTrees(D);\n24\ngap> NrSpanningTrees(EmptyDigraph(0));\n\ 0\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> NrSpanningTrees(D);\n1134225\n", count := [ 5, 3, 29, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 29, 5 ], mathmode := "Text", name := "Example", next := 215615, root := ~, start := 215218, stop := 215614 ), rec( content := "\n ", count := [ 5, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "Description", next := 215632, root := ~, start := 214896, stop := 215631 ) ], count := [ 5, 3, 29, 1 ], mathmode := "Text", name := "ManSection", next := 215646, root := ~, start := 214801, stop := 215645 ), rec( content := "\n\n ", count := [ 5, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, source,\ target", Label := "for a source and target", Name := "DigraphDijkstra" ), content := 0, count := [ 5, 3, 30, 2 ], mathmode := "Text", name := "Oper", next := 215759, root := ~, start := 215667, stop := 215758 ), rec( attributes := rec( Arg := "digraph, source" , Label := "for a source", Name := "DigraphDijkstra\ " ), content := 0, count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "Oper", next := 215835, root := ~, start := 215762, stop := 215834 ), rec( attributes := rec( ), content := [ rec( content := "Two lists\ .", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "Returns", next := 215867, root := ~, start := 215838, stop := 215866 ), rec( attributes := rec( ), content := [ rec( content := "\n If ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 215903, root := ~, start := 215889, stop := 215902 ), rec( content := " is a di\ graph and ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 215934, root := ~, start := 215921, stop := 215933 ), rec( content := " and ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 215952, root := ~, start := 215939, stop := 215951 ), rec( content := " are\n \ vertices of ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 215986, root := ~, start := 215972, stop := 215985 ), rec( content := ", then " , count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDijkstra", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "C", next := 216015, root := ~, start := 215993, stop := 216014 ), rec( content := " calcula\ tes the\n length of the shortest path from ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 216079, root := ~, start := 216066, stop := 216078 ), rec( content := " to ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 216096, root := ~, start := 216083, stop := 216095 ), rec( content := " and ret\ urns\n two lists. Each element of the first list is the distance of the\n co\ rresponding element from ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 216218, root := ~, start := 216205, stop := 216217 ), rec( content := ". If a \ vertex was not visited in\n the process of calculating the shortest distance \ to ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 216319, root := ~, start := 216306, stop := 216318 ), rec( content := " or if\n\ there is no path connecting that vertex with ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 216386, root := ~, start := 216373, stop := 216385 ), rec( content := ", then\n\ the corresponding distance is ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "K", next := 216440, root := ~, start := 216425, stop := 216439 ), rec( content := ". Each \ element of the\n second list gives the previous vertex in the shortest path\n\ from ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 216544, root := ~, start := 216531, stop := 216543 ), rec( content := " to the \ corresponding vertex. For\n ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "A", next := 216594, root := ~, start := 216581, stop := 216593 ), rec( content := " and for\ any vertices that remained unvisited this\n will be ", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "-\ 1", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "C", next := 216666, root := ~, start := 216657, stop := 216665 ), rec( content := ".", count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "P", next := 216671, root := ~, start := 216667, stop := 216670 ), rec( content := "\n\n \ If the optional second argument ", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "A", next := 216722, root := ~, start := 216709, stop := 216721 ), rec( content := " is not \ present, then\n ", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDijkstra", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "C", next := 216770, root := ~, start := 216748, stop := 216769 ), rec( content := " returns\ the shortest path from ", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "A", next := 216815, root := ~, start := 216802, stop := 216814 ), rec( content := " to\n \ every vertex that is reachable from ", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "A", next := 216872, root := ~, start := 216859, stop := 216871 ), rec( content := ".\n\n " , count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> mat := [[0, 1, 1], [0, 0, 1], [0, 0, 0]];\n[ [ 0, 1, 1 ], [ 0, 0, 1 ], [\ 0, 0, 0 ] ]\ngap> D := DigraphByAdjacencyMatrix(mat);\n\ngap> DigraphDijkstra(D, 2, 3);\n[ [ infinity, 0, 1 ], \ [ -1, -1, 2 ] ]\ngap> DigraphDijkstra(D, 1, 3);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\ \ngap> DigraphDijkstra(D, 1, 2);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\n", count := [ 5, 3, 30, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 30, 7 ], mathmode := "Text", name := "Example", next := 217275, root := ~, start := 216877, stop := 217274 ), rec( content := "\n", count := [ 5, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "Description", next := 217290, root := ~, start := 215870, stop := 217289 ) ], count := [ 5, 3, 30, 1 ], mathmode := "Text", name := "ManSection", next := 217304, root := ~, start := 215652, stop := 217303 ), rec( content := "\n\n ", count := [ 5, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 217318, root := ~, start := 172687, stop := 217317 ) , rec( content := "\n\n ", count := [ 5, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Cayley graphs of \ groups", count := [ 5, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 217373, root := ~, start := 217331, stop := 217372 ), rec( content := "\n ", count := [ 5, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "GroupOfCayleyDi\ graph" ), content := 0, count := [ 5, 4, 1, 2 ], mathmode := "Text", name := "Attr", next := 217442, root := ~, start := 217393, stop := 217441 ), rec( attributes := rec( Arg := "digraph", Name := "SemigroupOfCayl\ eyDigraph" ), content := 0, count := [ 5, 4, 1, 4 ], mathmode := "Text", name := "Attr", next := 217498, root := ~, start := 217445, stop := 217497 ), rec( attributes := rec( ), content := [ rec( content := "A group o\ r semigroup.", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Returns", next := 217541, root := ~, start := 217501, stop := 217540 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "A", next := 217579, root := ~, start := 217565, stop := 217578 ), rec( content := " is an i\ mmutable Cayley graph of a group ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "C", next := 217628, root := ~, start := 217620, stop := 217627 ), rec( content := " and\n \ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "A", next := 217651, root := ~, start := 217637, stop := 217650 ), rec( content := " belongs\ to the category ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Ref", next := 217705, root := ~, start := 217676, stop := 217704 ), rec( content := ", then\n\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ roupOfCayleyDigraph", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "C", next := 217743, root := ~, start := 217716, stop := 217742 ), rec( content := " returns\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "C", next := 217760, root := ~, start := 217752, stop := 217759 ), rec( content := ".\n " , count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "P", next := 217770, root := ~, start := 217766, stop := 217769 ), rec( content := "\n\n \ If ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "A", next := 217793, root := ~, start := 217779, stop := 217792 ), rec( content := " is a Ca\ yley graph of a semigroup ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "C", next := 217835, root := ~, start := 217827, stop := 217834 ), rec( content := " and\n \ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "A", next := 217858, root := ~, start := 217844, stop := 217857 ), rec( content := " belongs\ to the category ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "Ref", next := 217912, root := ~, start := 217883, stop := 217911 ), rec( content := ", then\n\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ emigroupOfCayleyDigraph", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "C", next := 217954, root := ~, start := 217923, stop := 217953 ), rec( content := " returns\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "S\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "C", next := 217971, root := ~, start := 217963, stop := 217970 ), rec( content := ".\n " , count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "P", next := 217981, root := ~, start := 217977, stop := 217980 ), rec( content := "\n\n \ See also ", count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Generat\ orsOfCayleyDigraph" ), content := 0, count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "Ref", next := 218035, root := ~, start := 217996, stop := 218034 ), rec( content := ".\n " , count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := DihedralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> d\ igraph := CayleyDigraph(G);\n\nga\ p> GroupOfCayleyDigraph(digraph) = G;\ntrue\n", count := [ 5, 4, 1, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 1, 8 ], mathmode := "Text", name := "Example", next := 218267, root := ~, start := 218041, stop := 218266 ), rec( content := "\n ", count := [ 5, 4, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Description", next := 218284, root := ~, start := 217544, stop := 218283 ) ], count := [ 5, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 218298, root := ~, start := 217378, stop := 218297 ), rec( content := "\n\n ", count := [ 5, 4, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "GeneratorsOfCay\ leyDigraph" ), content := 0, count := [ 5, 4, 2, 2 ], mathmode := "Text", name := "Attr", next := 218373, root := ~, start := 218319, stop := 218372 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ generators.", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 218416, root := ~, start := 218376, stop := 218415 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "A", next := 218454, root := ~, start := 218440, stop := 218453 ), rec( content := " is an i\ mmutable Cayley graph of a group or semigroup with\n respect to a set of ge\ nerators ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "C", next := 218559, root := ~, start := 218548, stop := 218558 ), rec( content := " and ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "A", next := 218578, root := ~, start := 218564, stop := 218577 ), rec( content := " belongs\ to\n the category ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Ref", next := 218636, root := ~, start := 218607, stop := 218635 ), rec( content := ", then\n\ ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfCayleyDigraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "C", next := 218679, root := ~, start := 218647, stop := 218678 ), rec( content := " return \ the list of generators ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "C", next := 218721, root := ~, start := 218710, stop := 218720 ), rec( content := "\n ov\ er which ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "A", next := 218751, root := ~, start := 218737, stop := 218750 ), rec( content := " is defi\ ned.\n ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "P", next := 218772, root := ~, start := 218768, stop := 218771 ), rec( content := "\n\n \ See also ", count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "GroupOf\ CayleyDigraph" ), content := 0, count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "Ref", next := 218821, root := ~, start := 218787, stop := 218820 ), rec( content := " \n o\ r ", count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Semigro\ upOfCayleyDigraph" ), content := 0, count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "Ref", next := 218868, root := ~, start := 218830, stop := 218867 ), rec( content := ".\n " , count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := DihedralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> d\ igraph := CayleyDigraph(G);\n\nga\ p> GeneratorsOfCayleyDigraph(digraph) = GeneratorsOfGroup(G);\ntrue\ngap> digr\ aph := CayleyDigraph(G, [()]);\n\n\ gap> GeneratorsOfCayleyDigraph(digraph) = [()];\ntrue", count := [ 5, 4, 2, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 2, 5 ], mathmode := "Text", name := "Example", next := 219261, root := ~, start := 218874, stop := 219260 ), rec( content := "\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 219278, root := ~, start := 218419, stop := 219277 ) ], count := [ 5, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 219292, root := ~, start := 218304, stop := 219291 ), rec( content := "\n\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 219306, root := ~, start := 217322, stop := 219305 ) , rec( content := "\n\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Associated semigr\ oups", count := [ 5, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 219359, root := ~, start := 219319, stop := 219358 ), rec( content := "\n ", count := [ 5, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filt, digraph", Label := "for a filter a\ nd a digraph", Name := "AsSemigroup" ), content := 0, count := [ 5, 5, 1, 2 ], mathmode := "Text", name := "Oper", next := 219470, root := ~, start := 219383, stop := 219469 ), rec( attributes := rec( Arg := "filt, digraph", Name := "AsMonoid" ), content := 0, count := [ 5, 5, 1, 4 ], mathmode := "Text", name := "Oper", next := 219518, root := ~, start := 219475, stop := 219517 ), rec( attributes := rec( ), content := [ rec( content := "A semilat\ tice of partial perms.", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Returns", next := 219573, root := ~, start := 219523, stop := 219572 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "C", next := 219630, root := ~, start := 219612, stop := 219629 ), rec( content := " require\ s that ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 219656, root := ~, start := 219645, stop := 219655 ), rec( content := " be equa\ l to\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsPartialPermSemigroup" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 219726, root := ~, start := 219675, stop := 219725 ), rec( content := ".\n \ If ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 219751, root := ~, start := 219737, stop := 219750 ), rec( content := " is a\n \ ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 219801, root := ~, start := 219763, stop := 219800 ), rec( content := " or ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsLatti\ ceDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 219835, root := ~, start := 219805, stop := 219834 ), rec( content := "\n \ then ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "C", next := 219865, root := ~, start := 219847, stop := 219864 ), rec( content := " returns\ a semigroup of partial perms which is\n isomorphic to the semigroup whos\ e elements are the vertices of\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 220001, root := ~, start := 219987, stop := 220000 ), rec( content := " with th\ e binary operation\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Partial\ OrderDigraphJoinOfVertices" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 220081, root := ~, start := 220034, stop := 220080 ), rec( content := ".\n \ If ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 220106, root := ~, start := 220092, stop := 220105 ), rec( content := "\n \ satisfies ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMeetS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 220162, root := ~, start := 220124, stop := 220161 ), rec( content := " but not\ \n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 220215, root := ~, start := 220177, stop := 220214 ), rec( content := " then\n \ ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "C", next := 220245, root := ~, start := 220227, stop := 220244 ), rec( content := " returns\ a semigroup of partial perms which is\n isomorphic to the semigroup whos\ e elements are the vertices of\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 220381, root := ~, start := 220367, stop := 220380 ), rec( content := " with th\ e binary operation\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Partial\ OrderDigraphMeetOfVertices" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 220461, root := ~, start := 220414, stop := 220460 ), rec( content := ".\n \ ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "P", next := 220473, root := ~, start := 220469, stop := 220472 ), rec( content := "\n\n \ The operation ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sMonoid", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "C", next := 220510, root := ~, start := 220495, stop := 220509 ), rec( content := " behaves\ similarly to ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "C", next := 220550, root := ~, start := 220532, stop := 220549 ), rec( content := "\n \ except that ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "A", next := 220580, root := ~, start := 220569, stop := 220579 ), rec( content := " may als\ o be equal to\n ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsPartialPermMonoid" ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "Ref", next := 220656, root := ~, start := 220608, stop := 220655 ), rec( content := ",\n \ ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "A", next := 220678, root := ~, start := 220664, stop := 220677 ), rec( content := " must sa\ tisfy ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsLatti\ ceDigraph" ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "Ref", next := 220722, root := ~, start := 220692, stop := 220721 ), rec( content := ", and th\ e\n output satisfies ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsMonoid" ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "Ref", next := 220792, root := ~, start := 220755, stop := 220791 ), rec( content := ". ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "P", next := 220798, root := ~, start := 220794, stop := 220797 ), rec( content := "\n\n \ The output of both of these operations is guaranteed to be of minimal\n \ degree (see ", count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DegreeO\ fPartialPermSemigroup", BookName := "ref" ), content := 0, count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "Ref", next := 220951, root := ~, start := 220894, stop := 220950 ), rec( content := ").\n \ Furthermore the ", count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Generat\ orsOfSemigroup", BookName := "ref" ), content := 0, count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "Ref", next := 221026, root := ~, start := 220976, stop := 221025 ), rec( content := " of\n \ the output is guaranteed to be the unique generating set of minimal size.\n\ ", count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> di := Digraph([[1], [1, 2], [1, 3], [1, 4], [1, 2, 3, 5]]);\n\ngap> S := AsSemigroup(IsPartialPermSemigro\ up, di);\n\ngap> ForAll(El\ ements(S), IsIdempotent);\ntrue\ngap> IsInverseSemigroup(S);\ntrue\ngap> Size(\ S);\n5\ngap> di := Digraph([[1], [1, 2], [1, 2, 3]]);\n\ngap> M := AsMonoid(IsPartialPermMonoid, di);\n\ngap> Size(M);\n3", count := [ 5, 5, 1, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 5, 1, 8 ], mathmode := "Text", name := "Example", next := 221658, root := ~, start := 221116, stop := 221657 ), rec( content := "\n ", count := [ 5, 5, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Description", next := 221677, root := ~, start := 219578, stop := 221676 ) ], count := [ 5, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 221693, root := ~, start := 219366, stop := 221692 ), rec( content := "\n\n ", count := [ 5, 5, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filt, Y, gps, ho\ ms", Label := "for a filter, semilattice digraph, and two lists", Name := "AsSemigroup" ), content := 0, count := [ 5, 5, 2, 2 ], mathmode := "Text", name := "Oper", next := 221832, root := ~, start := 221718, stop := 221831 ), rec( attributes := rec( ), content := [ rec( content := " A Cliffo\ rd semigroup of partial perms. ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 221896, root := ~, start := 221837, stop := 221895 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 221953, root := ~, start := 221935, stop := 221952 ), rec( content := " require\ s that ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 221979, root := ~, start := 221968, stop := 221978 ), rec( content := " be equa\ l to\n ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsPartialPermSemigroup" ), content := 0, count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 222049, root := ~, start := 221998, stop := 222048 ), rec( content := ".\n \ If ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Y\ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 222068, root := ~, start := 222060, stop := 222067 ), rec( content := " is a ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 222112, root := ~, start := 222074, stop := 222111 ), rec( content := " or\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMeetS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 222160, root := ~, start := 222122, stop := 222159 ), rec( content := ",\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "g\ ps", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 222178, root := ~, start := 222168, stop := 222177 ), rec( content := " is a li\ st of groups corresponding to each vertex, and\n ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ oms", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 222250, root := ~, start := 222239, stop := 222249 ), rec( content := " is a li\ st containing for each edge ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ i, j)", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 222299, root := ~, start := 222286, stop := 222298 ), rec( content := " in\n \ the transitive reduction of ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 222351, root := ~, start := 222337, stop := 222350 ), rec( content := " a tripl\ e ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j, hom]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 222379, root := ~, start := 222361, stop := 222378 ), rec( content := "\n \ where ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ om", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 222402, root := ~, start := 222392, stop := 222401 ), rec( content := " is a gr\ oup homomorphism from ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ps[i]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 222445, root := ~, start := 222432, stop := 222444 ), rec( content := " to\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ps[j]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 222468, root := ~, start := 222455, stop := 222467 ), rec( content := ", and th\ e diagram of homomorphisms commutes, then\n ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 222542, root := ~, start := 222524, stop := 222541 ), rec( content := " returns\ a semigroup of partial perms which is\n isomorphic to the strong semilat\ tice of groups ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ [Y; gps; homs]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "M", next := 222664, root := ~, start := 222642, stop := 222663 ), rec( content := ".\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G1 := AlternatingGroup(4);;\ngap> G2 := SymmetricGroup(2);;\ngap> G3 := \ SymmetricGroup(3);;\ngap> gr := Digraph([[1, 3], [2, 3], [3]]);;\ngap> sgn := \ function(x)\n> if SignPerm(x) = 1 then\n> return ();\n> fi;\n> return (1, 2);\ \n> end;;\ngap> hom13 := GroupHomomorphismByFunction(G1, G3, sgn);;\ngap> hom2\ 3 := GroupHomomorphismByFunction(G2, G3, sgn);;\ngap> T := AsSemigroup(IsParti\ alPermSemigroup,\n> gr,\n> [G1, G2, G3], [[1, 3, hom13], [2, 3, hom23]]);;\nga\ p> Size(T);\n20\ngap> D := GreensDClasses(T);;\ngap> List(D, x -> Size(x));\n[\ 6, 12, 2 ]\n", count := [ 5, 5, 2, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 4 ], mathmode := "Text", name := "Example", next := 223241, root := ~, start := 222672, stop := 223240 ), rec( content := "\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 223260, root := ~, start := 221901, stop := 223259 ) ], count := [ 5, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 223276, root := ~, start := 221701, stop := 223275 ), rec( content := "\n\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 223290, root := ~, start := 219310, stop := 223289 ) , rec( content := "\n\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Planarity", count := [ 5, 6, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 0, 2 ], mathmode := "Text", name := "Heading", next := 223331, root := ~, start := 223303, stop := 223330 ), rec( content := "\n ", count := [ 5, 6, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "KuratowskiPlana\ rSubdigraph" ), content := 0, count := [ 5, 6, 1, 2 ], mathmode := "Text", name := "Attr", next := 223406, root := ~, start := 223351, stop := 223405 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "K", next := 223439, root := ~, start := 223428, stop := 223438 ), rec( content := ".", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Returns", next := 223450, root := ~, start := 223409, stop := 223449 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiPlanarSubdigraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 223504, root := ~, start := 223471, stop := 223503 ), rec( content := " returns\ the immutable list of\n lists of out-neighbours of a (not necessarily indu\ ced) subdigraph of the\n digraph ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 223638, root := ~, start := 223624, stop := 223637 ), rec( content := " that wi\ tnesses the fact that ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 223682, root := ~, start := 223668, stop := 223681 ), rec( content := " is not\\ n planar, or ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "K", next := 223716, root := ~, start := 223705, stop := 223715 ), rec( content := " if ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 223734, root := ~, start := 223720, stop := 223733 ), rec( content := " is plan\ ar. In other words,\n ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiPlanarSubdigraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 223800, root := ~, start := 223767, stop := 223799 ), rec( content := " returns\ the out-neighbours of a\n subdigraph of ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 223865, root := ~, start := 223851, stop := 223864 ), rec( content := " that is\ homeomorphic to the complete graph\n with ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "5\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 223926, root := ~, start := 223918, stop := 223925 ), rec( content := " vertice\ s, or to the complete bipartite graph with vertex sets\n of sizes ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 224010, root := ~, start := 224002, stop := 224009 ), rec( content := " and ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 224023, root := ~, start := 224015, stop := 224022 ), rec( content := ". ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "P", next := 224029, root := ~, start := 224025, stop := 224028 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "A", next := 224103, root := ~, start := 224089, stop := 224102 ), rec( content := " are\n \ ignored when considering whether or not ", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "A", next := 224166, root := ~, start := 224152, stop := 224165 ), rec( content := " is plan\ ar. ", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "P", next := 224182, root := ~, start := 224178, stop := 224181 ), rec( content := "\n\n \ See also ", count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "Ref", next := 224226, root := ~, start := 224197, stop := 224225 ), rec( content := "\n an\ d ", count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Subdigr\ aphHomeomorphicToK33" ), content := 0, count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "Ref", next := 224276, root := ~, start := 224235, stop := 224275 ), rec( content := ".\n " , count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "P", next := 224286, root := ~, start := 224282, stop := 224285 ), rec( content := "\n \n\ This method uses the reference implementation in\n ", count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 1, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "Cite", next := 224448, root := ~, start := 224430, stop := 224447 ), rec( content := ".\n\n", count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiPlanarSubdigraph(D);\nfail\ngap> D := Digra\ ph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3\ , 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, \ 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsPlanarDigraph(D);\nfalse\ngap> KuratowskiPlanarSubdigraph(D);\n[ [\ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 ], [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ],\ [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\ \n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiPlanarSubdigraph(D)\ ;\nfail\ngap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6\ , 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9,\ 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ \ngap> IsPlanarDigraph(D);\nfalse\ \ngap> KuratowskiPlanarSubdigraph(D);\n[ [ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 ], \ [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ], [ ] ]\n", count := [ 5, 6, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 1, 7 ], mathmode := "Text", name := "Example", next := 225750, root := ~, start := 224451, stop := 225749 ), rec( content := "\n ", count := [ 5, 6, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Description", next := 225767, root := ~, start := 223453, stop := 225766 ) ], count := [ 5, 6, 1, 1 ], mathmode := "Text", name := "ManSection", next := 225781, root := ~, start := 223336, stop := 225780 ), rec( content := "\n\n ", count := [ 5, 6, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "KuratowskiOuter\ PlanarSubdigraph" ), content := 0, count := [ 5, 6, 2, 2 ], mathmode := "Text", name := "Attr", next := 225862, root := ~, start := 225802, stop := 225861 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "K", next := 225895, root := ~, start := 225884, stop := 225894 ), rec( content := ".", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Returns", next := 225906, root := ~, start := 225865, stop := 225905 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiOuterPlanarSubdigraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 225965, root := ~, start := 225927, stop := 225964 ), rec( content := " returns\ the immutable list of\n immutable lists of out-neighbours of a (not necess\ arily induced)\n subdigraph of the digraph ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 226109, root := ~, start := 226095, stop := 226108 ), rec( content := " that wi\ tnesses the fact\n that ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 226157, root := ~, start := 226143, stop := 226156 ), rec( content := " is not \ outer planar, or ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "K", next := 226193, root := ~, start := 226182, stop := 226192 ), rec( content := " if\n \ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 226215, root := ~, start := 226201, stop := 226214 ), rec( content := " is oute\ r planar. In other words,\n ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiOuterPlanarSubdigraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 226292, root := ~, start := 226254, stop := 226291 ), rec( content := " returns\ the out-neighbours of a\n subdigraph of ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 226357, root := ~, start := 226343, stop := 226356 ), rec( content := " that is\ homeomorphic to the complete graph\n with ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "4\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 226418, root := ~, start := 226410, stop := 226417 ), rec( content := " vertice\ s, or to the complete bipartite graph with vertex sets\n of sizes ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 226502, root := ~, start := 226494, stop := 226501 ), rec( content := " and ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 226515, root := ~, start := 226507, stop := 226514 ), rec( content := ". ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "P", next := 226521, root := ~, start := 226517, stop := 226520 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "A", next := 226595, root := ~, start := 226581, stop := 226594 ), rec( content := " are\n \ ignored when considering whether or not ", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "A", next := 226658, root := ~, start := 226644, stop := 226657 ), rec( content := " is oute\ r planar. \n ", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "P", next := 226685, root := ~, start := 226681, stop := 226684 ), rec( content := "\n\n \ See also \n ", count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "Ref", next := 226739, root := ~, start := 226705, stop := 226738 ), rec( content := ",\n " , count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Subdigr\ aphHomeomorphicToK4" ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "Ref", next := 226785, root := ~, start := 226745, stop := 226784 ), rec( content := ", and \n\ ", count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Subdigr\ aphHomeomorphicToK23" ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "Ref", next := 226837, root := ~, start := 226796, stop := 226836 ), rec( content := ".\n " , count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "P", next := 226847, root := ~, start := 226843, stop := 226846 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 2, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "Cite", next := 227005, root := ~, start := 226987, stop := 227004 ), rec( content := ".\n\n", count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], \ [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ]\ ]\ngap> D := Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2,\ 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9,\ 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsOuterPlanarDigraph(D);\nfalse\ngap> KuratowskiO\ uterPlanarSubdigraph(D);\n[ [ ], [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], \ [ 7, 9 ], [ ], \n [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [\ 8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1,\ 2], [5, 9]]);\n\ngap> KuratowskiO\ uterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], \ [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDigra\ ph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2\ , 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7\ , 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsOuterPlanarDigraph(D);\nfalse\ngap> KuratowskiOuterPlanarSubdigrap\ h(D);\n[ [ ], [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ], \n \ [ ] ]", count := [ 5, 6, 2, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 7 ], mathmode := "Text", name := "Example", next := 228498, root := ~, start := 227008, stop := 228497 ), rec( content := "\n ", count := [ 5, 6, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Description", next := 228515, root := ~, start := 225909, stop := 228514 ) ], count := [ 5, 6, 2, 1 ], mathmode := "Text", name := "ManSection", next := 228529, root := ~, start := 225787, stop := 228528 ), rec( content := "\n\n ", count := [ 5, 6, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "PlanarEmbedding\ " ), content := 0, count := [ 5, 6, 3, 2 ], mathmode := "Text", name := "Attr", next := 228594, root := ~, start := 228550, stop := 228593 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "K", next := 228627, root := ~, start := 228616, stop := 228626 ), rec( content := ".", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Returns", next := 228638, root := ~, start := 228597, stop := 228637 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "A", next := 228676, root := ~, start := 228662, stop := 228675 ), rec( content := " is a pl\ anar digraph, then \n ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lanarEmbedding", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "C", next := 228730, root := ~, start := 228708, stop := 228729 ), rec( content := " returns\ the immutable list of lists of\n out-neighbours of a subdigraph of ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "A", next := 228822, root := ~, start := 228808, stop := 228821 ), rec( content := " such th\ at each vertex's\n neighbours are given in clockwise order. If ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "A", next := 228909, root := ~, start := 228895, stop := 228908 ), rec( content := " is not \ planar,\n then ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "K", next := 228945, root := ~, start := 228934, stop := 228944 ), rec( content := " is retu\ rned. ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "P", next := 228963, root := ~, start := 228959, stop := 228962 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "A", next := 229037, root := ~, start := 229023, stop := 229036 ), rec( content := " are\n \ ignored by ", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lanarEmbedding", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "C", next := 229079, root := ~, start := 229057, stop := 229078 ), rec( content := ".\n " , count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "P", next := 229089, root := ~, start := 229085, stop := 229088 ), rec( content := "\n\n \ See also \n ", count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "Ref", next := 229138, root := ~, start := 229109, stop := 229137 ), rec( content := ".\n " , count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "P", next := 229148, root := ~, start := 229144, stop := 229147 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 3, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "Cite", next := 229306, root := ~, start := 229288, stop := 229305 ), rec( content := ".\n\n", count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ \ 4 ], [ 6 ], [ 11, 7 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D :=\ Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1\ , 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, \ 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> PlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDigraph,\ [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7\ ], [2, 11], [1, 2], [5, 9]]);\n\ng\ ap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11, 7 ]\ , [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDig\ raph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1,\ 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5,\ 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> PlanarEmbedding(D);\nfail\n", count := [ 5, 6, 3, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 3, 7 ], mathmode := "Text", name := "Example", next := 230524, root := ~, start := 229309, stop := 230523 ), rec( content := "\n ", count := [ 5, 6, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Description", next := 230541, root := ~, start := 228641, stop := 230540 ) ], count := [ 5, 6, 3, 1 ], mathmode := "Text", name := "ManSection", next := 230555, root := ~, start := 228535, stop := 230554 ), rec( content := "\n\n ", count := [ 5, 6, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "OuterPlanarEmbe\ dding" ), content := 0, count := [ 5, 6, 4, 2 ], mathmode := "Text", name := "Attr", next := 230625, root := ~, start := 230576, stop := 230624 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "K", next := 230658, root := ~, start := 230647, stop := 230657 ), rec( content := ".", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Returns", next := 230669, root := ~, start := 230628, stop := 230668 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "A", next := 230707, root := ~, start := 230693, stop := 230706 ), rec( content := " is an o\ uter planar digraph, then\n ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ uterPlanarEmbedding", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "C", next := 230772, root := ~, start := 230745, stop := 230771 ), rec( content := " returns\ the immutable list of lists\n of out-neighbours of a subdigraph of ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "A", next := 230864, root := ~, start := 230850, stop := 230863 ), rec( content := " such th\ at each\n vertex's neighbours are given in clockwise order. If ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "A", next := 230951, root := ~, start := 230937, stop := 230950 ), rec( content := " is not\\ n outer planar, then ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "K", next := 230993, root := ~, start := 230982, stop := 230992 ), rec( content := " is retu\ rned. ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "P", next := 231011, root := ~, start := 231007, stop := 231010 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "A", next := 231085, root := ~, start := 231071, stop := 231084 ), rec( content := " are\n \ ignored by ", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ uterPlanarEmbedding", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "C", next := 231132, root := ~, start := 231105, stop := 231131 ), rec( content := ".\n " , count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "P", next := 231142, root := ~, start := 231138, stop := 231141 ), rec( content := "\n\n \ See also ", count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "Ref", next := 231191, root := ~, start := 231157, stop := 231190 ), rec( content := ".\n " , count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "P", next := 231201, root := ~, start := 231197, stop := 231200 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 4, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 4, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "Cite", next := 231359, root := ~, start := 231341, stop := 231358 ), rec( content := ".\n\n", count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph([[2\ , 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, 6\ , 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2\ , 3, 5, 6, 7, 8], [3, 5]]);\n\ng\ ap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(CompleteBipartit\ eDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\ngap> D := Digraph(IsMuta\ bleDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6\ , 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDigr\ aph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, \ 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, \ 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(CompleteB\ ipartiteDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\n", count := [ 5, 6, 4, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 4, 7 ], mathmode := "Text", name := "Example", next := 232597, root := ~, start := 231362, stop := 232596 ), rec( content := "\n ", count := [ 5, 6, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Description", next := 232614, root := ~, start := 230672, stop := 232613 ) ], count := [ 5, 6, 4, 1 ], mathmode := "Text", name := "ManSection", next := 232628, root := ~, start := 230561, stop := 232627 ), rec( content := "\n\n ", count := [ 5, 6, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK23" ), content := 0, count := [ 5, 6, 5, 2 ], mathmode := "Text", name := "Attr", next := 232705, root := ~, start := 232649, stop := 232704 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK33" ), content := 0, count := [ 5, 6, 5, 4 ], mathmode := "Text", name := "Attr", next := 232764, root := ~, start := 232708, stop := 232763 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK4" ), content := 0, count := [ 5, 6, 5, 6 ], mathmode := "Text", name := "Attr", next := 232822, root := ~, start := 232767, stop := 232821 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "K", next := 232855, root := ~, start := 232844, stop := 232854 ), rec( content := ".", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Returns", next := 232866, root := ~, start := 232825, stop := 232865 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se attributes return the immutable list of lists of\n out-neighbours of a s\ ubdigraph of the digraph ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "A", next := 233006, root := ~, start := 232992, stop := 233005 ), rec( content := " which i\ s\n homeomorphic to one of the following:\n the complete bipartite graph\ with vertex sets of sizes ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 233125, root := ~, start := 233117, stop := 233124 ), rec( content := " and\n \ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 233142, root := ~, start := 233134, stop := 233141 ), rec( content := "; the co\ mplete bipartite graph with vertex sets of sizes ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 233207, root := ~, start := 233199, stop := 233206 ), rec( content := "\n an\ d ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 233224, root := ~, start := 233216, stop := 233223 ), rec( content := "; or the\ complete graph with ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "4\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 233261, root := ~, start := 233253, stop := 233260 ), rec( content := " vertice\ s. If\n ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "A", next := 233293, root := ~, start := 233279, stop := 233292 ), rec( content := " has no \ such subdigraphs, then ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "K", next := 233335, root := ~, start := 233324, stop := 233334 ), rec( content := " is retu\ rned. \n ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "P", next := 233358, root := ~, start := 233354, stop := 233357 ), rec( content := "\n\n \ See also ", count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "Ref", next := 233402, root := ~, start := 233373, stop := 233401 ), rec( content := " and ", count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "Ref", next := 233447, root := ~, start := 233407, stop := 233446 ), rec( content := " for mor\ e details.", count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "P", next := 233469, root := ~, start := 233465, stop := 233468 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 9 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 5, 9 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 5, 9 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 9 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "Cite", next := 233627, root := ~, start := 233609, stop := 233626 ), rec( content := ".\n\n", count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 7, 11], \n> [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9, \ 8, 10 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ \ngap> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [\ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(\ [[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 11], \n> [4, 7], [6, 8], [2, 7],\ [2, 11], [1, 2], [5, 9]]);\n\ng\ ap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphicToK23(D);\ \n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], \ [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\ngap> Subdigr\ aphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomorphicToK33(\ CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(CompleteBipartite\ Digraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ], [ ] ]\n\ gap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\ngap> D := Digraph(\ IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, \ 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9, 8, 1\ 0 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\nga\ p> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ]\ , [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMu\ tableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 11], [4, 7], [6,\ 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphi\ cToK23(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \ \n [ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\nga\ p> SubdigraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomor\ phicToK33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(Complet\ eBipartiteDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ]\ , [ ] ]\ngap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\n", count := [ 5, 6, 5, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 5, 10 ], mathmode := "Text", name := "Example", next := 235939, root := ~, start := 233630, stop := 235938 ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Description", next := 235956, root := ~, start := 232869, stop := 235955 ) ], count := [ 5, 6, 5, 1 ], mathmode := "Text", name := "ManSection", next := 235970, root := ~, start := 232634, stop := 235969 ), rec( content := "\n\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 0, 1 ], mathmode := "Text", name := "Section", next := 235984, root := ~, start := 223294, stop := 235983 ) , rec( content := "\n\n", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 235996, root := ~, start := 131966, stop := 235995 ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " attributes, operations ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Properties of digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Properties of digraphs", count := [ 6, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 236113, root := ~, start := 236072, stop := 236112 ) , rec( content := "\n\n ", count := [ 6, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Edge properties", count := [ 6, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 236160, root := ~, start := 236126, stop := 236159 ), rec( content := "\n ", count := [ 6, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphHasLoops\ " ), content := 0, count := [ 6, 1, 1, 2 ], mathmode := "Text", name := "Prop", next := 236224, root := ~, start := 236180, stop := 236223 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 236247, root := ~, start := 236236, stop := 236246 ), rec( content := " or ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 236263, root := ~, start := 236251, stop := 236262 ), rec( content := ".", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 236274, root := ~, start := 236227, stop := 236273 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 236314, root := ~, start := 236303, stop := 236313 ), rec( content := " if the \ digraph ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "A", next := 236344, root := ~, start := 236330, stop := 236343 ), rec( content := " has loo\ ps, and\n ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 236376, root := ~, start := 236364, stop := 236375 ), rec( content := " if it d\ oes not. A loop is an edge with equal source and range.\n ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "P", next := 236448, root := ~, start := 236444, stop := 236447 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "A", next := 236490, root := ~, start := 236476, stop := 236489 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 2 ] ]\ngap> DigraphHasLo\ ops(D);\ntrue\ngap> D := Digraph([[2, 3], [1], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], \ [ 3, 2 ] ]\ngap> DigraphHasLoops(D);\nfalse\ngap> D := CompleteDigraph(IsMutab\ leDigraph, 4);\n\ngap> DigraphHasLo\ ops(D);\nfalse", count := [ 6, 1, 1, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 5 ], mathmode := "Text", name := "Example", next := 237085, root := ~, start := 236587, stop := 237084 ), rec( content := "\n ", count := [ 6, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 237102, root := ~, start := 236277, stop := 237101 ) ], count := [ 6, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 237116, root := ~, start := 236165, stop := 237115 ), rec( content := "\n\n ", count := [ 6, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAntiSymmetric\ Digraph" ), content := 0, count := [ 6, 1, 2, 2 ], mathmode := "Text", name := "Prop", next := 237188, root := ~, start := 237137, stop := 237187 ), rec( attributes := rec( Arg := "digraph", Name := "IsAntisymmetric\ Digraph" ), content := 0, count := [ 6, 1, 2, 4 ], mathmode := "Text", name := "Prop", next := 237242, root := ~, start := 237191, stop := 237241 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 237265, root := ~, start := 237254, stop := 237264 ), rec( content := " or ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 237281, root := ~, start := 237269, stop := 237280 ), rec( content := ".", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 237292, root := ~, start := 237245, stop := 237291 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 237341, root := ~, start := 237330, stop := 237340 ), rec( content := " if the \ digraph ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "A", next := 237371, root := ~, start := 237357, stop := 237370 ), rec( content := "\n is\ antisymmetric, and ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 237410, root := ~, start := 237398, stop := 237409 ), rec( content := " if it i\ s not.\n ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "P", next := 237433, root := ~, start := 237429, stop := 237432 ), rec( content := "\n\n \ A digraph is ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ ntisymmetric", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "E", next := 237472, root := ~, start := 237452, stop := 237471 ), rec( content := " if when\ ever there is an edge with source\n ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 237526, root := ~, start := 237518, stop := 237525 ), rec( content := " and ran\ ge ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 237545, root := ~, start := 237537, stop := 237544 ), rec( content := ", and an\ edge with source ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 237579, root := ~, start := 237571, stop := 237578 ), rec( content := " and ran\ ge\n ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 237602, root := ~, start := 237594, stop := 237601 ), rec( content := ", then t\ he vertices ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 237630, root := ~, start := 237622, stop := 237629 ), rec( content := " and ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 237643, root := ~, start := 237635, stop := 237642 ), rec( content := " are equ\ al.\n ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "P", next := 237663, root := ~, start := 237659, stop := 237662 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "A", next := 237705, root := ~, start := 237691, stop := 237704 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := Digraph([[2], [1, 3], [2, 3]]);\n\ngap> IsAntisymmetricDigraph(gr1);\nfalse\ngap> DigraphEdges(gr1)\ {[1, 2]};\n[ [ 1, 2 ], [ 2, 1 ] ]\ngap> gr2 := Digraph([[1, 2], [3, 3], [1]]);\ \n\ngap> IsAntisymmetricDigra\ ph(gr2);\ntrue\ngap> DigraphEdges(gr2);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 3 ], [ 2, \ 3 ], [ 3, 1 ] ]\n", count := [ 6, 1, 2, 8 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 8 ], mathmode := "Text", name := "Example", next := 238232, root := ~, start := 237806, stop := 238231 ), rec( content := "\n ", count := [ 6, 1, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 238249, root := ~, start := 237295, stop := 238248 ) ], count := [ 6, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 238263, root := ~, start := 237122, stop := 238262 ), rec( content := "\n\n ", count := [ 6, 1, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsBipartiteDigr\ aph" ), content := 0, count := [ 6, 1, 3, 2 ], mathmode := "Text", name := "Prop", next := 238331, root := ~, start := 238284, stop := 238330 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 238354, root := ~, start := 238343, stop := 238353 ), rec( content := " or ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 238370, root := ~, start := 238358, stop := 238369 ), rec( content := ".", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 238381, root := ~, start := 238334, stop := 238380 ), rec( attributes := rec( ), content := [ rec( content := "\n This \ property is ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 238428, root := ~, start := 238417, stop := 238427 ), rec( content := " if the \ digraph ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "A", next := 238458, root := ~, start := 238444, stop := 238457 ), rec( content := " is bipa\ rtite, and\n ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 238491, root := ~, start := 238479, stop := 238490 ), rec( content := " if it i\ s not. A digraph is bipartite if and only if the\n vertices of ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "A", next := 238577, root := ~, start := 238563, stop := 238576 ), rec( content := " can be \ partitioned into two non-empty sets such\n that the source and range of any e\ dge of ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "A", next := 238683, root := ~, start := 238669, stop := 238682 ), rec( content := " lie in \ distinct sets.\n Equivalently, a digraph is bipartite if and only if it is 2-\ colorable; see\n ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ GreedyColouring", Label := "for a digraph" ), content := 0, count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 238843, root := ~, start := 238785, stop := 238842 ), rec( content := ". ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "P", next := 238849, root := ~, start := 238845, stop := 238848 ), rec( content := "\n\n Se\ e also ", count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Bicomponents" ), content := 0, count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "Ref", next := 238895, root := ~, start := 238862, stop := 238894 ), rec( content := ".\n " , count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "P", next := 238905, root := ~, start := 238901, stop := 238904 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "A", next := 238947, root := ~, start := 238933, stop := 238946 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := ChainDigraph(4);\n\ngap> I\ sBipartiteDigraph(D);\ntrue\ngap> D := CycleDigraph(3);\n\ngap> IsBipartiteDigraph(D);\nfalse\ngap> D := CompleteBi\ partiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBipartiteDigraph(D);\ntrue", count := [ 6, 1, 3, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 3, 6 ], mathmode := "Text", name := "Example", next := 239420, root := ~, start := 239047, stop := 239419 ), rec( content := "\n ", count := [ 6, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 239437, root := ~, start := 238384, stop := 239436 ) ], count := [ 6, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 239451, root := ~, start := 238269, stop := 239450 ), rec( content := "\n\n ", count := [ 6, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCompleteBipar\ titeDigraph" ), content := 0, count := [ 6, 1, 4, 2 ], mathmode := "Text", name := "Prop", next := 239527, root := ~, start := 239472, stop := 239526 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 239550, root := ~, start := 239539, stop := 239549 ), rec( content := " or ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 239566, root := ~, start := 239554, stop := 239565 ), rec( content := ".", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 239577, root := ~, start := 239530, stop := 239576 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 239617, root := ~, start := 239606, stop := 239616 ), rec( content := " if the \ digraph ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "A", next := 239647, root := ~, start := 239633, stop := 239646 ), rec( content := " is a co\ mplete bipartite\n digraph, and ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 239701, root := ~, start := 239689, stop := 239700 ), rec( content := " if it i\ s not. ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "P", next := 239720, root := ~, start := 239716, stop := 239719 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplete bipartite digraph", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "E", next := 239774, root := ~, start := 239741, stop := 239773 ), rec( content := " if it i\ s bipartite, see\n ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 239835, root := ~, start := 239803, stop := 239834 ), rec( content := ", and th\ ere exists a unique edge with\n source ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 239892, root := ~, start := 239884, stop := 239891 ), rec( content := " and ran\ ge ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 239911, root := ~, start := 239903, stop := 239910 ), rec( content := " if and \ only if ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 239935, root := ~, start := 239927, stop := 239934 ), rec( content := " and ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 239948, root := ~, start := 239940, stop := 239947 ), rec( content := " lie\n \ in different bicomponents of ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "A", next := 240000, root := ~, start := 239986, stop := 239999 ), rec( content := ", see ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Bicomponents" ), content := 0, count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 240045, root := ~, start := 240006, stop := 240044 ), rec( content := ". ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "P", next := 240051, root := ~, start := 240047, stop := 240050 ), rec( content := "\n\n \ Equivalently, a bipartite digraph with bicomponents of size ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "M", next := 240125, root := ~, start := 240117, stop := 240124 ), rec( content := " and\n \ ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "M", next := 240142, root := ~, start := 240134, stop := 240141 ), rec( content := " is comp\ lete precisely when it has ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ mn", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "M", next := 240187, root := ~, start := 240177, stop := 240186 ), rec( content := " edges, \ none of which\n are multiple edges. ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "P", next := 240237, root := ~, start := 240233, stop := 240236 ), rec( content := "\n\n \ See also ", count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Complet\ eBipartiteDigraph" ), content := 0, count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "Ref", next := 240290, root := ~, start := 240252, stop := 240289 ), rec( content := ".\n " , count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "P", next := 240300, root := ~, start := 240296, stop := 240299 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "A", next := 240342, root := ~, start := 240328, stop := 240341 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(2);\n\ngap> I\ sCompleteBipartiteDigraph(D);\ntrue\ngap> D := CycleDigraph(4);\n\ngap> IsBipartiteDigraph(D);\ntrue\ngap> IsComple\ teBipartiteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDig\ raph, 5, 4);\n\ngap> IsCompleteBipa\ rtiteDigraph(D);\ntrue", count := [ 6, 1, 4, 8 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 8 ], mathmode := "Text", name := "Example", next := 240873, root := ~, start := 240443, stop := 240872 ), rec( content := "\n ", count := [ 6, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 240890, root := ~, start := 239580, stop := 240889 ) ], count := [ 6, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 240904, root := ~, start := 239457, stop := 240903 ), rec( content := "\n\n ", count := [ 6, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCompleteDigra\ ph" ), content := 0, count := [ 6, 1, 5, 2 ], mathmode := "Text", name := "Prop", next := 240971, root := ~, start := 240925, stop := 240970 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 240994, root := ~, start := 240983, stop := 240993 ), rec( content := " or ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 241010, root := ~, start := 240998, stop := 241009 ), rec( content := ".", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 241021, root := ~, start := 240974, stop := 241020 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 241061, root := ~, start := 241050, stop := 241060 ), rec( content := " if the \ digraph ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "A", next := 241091, root := ~, start := 241077, stop := 241090 ), rec( content := " is comp\ lete, and\n ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 241125, root := ~, start := 241113, stop := 241124 ), rec( content := " if it i\ s not. ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "P", next := 241144, root := ~, start := 241140, stop := 241143 ), rec( content := "\n\n \ A digraph is ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplete", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "E", next := 241178, root := ~, start := 241163, stop := 241177 ), rec( content := " if it h\ as no loops, and for all\n ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istinct", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "E", next := 241230, root := ~, start := 241215, stop := 241229 ), rec( content := " vertice\ s ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 241248, root := ~, start := 241240, stop := 241247 ), rec( content := " and ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 241261, root := ~, start := 241253, stop := 241260 ), rec( content := ",\n t\ here is exactly one edge with source ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 241313, root := ~, start := 241305, stop := 241312 ), rec( content := " and ran\ ge ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 241332, root := ~, start := 241324, stop := 241331 ), rec( content := ".\n\n \ Equivalently, a digraph with ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "M", next := 241376, root := ~, start := 241368, stop := 241375 ), rec( content := " vertice\ s is complete precisely when\n it has ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ (n - 1)", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "M", next := 241439, root := ~, start := 241424, stop := 241438 ), rec( content := " edges, \ no loops, and no multiple edges.\n ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "P", next := 241488, root := ~, start := 241484, stop := 241487 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "A", next := 241530, root := ~, start := 241516, stop := 241529 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3], [1, 3], [1, 2]]);\n\ngap> IsCompleteDigraph(D);\ntrue\ngap> D := Digraph([[2, 2], [1\ ]]);\n\ngap> IsCompleteDigrap\ h(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteDigraph(D);\nfalse", count := [ 6, 1, 5, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 5, 6 ], mathmode := "Text", name := "Example", next := 242038, root := ~, start := 241631, stop := 242037 ), rec( content := "\n ", count := [ 6, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 242055, root := ~, start := 241024, stop := 242054 ) ], count := [ 6, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 242069, root := ~, start := 240910, stop := 242068 ), rec( content := "\n\n ", count := [ 6, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCompleteMulti\ partiteDigraph" ), content := 0, count := [ 6, 1, 6, 2 ], mathmode := "Text", name := "Prop", next := 242148, root := ~, start := 242090, stop := 242147 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 242171, root := ~, start := 242160, stop := 242170 ), rec( content := " or ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 242187, root := ~, start := 242175, stop := 242186 ), rec( content := ".", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 242198, root := ~, start := 242151, stop := 242197 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 242252, root := ~, start := 242241, stop := 242251 ), rec( content := " if ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "A", next := 242270, root := ~, start := 242256, stop := 242269 ), rec( content := " is a co\ mplete\n multipartite digraph, and ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 242327, root := ~, start := 242315, stop := 242326 ), rec( content := " if not.\ ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "P", next := 242340, root := ~, start := 242336, stop := 242339 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplete multipartite digraph", count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "E", next := 242397, root := ~, start := 242361, stop := 242396 ), rec( content := " if and \ only if\n its vertices can be partitioned into at least two maximal indepen\ dent sets,\n where every possible edge between these independent sets occur\ s in the\n digraph exactly once.\n ", count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "P", next := 242602, root := ~, start := 242598, stop := 242601 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "A", next := 242644, root := ~, start := 242630, stop := 242643 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteMultipartiteDigraph([2, 4, 6]);\n\ngap> IsCompleteMultipartiteDigraph\ (D);\ntrue\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteMultipartiteDigraph(D);\ \ntrue", count := [ 6, 1, 6, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 6 ], mathmode := "Text", name := "Example", next := 243087, root := ~, start := 242745, stop := 243086 ), rec( content := "\n ", count := [ 6, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 243104, root := ~, start := 242201, stop := 243103 ) ], count := [ 6, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 243118, root := ~, start := 242075, stop := 243117 ), rec( content := "\n\n ", count := [ 6, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEmptyDigraph"\ ), content := 0, count := [ 6, 1, 7, 2 ], mathmode := "Text", name := "Prop", next := 243182, root := ~, start := 243139, stop := 243181 ), rec( attributes := rec( Arg := "digraph", Name := "IsNullDigraph" ), content := 0, count := [ 6, 1, 7, 4 ], mathmode := "Text", name := "Prop", next := 243227, root := ~, start := 243185, stop := 243226 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 243250, root := ~, start := 243239, stop := 243249 ), rec( content := " or ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 243266, root := ~, start := 243254, stop := 243265 ), rec( content := ".", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Returns", next := 243277, root := ~, start := 243230, stop := 243276 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 243317, root := ~, start := 243306, stop := 243316 ), rec( content := " if the \ digraph ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "A", next := 243347, root := ~, start := 243333, stop := 243346 ), rec( content := " is empt\ y, and\n ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 243378, root := ~, start := 243366, stop := 243377 ), rec( content := " if it i\ s not. A digraph is ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ mpty", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "E", next := 243418, root := ~, start := 243406, stop := 243417 ), rec( content := " if it h\ as no\n edges.", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "P", next := 243446, root := ~, start := 243442, stop := 243445 ), rec( content := "\n\n \ ", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sNullDigraph", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "C", next := 243472, root := ~, start := 243452, stop := 243471 ), rec( content := " is a sy\ nonym for ", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sEmptyDigraph", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "C", next := 243511, root := ~, start := 243490, stop := 243510 ), rec( content := ".\n " , count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "P", next := 243521, root := ~, start := 243517, stop := 243520 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "A", next := 243563, root := ~, start := 243549, stop := 243562 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], []]);\n\ngap>\ IsEmptyDigraph(D);\ntrue\ngap> IsNullDigraph(D);\ntrue\ngap> D := Digraph([[]\ , [1]]);\n\ngap> IsEmptyDigraph(D);\ \nfalse\ngap> IsNullDigraph(D);\nfalse", count := [ 6, 1, 7, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 7, 8 ], mathmode := "Text", name := "Example", next := 243956, root := ~, start := 243664, stop := 243955 ), rec( content := "\n ", count := [ 6, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Description", next := 243973, root := ~, start := 243280, stop := 243972 ) ], count := [ 6, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 243987, root := ~, start := 243124, stop := 243986 ), rec( content := "\n\n ", count := [ 6, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEquivalenceDi\ graph" ), content := 0, count := [ 6, 1, 8, 2 ], mathmode := "Text", name := "Prop", next := 244057, root := ~, start := 244008, stop := 244056 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "K", next := 244080, root := ~, start := 244069, stop := 244079 ), rec( content := " or ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "K", next := 244096, root := ~, start := 244084, stop := 244095 ), rec( content := ".", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 244107, root := ~, start := 244060, stop := 244106 ), rec( attributes := rec( ), content := [ rec( content := "\n A d\ igraph is an equivalence digraph if and only if the digraph satisifies\n al\ l of ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRefle\ xiveDigraph" ), content := 0, count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 244245, root := ~, start := 244213, stop := 244244 ), rec( content := ",\n " , count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 244283, root := ~, start := 244251, stop := 244282 ), rec( content := " and ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTrans\ itiveDigraph" ), content := 0, count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 244321, root := ~, start := 244288, stop := 244320 ), rec( content := ".\n A\ partial order ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "A", next := 244357, root := ~, start := 244343, stop := 244356 ), rec( content := " corresp\ onds to an equivalence relation.\n ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "P", next := 244406, root := ~, start := 244402, stop := 244405 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "A", next := 244448, root := ~, start := 244434, stop := 244447 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2], [1, 3]]);\n\ngap> IsEquivalenceDigraph(D);\ntrue\n", count := [ 6, 1, 8, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 8, 5 ], mathmode := "Text", name := "Example", next := 244699, root := ~, start := 244545, stop := 244698 ), rec( content := "\n ", count := [ 6, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 244716, root := ~, start := 244110, stop := 244715 ) ], count := [ 6, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 244730, root := ~, start := 243993, stop := 244729 ), rec( content := "\n\n ", count := [ 6, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsFunctionalDig\ raph" ), content := 0, count := [ 6, 1, 9, 2 ], mathmode := "Text", name := "Prop", next := 244799, root := ~, start := 244751, stop := 244798 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "K", next := 244822, root := ~, start := 244811, stop := 244821 ), rec( content := " or ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "K", next := 244838, root := ~, start := 244826, stop := 244837 ), rec( content := ".", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Returns", next := 244849, root := ~, start := 244802, stop := 244848 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "K", next := 244898, root := ~, start := 244887, stop := 244897 ), rec( content := " if the \ digraph ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "A", next := 244928, root := ~, start := 244914, stop := 244927 ), rec( content := " is\n \ functional. ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "P", next := 244952, root := ~, start := 244948, stop := 244951 ), rec( content := "\n\n \ A digraph is ", count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unctional", count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "E", next := 244988, root := ~, start := 244971, stop := 244987 ), rec( content := " if ever\ y vertex is the source of a\n unique edge.\n ", count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "P", next := 245049, root := ~, start := 245045, stop := 245048 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "A", next := 245091, root := ~, start := 245077, stop := 245090 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := Digraph([[3], [2], [2], [1], [6], [5]]);\n\ngap> IsFunctionalDigraph(gr1);\ntrue\ngap> gr2 := Digra\ ph([[1, 2], [1]]);\n\ngap> IsFunct\ ionalDigraph(gr2);\nfalse\ngap> gr3 := Digraph(3, [1, 2, 3], [2, 3, 1]);\n\ngap> IsFunctionalDigraph(gr3);\ntrue\ \n", count := [ 6, 1, 9, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 6 ], mathmode := "Text", name := "Example", next := 245602, root := ~, start := 245191, stop := 245601 ), rec( content := "\n ", count := [ 6, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 245619, root := ~, start := 244852, stop := 245618 ) ], count := [ 6, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 245633, root := ~, start := 244736, stop := 245632 ), rec( content := "\n\n ", count := [ 6, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsMultiDigraph"\ ), content := 0, count := [ 6, 1, 10, 2 ], mathmode := "Text", name := "Prop", next := 245697, root := ~, start := 245654, stop := 245696 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "K", next := 245720, root := ~, start := 245709, stop := 245719 ), rec( content := " or ", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "K", next := 245736, root := ~, start := 245724, stop := 245735 ), rec( content := ".", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Returns", next := 245747, root := ~, start := 245700, stop := 245746 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ultidigraph", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "E", next := 245789, root := ~, start := 245770, stop := 245788 ), rec( content := " is one \ that has at least two\n edges with equal source and range.", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "P", next := 245861, root := ~, start := 245857, stop := 245860 ), rec( content := "\n\n \ If the argument ", count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "A", next := 245897, root := ~, start := 245883, stop := 245896 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\"a\", \"b\", \"c\"], [\"a\", \"b\", \"b\"], [\"b\", \"c\"\ , \"a\"]);\n\ngap> IsMultiDigraph(\ D);\nfalse\ngap> D := DigraphFromDigraph6String(\"&Bug\");\n\ngap> IsDuplicateFree(DigraphEdges(D));\ntrue\ngap>\ IsMultiDigraph(D);\nfalse\ngap> D := Digraph([[1, 2, 3, 2], [2, 1], [3]]);\n<\ immutable multidigraph with 3 vertices, 7 edges>\ngap> IsDuplicateFree(Digraph\ Edges(D));\nfalse\ngap> IsMultiDigraph(D);\ntrue\ngap> D := DigraphMutableCopy\ (D); \n\ngap> IsMultiDigraph(D)\ ;\ntrue", count := [ 6, 1, 10, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 5 ], mathmode := "Text", name := "Example", next := 246621, root := ~, start := 245998, stop := 246620 ), rec( content := "\n ", count := [ 6, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 246638, root := ~, start := 245750, stop := 246637 ) ], count := [ 6, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 246652, root := ~, start := 245639, stop := 246651 ), rec( content := "\n\n ", count := [ 6, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsReflexiveDigr\ aph" ), content := 0, count := [ 6, 1, 11, 2 ], mathmode := "Text", name := "Prop", next := 246720, root := ~, start := 246673, stop := 246719 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 246743, root := ~, start := 246732, stop := 246742 ), rec( content := " or ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 246759, root := ~, start := 246747, stop := 246758 ), rec( content := ".", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Returns", next := 246770, root := ~, start := 246723, stop := 246769 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 246819, root := ~, start := 246808, stop := 246818 ), rec( content := " if the \ digraph ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "A", next := 246849, root := ~, start := 246835, stop := 246848 ), rec( content := " is\n \ reflexive, and ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 246884, root := ~, start := 246872, stop := 246883 ), rec( content := " if it i\ s not.\n A digraph is ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eflexive", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "E", next := 246932, root := ~, start := 246916, stop := 246931 ), rec( content := " if it h\ as a loop at every vertex. ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 11, 4 ], mathmode := "Text", name := "P", next := 246971, root := ~, start := 246967, stop := 246970 ), rec( content := "\n ", count := [ 6, 1, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "P", next := 246980, root := ~, start := 246976, stop := 246979 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "A", next := 247022, root := ~, start := 247008, stop := 247021 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [2]]);\n\ngap> IsReflexiveDigraph(D);\ntrue\ngap> D := Digraph([[3, 1], [4, 2], [3],\ [2, 1]]);\n\ngap> IsReflexiveDigr\ aph(D);\nfalse\n", count := [ 6, 1, 11, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 6 ], mathmode := "Text", name := "Example", next := 247395, root := ~, start := 247122, stop := 247394 ), rec( content := "\n ", count := [ 6, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Description", next := 247412, root := ~, start := 246773, stop := 247411 ) ], count := [ 6, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 247426, root := ~, start := 246658, stop := 247425 ), rec( content := "\n\n ", count := [ 6, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsSymmetricDigr\ aph" ), content := 0, count := [ 6, 1, 12, 2 ], mathmode := "Text", name := "Prop", next := 247494, root := ~, start := 247447, stop := 247493 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 247517, root := ~, start := 247506, stop := 247516 ), rec( content := " or ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 247533, root := ~, start := 247521, stop := 247532 ), rec( content := ".", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 247544, root := ~, start := 247497, stop := 247543 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 247593, root := ~, start := 247582, stop := 247592 ), rec( content := " if the \ digraph ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "A", next := 247623, root := ~, start := 247609, stop := 247622 ), rec( content := "\n is\ symmetric, and ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 247658, root := ~, start := 247646, stop := 247657 ), rec( content := " if it i\ s not.", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "P", next := 247676, root := ~, start := 247672, stop := 247675 ), rec( content := "\n\n \ A ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ymmetric digraph", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "E", next := 247708, root := ~, start := 247684, stop := 247707 ), rec( content := " is one \ where for each non-loop edge, having\n source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 247772, root := ~, start := 247764, stop := 247771 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 247791, root := ~, start := 247783, stop := 247790 ), rec( content := ", there \ is a corresponding edge with\n source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 247847, root := ~, start := 247839, stop := 247846 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 247866, root := ~, start := 247858, stop := 247865 ), rec( content := ". If th\ ere are ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 247890, root := ~, start := 247882, stop := 247889 ), rec( content := " edges w\ ith\n source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 247921, root := ~, start := 247913, stop := 247920 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 247940, root := ~, start := 247932, stop := 247939 ), rec( content := ", then t\ here must be precisely ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 247979, root := ~, start := 247971, stop := 247978 ), rec( content := "\n ed\ ges with source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 248010, root := ~, start := 248002, stop := 248009 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 248029, root := ~, start := 248021, stop := 248028 ), rec( content := ". In ot\ her words, a symmetric\n digraph has a symmetric adjacency matrix ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "Ref", next := 248134, root := ~, start := 248105, stop := 248133 ), rec( content := ".\n " , count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "P", next := 248144, root := ~, start := 248140, stop := 248143 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "A", next := 248186, root := ~, start := 248172, stop := 248185 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := Digraph([[2], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr1);\ntrue\ngap> adj1 := AdjacencyMatri\ x(gr1);;\ngap> Display(adj1);\n[ [ 0, 1, 0 ],\n [ 1, 0, 1 ],\n [ 0, \ 1, 1 ] ]\ngap> adj1 = TransposedMat(adj1);\ntrue\ngap> gr1 = DigraphReverse(g\ r1);\ntrue\ngap> gr2 := Digraph([[2, 3], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr2);\nfalse\ngap> adj2 :=\ AdjacencyMatrix(gr2);;\ngap> Display(adj2);\n[ [ 0, 1, 1 ],\n [ 1, 0, \ 1 ],\n [ 0, 1, 1 ] ]\ngap> adj2 = TransposedMat(adj2);\nfalse\n", count := [ 6, 1, 12, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 12, 6 ], mathmode := "Text", name := "Example", next := 248906, root := ~, start := 248286, stop := 248905 ), rec( content := "\n ", count := [ 6, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 248923, root := ~, start := 247547, stop := 248922 ) ], count := [ 6, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 248937, root := ~, start := 247432, stop := 248936 ), rec( content := "\n\n ", count := [ 6, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsTournament" ) , content := 0, count := [ 6, 1, 13, 2 ], mathmode := "Text", name := "Prop", next := 248999, root := ~, start := 248958, stop := 248998 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 249022, root := ~, start := 249011, stop := 249021 ), rec( content := " or ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 249038, root := ~, start := 249026, stop := 249037 ), rec( content := ".", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Returns", next := 249049, root := ~, start := 249002, stop := 249048 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 249098, root := ~, start := 249087, stop := 249097 ), rec( content := " if the \ digraph ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "A", next := 249128, root := ~, start := 249114, stop := 249127 ), rec( content := " is a to\ urnament,\n and ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 249166, root := ~, start := 249154, stop := 249165 ), rec( content := " if it i\ s not. ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 13, 4 ], mathmode := "Text", name := "P", next := 249185, root := ~, start := 249181, stop := 249184 ), rec( content := "\n\n \ A tournament is an orientation of a complete (undirected) graph.\n Specific\ ally, a tournament is a digraph which has a unique directed edge\n (of some\ orientation) between any pair of distinct vertices, and no loops.\n ", count := [ 6, 1, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "P", next := 249420, root := ~, start := 249416, stop := 249419 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "A", next := 249462, root := ~, start := 249448, stop := 249461 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3, 4], [3, 4], [4], []]);\n\ngap> IsTournament(D);\ntrue\ngap> D := Digraph([[2], [1], [\ 3]]);\n\ngap> IsTournament(D);\nfa\ lse\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> IsTournament(D);\ntrue\ngap> DigraphRemoveEdge(D, 1, 2);\ \n\ngap> IsTournament(D);\nfalse\n", count := [ 6, 1, 13, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 13, 6 ], mathmode := "Text", name := "Example", next := 250043, root := ~, start := 249562, stop := 250042 ), rec( content := "\n ", count := [ 6, 1, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 250060, root := ~, start := 249052, stop := 250059 ) ], count := [ 6, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 250074, root := ~, start := 248943, stop := 250073 ), rec( content := "\n\n ", count := [ 6, 1, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsTransitiveDig\ raph" ), content := 0, count := [ 6, 1, 14, 2 ], mathmode := "Text", name := "Prop", next := 250143, root := ~, start := 250095, stop := 250142 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 250166, root := ~, start := 250155, stop := 250165 ), rec( content := " or ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 250182, root := ~, start := 250170, stop := 250181 ), rec( content := ".", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Returns", next := 250193, root := ~, start := 250146, stop := 250192 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 250242, root := ~, start := 250231, stop := 250241 ), rec( content := " if the \ digraph ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "A", next := 250272, root := ~, start := 250258, stop := 250271 ), rec( content := "\n is\ transitive, and ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 250308, root := ~, start := 250296, stop := 250307 ), rec( content := " if it i\ s not.\n\n A digraph is ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ransitive", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "E", next := 250358, root := ~, start := 250341, stop := 250357 ), rec( content := " if when\ ever ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j ]", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "C", next := 250386, root := ~, start := 250371, stop := 250385 ), rec( content := " and\n \ ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j, k ]", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "C", next := 250410, root := ~, start := 250395, stop := 250409 ), rec( content := " are edg\ es of the digraph, then ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, k ]", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "C", next := 250457, root := ~, start := 250442, stop := 250456 ), rec( content := " is also\ an\n edge of the digraph. ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "P", next := 250498, root := ~, start := 250494, stop := 250497 ), rec( content := "\n\n \ Let ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 250516, root := ~, start := 250508, stop := 250515 ), rec( content := " be the \ number of vertices of an arbitrary digraph, and let\n ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 250592, root := ~, start := 250584, stop := 250591 ), rec( content := " be the \ number of edges.\n For general digraphs, the methods used for this property\ use a version\n of the Floyd-Warshall algorithm, and have complexity ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (n^3)", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 250762, root := ~, start := 250749, stop := 250761 ), rec( content := ".\n\n \ However for digraphs which are topologically sortable\n [", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "Ref", next := 250864, root := ~, start := 250828, stop := 250863 ), rec( content := "], then \ methods with\n complexity ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n + m \\cdot n)", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 250927, root := ~, start := 250900, stop := 250926 ), rec( content := " will be\ used when appropriate.\n ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "P", next := 250967, root := ~, start := 250963, stop := 250966 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "A", next := 251009, root := ~, start := 250995, stop := 251008 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 14, 6 ], mathmode := "Text", name := "P", next := 251113, root := ~, start := 251109, stop := 251112 ), rec( content := "\n ", count := [ 6, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], [3]]);\n\ngap> IsTransitiveDigraph(D);\nfalse\ngap> gr2 := Digraph([[1, 2, 3], \ [3], [3]]);\n\ngap> IsTransitiveDi\ graph(gr2);\ntrue\ngap> gr2 = DigraphTransitiveClosure(D);\ntrue\ngap> gr3 := \ Digraph([[1, 2, 2, 3], [3, 3], [3]]);\n\ngap> IsTransitiveDigraph(gr3);\ntrue\n", count := [ 6, 1, 14, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 14, 7 ], mathmode := "Text", name := "Example", next := 251575, root := ~, start := 251118, stop := 251574 ), rec( content := "\n ", count := [ 6, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Description", next := 251592, root := ~, start := 250196, stop := 251591 ) ], count := [ 6, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 251606, root := ~, start := 250080, stop := 251605 ), rec( content := "\n\n ", count := [ 6, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsPreorderDigra\ ph" ), content := 0, count := [ 6, 1, 15, 2 ], mathmode := "Text", name := "Prop", next := 251673, root := ~, start := 251627, stop := 251672 ), rec( attributes := rec( Arg := "digraph", Name := "IsQuasiorderDig\ raph" ), content := 0, count := [ 6, 1, 15, 4 ], mathmode := "Text", name := "Prop", next := 251724, root := ~, start := 251676, stop := 251723 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "K", next := 251747, root := ~, start := 251736, stop := 251746 ), rec( content := " or ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "K", next := 251763, root := ~, start := 251751, stop := 251762 ), rec( content := ".", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Returns", next := 251774, root := ~, start := 251727, stop := 251773 ), rec( attributes := rec( ), content := [ rec( content := "\n A d\ igraph is a preorder digraph if and only if the digraph satisifies both\n " , count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRefle\ xiveDigraph" ), content := 0, count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Ref", next := 251906, root := ~, start := 251874, stop := 251905 ), rec( content := " and ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTrans\ itiveDigraph" ), content := 0, count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Ref", next := 251944, root := ~, start := 251911, stop := 251943 ), rec( content := ".\n A\ preorder digraph (or quasiorder digraph) ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "A", next := 252007, root := ~, start := 251993, stop := 252006 ), rec( content := " corresp\ onds to\n the preorder relation ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \leq", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "M", next := 252060, root := ~, start := 252049, stop := 252059 ), rec( content := " defined\ by ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ \\leq y", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "M", next := 252087, root := ~, start := 252072, stop := 252086 ), rec( content := " if and \ only\n if ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ x, y]", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "C", next := 252120, root := ~, start := 252107, stop := 252119 ), rec( content := " is an e\ dge of ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "A", next := 252149, root := ~, start := 252135, stop := 252148 ), rec( content := ".\n " , count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "P", next := 252159, root := ~, start := 252155, stop := 252158 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "A", next := 252201, root := ~, start := 252187, stop := 252200 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [2, 3], [2, 3]]);\n\ngap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[1 .. 4], [1 \ .. 4], [1 .. 4], [1 .. 4]]);\n\ng\ ap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[2], [3], [4], [5], [1]]);\ \n\ngap> IsPreorderDigraph(D);\nfa\ lse\ngap> D := Digraph([[1], [1, 2], [2, 3]]);\n\ngap> IsQuasiorderDigraph(D);\nfalse\n", count := [ 6, 1, 15, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 15, 7 ], mathmode := "Text", name := "Example", next := 252834, root := ~, start := 252298, stop := 252833 ), rec( content := "\n ", count := [ 6, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Description", next := 252851, root := ~, start := 251777, stop := 252850 ) ], count := [ 6, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 252865, root := ~, start := 251612, stop := 252864 ), rec( content := "\n\n ", count := [ 6, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsPartialOrderD\ igraph" ), content := 0, count := [ 6, 1, 16, 2 ], mathmode := "Text", name := "Prop", next := 252936, root := ~, start := 252886, stop := 252935 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "K", next := 252959, root := ~, start := 252948, stop := 252958 ), rec( content := " or ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "K", next := 252975, root := ~, start := 252963, stop := 252974 ), rec( content := ".", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Returns", next := 252986, root := ~, start := 252939, stop := 252985 ), rec( attributes := rec( ), content := [ rec( content := "\n A d\ igraph is a partial order digraph if and only if the digraph satisifies\n a\ ll of ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRefle\ xiveDigraph" ), content := 0, count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Ref", next := 253125, root := ~, start := 253093, stop := 253124 ), rec( content := ",\n " , count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAntis\ ymmetricDigraph" ), content := 0, count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Ref", next := 253167, root := ~, start := 253131, stop := 253166 ), rec( content := " and ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTrans\ itiveDigraph" ), content := 0, count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Ref", next := 253205, root := ~, start := 253172, stop := 253204 ), rec( content := ".\n A\ partial order ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "A", next := 253241, root := ~, start := 253227, stop := 253240 ), rec( content := " corresp\ onds\n to the partial order relation ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \leq", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "M", next := 253299, root := ~, start := 253288, stop := 253298 ), rec( content := " defined\ by ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ \\leq y", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "M", next := 253326, root := ~, start := 253311, stop := 253325 ), rec( content := " if and\\ n only if ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ x, y]", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "C", next := 253359, root := ~, start := 253346, stop := 253358 ), rec( content := " is an e\ dge of ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "A", next := 253388, root := ~, start := 253374, stop := 253387 ), rec( content := ".\n " , count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "P", next := 253398, root := ~, start := 253394, stop := 253397 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "A", next := 253440, root := ~, start := 253426, stop := 253439 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsPartialOrderDigraph(D);\ntrue\ngap> D := CycleDigraph(5);\n\ \ngap> IsPartialOrderDigraph(D);\nfal\ se\ngap> D := Digraph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4]]);\n\ngap> IsPartialOrderDigraph(D);\ntrue\n", count := [ 6, 1, 16, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 16, 5 ], mathmode := "Text", name := "Example", next := 253944, root := ~, start := 253537, stop := 253943 ), rec( content := "\n ", count := [ 6, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Description", next := 253961, root := ~, start := 252989, stop := 253960 ) ], count := [ 6, 1, 16, 1 ], mathmode := "Text", name := "ManSection", next := 253975, root := ~, start := 252871, stop := 253974 ), rec( content := "\n\n ", count := [ 6, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsMeetSemilatti\ ceDigraph" ), content := 0, count := [ 6, 1, 17, 2 ], mathmode := "Text", name := "Prop", next := 254049, root := ~, start := 253996, stop := 254048 ), rec( attributes := rec( Arg := "digraph", Name := "IsJoinSemilatti\ ceDigraph" ), content := 0, count := [ 6, 1, 17, 4 ], mathmode := "Text", name := "Prop", next := 254105, root := ~, start := 254052, stop := 254104 ), rec( attributes := rec( Arg := "digraph", Name := "IsLatticeDigrap\ h" ), content := 0, count := [ 6, 1, 17, 6 ], mathmode := "Text", name := "Prop", next := 254153, root := ~, start := 254108, stop := 254152 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 254176, root := ~, start := 254165, stop := 254175 ), rec( content := " or ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 254192, root := ~, start := 254180, stop := 254191 ), rec( content := ".", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Returns", next := 254203, root := ~, start := 254156, stop := 254202 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMeetSemilatticeDigraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "C", next := 254255, root := ~, start := 254224, stop := 254254 ), rec( content := " returns\ ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 254275, root := ~, start := 254264, stop := 254274 ), rec( content := " if the \ digraph\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "A", next := 254309, root := ~, start := 254295, stop := 254308 ), rec( content := " is a me\ et semilattice; ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sJoinSemilatticeDigraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "C", next := 254364, root := ~, start := 254333, stop := 254363 ), rec( content := "\n re\ turns ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 254388, root := ~, start := 254377, stop := 254387 ), rec( content := " if the \ digraph ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "A", next := 254418, root := ~, start := 254404, stop := 254417 ), rec( content := " is a jo\ in semilattice;\n and ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sLatticeDigraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "C", next := 254474, root := ~, start := 254451, stop := 254473 ), rec( content := " returns\ ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 254494, root := ~, start := 254483, stop := 254493 ), rec( content := " if the \ digraph\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "A", next := 254528, root := ~, start := 254514, stop := 254527 ), rec( content := " is both\ a meet and a join semilattice.\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "P", next := 254576, root := ~, start := 254572, stop := 254575 ), rec( content := "\n\n \ For a partial order digraph ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "Ref", next := 254645, root := ~, start := 254610, stop := 254644 ), rec( content := " the\n \ corresponding partial order is the relation ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \leq", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "M", next := 254709, root := ~, start := 254698, stop := 254708 ), rec( content := ", define\ d by\n ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ \\leq y", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "M", next := 254741, root := ~, start := 254726, stop := 254740 ), rec( content := " if and \ only if ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ x, y]", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "C", next := 254770, root := ~, start := 254757, stop := 254769 ), rec( content := " is an e\ dge.\n A digraph is a ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ eet semilattice", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "E", next := 254825, root := ~, start := 254802, stop := 254824 ), rec( content := " if it i\ s a partial order and every\n pair of vertices has a greatest lower bound (\ meet) with respect to the\n aforementioned relation. A ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ oin semilattice", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "E", next := 254990, root := ~, start := 254967, stop := 254989 ), rec( content := " is a pa\ rtial order where\n every pair of vertices has a least upper bound (join) w\ ith respect to\n the relation.\n ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "P", next := 255116, root := ~, start := 255112, stop := 255115 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "A", next := 255158, root := ~, start := 255144, stop := 255157 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsMeetSemilatticeDigraph(D);\nfalse\ngap> IsJoinSemilatticeDi\ graph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ngap> D := Digraph([[1], [2]\ , [1 .. 3]]);\n\ngap> IsJoinSemila\ tticeDigraph(D);\nfalse\ngap> IsMeetSemilatticeDigraph(D);\ntrue\ngap> IsLatti\ ceDigraph(D);\nfalse\ngap> D := Digraph([[1 .. 4], [2, 4], [3, 4], [4]]);\n\ngap> IsMeetSemilatticeDigraph(D);\n\ true\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\ntrue\ \ngap> D := Digraph([[1, 1, 1], [1, 1, 2, 2],\n> [1, 3, 3], \ [1, 2, 3, 3, 4]]);\n\ngap> I\ sMeetSemilatticeDigraph(D);\ntrue\ngap> IsJoinSemilatticeDigraph(D);\ntrue\nga\ p> IsLatticeDigraph(D);\ntrue\n", count := [ 6, 1, 17, 10 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 10 ], mathmode := "Text", name := "Example", next := 256146, root := ~, start := 255259, stop := 256145 ), rec( content := "\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Description", next := 256163, root := ~, start := 254206, stop := 256162 ) ], count := [ 6, 1, 17, 1 ], mathmode := "Text", name := "ManSection", next := 256177, root := ~, start := 253981, stop := 256176 ), rec( content := "\n\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 256191, root := ~, start := 236117, stop := 256190 ) , rec( content := "\n\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Regularity", count := [ 6, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 256233, root := ~, start := 256204, stop := 256232 ), rec( content := "\n ", count := [ 6, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsInRegularDigr\ aph" ), content := 0, count := [ 6, 2, 1, 2 ], mathmode := "Text", name := "Prop", next := 256300, root := ~, start := 256253, stop := 256299 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "K", next := 256323, root := ~, start := 256312, stop := 256322 ), rec( content := " or ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "K", next := 256339, root := ~, start := 256327, stop := 256338 ), rec( content := ".", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Returns", next := 256350, root := ~, start := 256303, stop := 256349 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "K", next := 256399, root := ~, start := 256388, stop := 256398 ), rec( content := " if ther\ e is an integer ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 256431, root := ~, start := 256423, stop := 256430 ), rec( content := " such th\ at\n for every vertex ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 256471, root := ~, start := 256463, stop := 256470 ), rec( content := " of digr\ aph ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "A", next := 256497, root := ~, start := 256483, stop := 256496 ), rec( content := " there a\ re exactly\n ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 256528, root := ~, start := 256520, stop := 256527 ), rec( content := " edges t\ erminating in ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 256558, root := ~, start := 256550, stop := 256557 ), rec( content := ".\n\n \ See also ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutRe\ gularDigraph" ), content := 0, count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Ref", next := 256607, root := ~, start := 256574, stop := 256606 ), rec( content := " and\n \ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRegul\ arDigraph" ), content := 0, count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Ref", next := 256646, root := ~, start := 256616, stop := 256645 ), rec( content := ".\n " , count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "P", next := 256656, root := ~, start := 256652, stop := 256655 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "A", next := 256698, root := ~, start := 256684, stop := 256697 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsInRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsInRegularDigraph(C\ hainDigraph(4));\nfalse\n", count := [ 6, 2, 1, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 5 ], mathmode := "Text", name := "Example", next := 256928, root := ~, start := 256798, stop := 256927 ), rec( content := "\n ", count := [ 6, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 256945, root := ~, start := 256353, stop := 256944 ) ], count := [ 6, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 256959, root := ~, start := 256238, stop := 256958 ), rec( content := "\n\n ", count := [ 6, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsOutRegularDig\ raph" ), content := 0, count := [ 6, 2, 2, 2 ], mathmode := "Text", name := "Prop", next := 257028, root := ~, start := 256980, stop := 257027 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "K", next := 257051, root := ~, start := 257040, stop := 257050 ), rec( content := " or ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "K", next := 257067, root := ~, start := 257055, stop := 257066 ), rec( content := ".", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 257078, root := ~, start := 257031, stop := 257077 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "K", next := 257127, root := ~, start := 257116, stop := 257126 ), rec( content := " if ther\ e is an integer ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 257159, root := ~, start := 257151, stop := 257158 ), rec( content := " such th\ at\n for every vertex ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 257199, root := ~, start := 257191, stop := 257198 ), rec( content := " of digr\ aph ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "A", next := 257225, root := ~, start := 257211, stop := 257224 ), rec( content := " there a\ re exactly\n ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 257256, root := ~, start := 257248, stop := 257255 ), rec( content := " edges s\ tarting at ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 257283, root := ~, start := 257275, stop := 257282 ), rec( content := ".\n " , count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "P", next := 257293, root := ~, start := 257289, stop := 257292 ), rec( content := "\n\n \ See also ", count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsInReg\ ularDigraph" ), content := 0, count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "Ref", next := 257340, root := ~, start := 257308, stop := 257339 ), rec( content := " and\n \ ", count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRegul\ arDigraph" ), content := 0, count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "Ref", next := 257379, root := ~, start := 257349, stop := 257378 ), rec( content := ".\n " , count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "P", next := 257389, root := ~, start := 257385, stop := 257388 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "A", next := 257431, root := ~, start := 257417, stop := 257430 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsOutRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsOutRegularDigraph\ (ChainDigraph(4));\nfalse\n", count := [ 6, 2, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 6 ], mathmode := "Text", name := "Example", next := 257663, root := ~, start := 257531, stop := 257662 ), rec( content := "\n ", count := [ 6, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 257680, root := ~, start := 257081, stop := 257679 ) ], count := [ 6, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 257694, root := ~, start := 256965, stop := 257693 ), rec( content := "\n\n ", count := [ 6, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsRegularDigrap\ h" ), content := 0, count := [ 6, 2, 3, 2 ], mathmode := "Text", name := "Prop", next := 257760, root := ~, start := 257715, stop := 257759 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 257783, root := ~, start := 257772, stop := 257782 ), rec( content := " or ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 257799, root := ~, start := 257787, stop := 257798 ), rec( content := ".", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 257810, root := ~, start := 257763, stop := 257809 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 257859, root := ~, start := 257848, stop := 257858 ), rec( content := " if ther\ e is an integer ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 257891, root := ~, start := 257883, stop := 257890 ), rec( content := " such th\ at\n for every vertex ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 257931, root := ~, start := 257923, stop := 257930 ), rec( content := " of digr\ aph ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "A", next := 257957, root := ~, start := 257943, stop := 257956 ), rec( content := " there a\ re exactly\n ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 257988, root := ~, start := 257980, stop := 257987 ), rec( content := " edges s\ tarting and terminating at ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 258031, root := ~, start := 258023, stop := 258030 ), rec( content := ". In oth\ er words,\n the property is ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 258080, root := ~, start := 258069, stop := 258079 ), rec( content := " if ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "A", next := 258098, root := ~, start := 258084, stop := 258097 ), rec( content := " is both\ in-regular and\n and out-regular.\n\n See also ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsInReg\ ularDigraph" ), content := 0, count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Ref", next := 258189, root := ~, start := 258157, stop := 258188 ), rec( content := " and\n \ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutRe\ gularDigraph" ), content := 0, count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Ref", next := 258231, root := ~, start := 258198, stop := 258230 ), rec( content := ".\n " , count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "P", next := 258241, root := ~, start := 258237, stop := 258240 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "A", next := 258283, root := ~, start := 258269, stop := 258282 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsRegularDigraph(Chain\ Digraph(4));\nfalse\n", count := [ 6, 2, 3, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 5 ], mathmode := "Text", name := "Example", next := 258509, root := ~, start := 258383, stop := 258508 ), rec( content := "\n ", count := [ 6, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 258526, root := ~, start := 257813, stop := 258525 ) ], count := [ 6, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 258540, root := ~, start := 257700, stop := 258539 ), rec( content := "\n\n ", count := [ 6, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDistanceRegul\ arDigraph" ), content := 0, count := [ 6, 2, 4, 2 ], mathmode := "Text", name := "Prop", next := 258614, root := ~, start := 258561, stop := 258613 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "K", next := 258637, root := ~, start := 258626, stop := 258636 ), rec( content := " or ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "K", next := 258653, root := ~, start := 258641, stop := 258652 ), rec( content := ".", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 258664, root := ~, start := 258617, stop := 258663 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "A", next := 258702, root := ~, start := 258688, stop := 258701 ), rec( content := " is a co\ nnected symmetric graph, this property returns\n ", count := [ 6, 2, 4, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "K", next := 258772, root := ~, start := 258761, stop := 258771 ), rec( content := " if for \ any two vertices ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 258805, root := ~, start := 258797, stop := 258804 ), rec( content := " and ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 258818, root := ~, start := 258810, stop := 258817 ), rec( content := " of ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "A", next := 258836, root := ~, start := 258822, stop := 258835 ), rec( content := "\n an\ d any two integers ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 258870, root := ~, start := 258862, stop := 258869 ), rec( content := " and ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 258883, root := ~, start := 258875, stop := 258882 ), rec( content := " between\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 258900, root := ~, start := 258892, stop := 258899 ), rec( content := " and the\ \n diameter of ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "A", next := 258939, root := ~, start := 258925, stop := 258938 ), rec( content := ", the nu\ mber of vertices at distance ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 258984, root := ~, start := 258976, stop := 258983 ), rec( content := "\n fr\ om ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 259002, root := ~, start := 258994, stop := 259001 ), rec( content := " and dis\ tance ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 259024, root := ~, start := 259016, stop := 259023 ), rec( content := " from ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 259038, root := ~, start := 259030, stop := 259037 ), rec( content := " depends\ only on\n ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 259067, root := ~, start := 259059, stop := 259066 ), rec( content := ", ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 259077, root := ~, start := 259069, stop := 259076 ), rec( content := ", and th\ e distance between vertices ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 259121, root := ~, start := 259113, stop := 259120 ), rec( content := " and\n \ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 259138, root := ~, start := 259130, stop := 259137 ), rec( content := ".", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "P", next := 259143, root := ~, start := 259139, stop := 259142 ), rec( content := "\n\n \ Alternatively, a distance regular graph is a graph for which there exist\n \ integers ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259245, root := ~, start := 259235, stop := 259244 ), rec( content := ", ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259257, root := ~, start := 259247, stop := 259256 ), rec( content := ", and ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259271, root := ~, start := 259263, stop := 259270 ), rec( content := " such th\ at for any two\n vertices ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259315, root := ~, start := 259307, stop := 259314 ), rec( content := ", ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259325, root := ~, start := 259317, stop := 259324 ), rec( content := " in ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "A", next := 259343, root := ~, start := 259329, stop := 259342 ), rec( content := " which a\ re distance ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259371, root := ~, start := 259363, stop := 259370 ), rec( content := "\n ap\ art, there are exactly ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259411, root := ~, start := 259401, stop := 259410 ), rec( content := " neighbo\ rs of ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259433, root := ~, start := 259425, stop := 259432 ), rec( content := " which a\ re at\n distance ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ - 1", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259472, root := ~, start := 259460, stop := 259471 ), rec( content := " away fr\ om ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259491, root := ~, start := 259483, stop := 259490 ), rec( content := ", and ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259507, root := ~, start := 259497, stop := 259506 ), rec( content := " neighbo\ rs of\n ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259533, root := ~, start := 259525, stop := 259532 ), rec( content := " which a\ re at distance ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ + 1", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259568, root := ~, start := 259556, stop := 259567 ), rec( content := " away fr\ om ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 259587, root := ~, start := 259579, stop := 259586 ), rec( content := ". This\n\ definition is used to check whether ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "A", next := 259648, root := ~, start := 259634, stop := 259647 ), rec( content := " is dist\ ance regular.", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "P", next := 259673, root := ~, start := 259669, stop := 259672 ), rec( content := "\n\n \ In the case where ", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "A", next := 259711, root := ~, start := 259697, stop := 259710 ), rec( content := " is not \ symmetric or not connected, the\n property is ", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "K", next := 259779, root := ~, start := 259767, stop := 259778 ), rec( content := ".\n " , count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "P", next := 259789, root := ~, start := 259785, stop := 259788 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "A", next := 259831, root := ~, start := 259817, stop := 259830 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> IsDistanceRegularD\ igraph(D);\nfalse\ngap> D := Digraph([[2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, \ 3]]);\n\ngap> IsDistanceRegularDi\ graph(D);\ntrue\n", count := [ 6, 2, 4, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 7 ], mathmode := "Text", name := "Example", next := 260207, root := ~, start := 259932, stop := 260206 ), rec( content := "\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 260224, root := ~, start := 258667, stop := 260223 ) ], count := [ 6, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 260238, root := ~, start := 258546, stop := 260237 ), rec( content := "\n\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 260252, root := ~, start := 256195, stop := 260251 ) , rec( content := "\n\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Connectivity and \ cycles", count := [ 6, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 260307, root := ~, start := 260265, stop := 260306 ), rec( content := "\n ", count := [ 6, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAcyclicDigrap\ h" ), content := 0, count := [ 6, 3, 1, 2 ], mathmode := "Text", name := "Prop", next := 260372, root := ~, start := 260327, stop := 260371 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 260395, root := ~, start := 260384, stop := 260394 ), rec( content := " or ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 260411, root := ~, start := 260399, stop := 260410 ), rec( content := ".", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 260422, root := ~, start := 260375, stop := 260421 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 260471, root := ~, start := 260460, stop := 260470 ), rec( content := " if the \ digraph ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "A", next := 260501, root := ~, start := 260487, stop := 260500 ), rec( content := " is acyc\ lic, and\n ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 260534, root := ~, start := 260522, stop := 260533 ), rec( content := " if it i\ s not. A digraph is ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ cyclic", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "E", next := 260576, root := ~, start := 260562, stop := 260575 ), rec( content := " if ever\ y directed\n cycle on the digraph is trivial. See Section ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Ref", next := 260694, root := ~, start := 260644, stop := 260693 ), rec( content := " for the\ definition of a directed cycle, and of a trivial\n directed cycle.", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "P", next := 260775, root := ~, start := 260771, stop := 260774 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "M", next := 260843, root := ~, start := 260830, stop := 260842 ), rec( content := " where\n\ ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "M", next := 260862, root := ~, start := 260854, stop := 260861 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "M", next := 260935, root := ~, start := 260927, stop := 260934 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "P", next := 260982, root := ~, start := 260978, stop := 260981 ), rec( content := "\n\n \ If the argument ", count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "A", next := 261018, root := ~, start := 261004, stop := 261017 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y)\ \n> return IsEmpty(Intersection(x, y));\n> end);;\ngap> D := Digraph(Peterse\ n);\n\ngap> IsAcyclicDigraph(D);\ \nfalse\ngap> D := DigraphFromDiSparse6String(\n> \".b_OGCIDBaPGkULEbQHCeRIdrH\ cuZMfRyDAbPhTi|zF\");\n\ngap> Is\ AcyclicDigraph(D);\ntrue\ngap> IsAcyclicDigraph(ChainDigraph(10));\ntrue\ngap>\ D := CompleteDigraph(IsMutableDigraph, 4);\n\ngap> IsAcyclicDigraph(D);\nfalse\ngap> IsAcyclicDigraph(CycleDigra\ ph(10));\nfalse", count := [ 6, 3, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 6 ], mathmode := "Text", name := "Example", next := 261767, root := ~, start := 261119, stop := 261766 ), rec( content := "\n ", count := [ 6, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 261784, root := ~, start := 260425, stop := 261783 ) ], count := [ 6, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 261798, root := ~, start := 260312, stop := 261797 ), rec( content := "\n\n ", count := [ 6, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsChainDigraph"\ ), content := 0, count := [ 6, 3, 2, 2 ], mathmode := "Text", name := "Prop", next := 261862, root := ~, start := 261819, stop := 261861 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 261885, root := ~, start := 261874, stop := 261884 ), rec( content := " or ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 261901, root := ~, start := 261889, stop := 261900 ), rec( content := ".", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 261912, root := ~, start := 261865, stop := 261911 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sChainDigraph", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "C", next := 261954, root := ~, start := 261933, stop := 261953 ), rec( content := " returns\ ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 261974, root := ~, start := 261963, stop := 261973 ), rec( content := " if the \ digraph ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "A", next := 262004, root := ~, start := 261990, stop := 262003 ), rec( content := " is\n \ isomorphic to the chain digraph with the same number of vertices as\n ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "A", next := 262098, root := ~, start := 262084, stop := 262097 ), rec( content := ", and ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 262116, root := ~, start := 262104, stop := 262115 ), rec( content := " if it i\ s not; see ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ChainDi\ graph" ), content := 0, count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Ref", next := 262167, root := ~, start := 262135, stop := 262166 ), rec( content := ".", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "P", next := 262172, root := ~, start := 262168, stop := 262171 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ hain", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "E", next := 262205, root := ~, start := 262193, stop := 262204 ), rec( content := " if and \ only if it is a directed tree, in which\n every vertex has out degree at mo\ st one; see ", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsDirec\ tedTree" ), content := 0, count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "Ref", next := 262330, root := ~, start := 262302, stop := 262329 ), rec( content := "\n an\ d ", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDegr\ ees" ), content := 0, count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "Ref", next := 262363, root := ~, start := 262339, stop := 262362 ), rec( content := ".\n " , count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "P", next := 262373, root := ~, start := 262369, stop := 262372 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "A", next := 262415, root := ~, start := 262401, stop := 262414 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsChainDigraph(D);\nfalse\ngap> D := ChainDigraph(5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> D := \ DigraphReverse(D);\n\ngap> IsChain\ Digraph(D);\ntrue\ngap> D := ChainDigraph(IsMutableDigraph, 5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> DigraphReve\ rse(D);\n\ngap> IsChainDigraph(D);\n\ true", count := [ 6, 3, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 6 ], mathmode := "Text", name := "Example", next := 263074, root := ~, start := 262512, stop := 263073 ), rec( content := "\n ", count := [ 6, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 263091, root := ~, start := 261915, stop := 263090 ) ], count := [ 6, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 263105, root := ~, start := 261804, stop := 263104 ), rec( content := "\n\n ", count := [ 6, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsConnectedDigr\ aph" ), content := 0, count := [ 6, 3, 3, 2 ], mathmode := "Text", name := "Prop", next := 263173, root := ~, start := 263126, stop := 263172 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 263196, root := ~, start := 263185, stop := 263195 ), rec( content := " or ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 263212, root := ~, start := 263200, stop := 263211 ), rec( content := ".", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 263223, root := ~, start := 263176, stop := 263222 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 263272, root := ~, start := 263261, stop := 263271 ), rec( content := " if the \ digraph ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "A", next := 263302, root := ~, start := 263288, stop := 263301 ), rec( content := "\n is\ weakly connected and ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 263343, root := ~, start := 263331, stop := 263342 ), rec( content := " if it i\ s not. A digraph\n ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "A", next := 263387, root := ~, start := 263373, stop := 263386 ), rec( content := " is ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "w\ eakly connected", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "E", next := 263414, root := ~, start := 263391, stop := 263413 ), rec( content := " if it i\ s possible to travel\n from any vertex to any other vertex by traversing ed\ ges in either\n direction (possibly against the orientation of some of them\ ). ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "P", next := 263583, root := ~, start := 263579, stop := 263582 ), rec( content := "\n\n \ The method used in this function has complexity ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m)", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 263648, root := ~, start := 263637, stop := 263647 ), rec( content := " if the\\ n digraph's ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Source" ), content := 0, count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "Ref", next := 263697, root := ~, start := 263670, stop := 263696 ), rec( content := " attribu\ te is set, otherwise it has\n complexity ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 263761, root := ~, start := 263748, stop := 263760 ), rec( content := " (where\\ n ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 263781, root := ~, start := 263773, stop := 263780 ), rec( content := " is the \ number of edges and\n ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 263821, root := ~, start := 263813, stop := 263820 ), rec( content := " is the \ number of vertices of the digraph).\n ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "P", next := 263873, root := ~, start := 263869, stop := 263872 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "A", next := 263915, root := ~, start := 263901, stop := 263914 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> \ D := Digraph([[1, 3], [4], [3], []]);;\ngap> IsConnectedDigraph(D);\nfalse\nga\ p> D := Digraph(IsMutableDigraph, [[2], [3], []]);;\ngap> IsConnectedDigraph(D\ );\ntrue\ngap> D := Digraph(IsMutableDigraph, [[1, 3], [4], [3], []]);;\ngap> \ IsConnectedDigraph(D);\nfalse", count := [ 6, 3, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 6 ], mathmode := "Text", name := "Example", next := 264377, root := ~, start := 264016, stop := 264376 ), rec( content := "\n ", count := [ 6, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 264394, root := ~, start := 263226, stop := 264393 ) ], count := [ 6, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 264408, root := ~, start := 263111, stop := 264407 ), rec( content := "\n\n ", count := [ 6, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsBiconnectedDi\ graph" ), content := 0, count := [ 6, 3, 4, 2 ], mathmode := "Text", name := "Prop", next := 264478, root := ~, start := 264429, stop := 264477 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "K", next := 264501, root := ~, start := 264490, stop := 264500 ), rec( content := " or ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "K", next := 264517, root := ~, start := 264505, stop := 264516 ), rec( content := ".", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 264528, root := ~, start := 264481, stop := 264527 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ iconnected", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "E", next := 264590, root := ~, start := 264572, stop := 264589 ), rec( content := " if it i\ s still connected (in the\n sense of ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 264669, root := ~, start := 264637, stop := 264668 ), rec( content := ") when a\ ny vertex is removed. \n If ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "A", next := 264717, root := ~, start := 264709, stop := 264716 ), rec( content := " has at \ least 3 vertices, then ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBiconnectedDigraph", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "C", next := 264775, root := ~, start := 264748, stop := 264774 ), rec( content := "\n \ implies ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 264823, root := ~, start := 264790, stop := 264822 ), rec( content := ";\n s\ ee ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 264865, root := ~, start := 264833, stop := 264864 ), rec( content := " or ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 264890, root := ~, start := 264869, stop := 264889 ), rec( content := " for a m\ ore\n detailed explanation.\n ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "P", next := 264936, root := ~, start := 264932, stop := 264935 ), rec( content := "\n\n \ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBiconnectedDigraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "C", next := 264969, root := ~, start := 264942, stop := 264968 ), rec( content := " returns\ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "K", next := 264989, root := ~, start := 264978, stop := 264988 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "A", next := 265023, root := ~, start := 265009, stop := 265022 ), rec( content := " is bico\ nnected, and ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "K", next := 265056, root := ~, start := 265044, stop := 265055 ), rec( content := " if it i\ s not. In\n particular, ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBiconnectedDigraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "C", next := 265117, root := ~, start := 265090, stop := 265116 ), rec( content := " returns\ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "K", next := 265138, root := ~, start := 265126, stop := 265137 ), rec( content := " if\n \ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "A", next := 265160, root := ~, start := 265146, stop := 265159 ), rec( content := " is not \ connected. ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "P", next := 265183, root := ~, start := 265179, stop := 265182 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "A", next := 265225, root := ~, start := 265211, stop := 265224 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 6 ], mathmode := "Text", name := "P", next := 265324, root := ~, start := 265320, stop := 265323 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 6, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "P", next := 265377, root := ~, start := 265373, stop := 265376 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "M", next := 265445, root := ~, start := 265432, stop := 265444 ), rec( content := " where\n\ ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "M", next := 265464, root := ~, start := 265456, stop := 265463 ), rec( content := " is the \ number of edges and ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "M", next := 265500, root := ~, start := 265492, stop := 265499 ), rec( content := " is the \ number of vertices in\n the digraph.\n ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "P", next := 265555, root := ~, start := 265551, stop := 265554 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Ref", next := 265591, root := ~, start := 265570, stop := 265590 ), rec( content := ", ", count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Ref", next := 265625, root := ~, start := 265593, stop := 265624 ), rec( content := ", and \n\ ", count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Ref", next := 265669, root := ~, start := 265636, stop := 265668 ), rec( content := ".\n " , count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 9 ], mathmode := "Text", name := "P", next := 265679, root := ~, start := 265675, stop := 265678 ), rec( content := "\n \ \n", count := [ 6, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsBiconnectedDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBic\ onnectedDigraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2],\ [3], [3, 3, 4, 4]]);;\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> D := Comple\ teBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBiconnectedDigraph(D);\ntrue", count := [ 6, 3, 4, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 10 ], mathmode := "Text", name := "Example", next := 266067, root := ~, start := 265687, stop := 266066 ), rec( content := "\n ", count := [ 6, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 266084, root := ~, start := 264531, stop := 266083 ) ], count := [ 6, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 266098, root := ~, start := 264414, stop := 266097 ), rec( content := "\n\n ", count := [ 6, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "IsBridgelessDig\ raph" ), content := 0, count := [ 6, 3, 5, 2 ], mathmode := "Text", name := "Prop", next := 266161, root := ~, start := 266119, stop := 266160 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "K", next := 266184, root := ~, start := 266173, stop := 266183 ), rec( content := " or ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "K", next := 266200, root := ~, start := 266188, stop := 266199 ), rec( content := ".", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Returns", next := 266211, root := ~, start := 266164, stop := 266210 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ ridgeless", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "E", next := 266272, root := ~, start := 266255, stop := 266271 ), rec( content := " if it i\ s still connected (in the\n sense of ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 266351, root := ~, start := 266319, stop := 266350 ), rec( content := ") when a\ ny edge is removed. \n If ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "A", next := 266395, root := ~, start := 266387, stop := 266394 ), rec( content := " has at \ least 3 vertices, then ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 266464, root := ~, start := 266426, stop := 266463 ), rec( content := " implies\ ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "I\ sBridgelessDigraph", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "C", next := 266499, root := ~, start := 266473, stop := 266498 ), rec( content := ";\n s\ ee ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 266541, root := ~, start := 266509, stop := 266540 ), rec( content := " or ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 266566, root := ~, start := 266545, stop := 266565 ), rec( content := " for a m\ ore\n detailed explanation.\n ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "P", next := 266612, root := ~, start := 266608, stop := 266611 ), rec( content := "\n\n \ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBridgelessDigraph", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "C", next := 266644, root := ~, start := 266618, stop := 266643 ), rec( content := " returns\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "K", next := 266664, root := ~, start := 266653, stop := 266663 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "A", next := 266692, root := ~, start := 266684, stop := 266691 ), rec( content := " is brid\ geless, and ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "K", next := 266724, root := ~, start := 266712, stop := 266723 ), rec( content := " if it i\ s not. In\n particular, ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBridgelessDigraph", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "C", next := 266784, root := ~, start := 266758, stop := 266783 ), rec( content := " returns\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "K", next := 266805, root := ~, start := 266793, stop := 266804 ), rec( content := " if\n \ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "A", next := 266821, root := ~, start := 266813, stop := 266820 ), rec( content := " is not \ connected. ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "P", next := 266844, root := ~, start := 266840, stop := 266843 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "A", next := 266880, root := ~, start := 266872, stop := 266879 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 6 ], mathmode := "Text", name := "P", next := 266979, root := ~, start := 266975, stop := 266978 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 6, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "P", next := 267032, root := ~, start := 267028, stop := 267031 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "M", next := 267100, root := ~, start := 267087, stop := 267099 ), rec( content := " where\n\ ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "M", next := 267119, root := ~, start := 267111, stop := 267118 ), rec( content := " is the \ number of edges and ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "M", next := 267155, root := ~, start := 267147, stop := 267154 ), rec( content := " is the \ number of vertices in\n the digraph.\n ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "P", next := 267210, root := ~, start := 267206, stop := 267209 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "Ref", next := 267246, root := ~, start := 267225, stop := 267245 ), rec( content := ", ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "Ref", next := 267280, root := ~, start := 267248, stop := 267279 ), rec( content := ", and \n\ ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "Ref", next := 267325, root := ~, start := 267291, stop := 267324 ), rec( content := ".\n \ \n", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsBridgelessDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBrid\ gelessDigraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2], [\ 3], [3, 3, 4, 4]]);;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> D := CompleteB\ ipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBridgelessDigraph(D);\ntrue\ngap> D := Digraph([[2, 5], [1, 3,\ 4, 5], [2, 4], [2, 3], [1, 2]]);\n\ngap> IsBridgelessDigraph(D);\ntrue\ngap> IsBiconnectedDigraph(D);\nfalse\n\ gap> D := Digraph([[2], [3], [4], [2]]);\n\ngap> IsBridgelessDigraph(D);\nfalse\ngap> IsBiconnectedDigraph(D);\n\ false\ngap> IsBridgelessDigraph(ChainDigraph(2));\nfalse\ngap> IsBiconnectedDi\ graph(ChainDigraph(2));\ntrue\n", count := [ 6, 3, 5, 9 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 9 ], mathmode := "Text", name := "Example", next := 268149, root := ~, start := 267334, stop := 268148 ), rec( content := "\n ", count := [ 6, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Description", next := 268166, root := ~, start := 266214, stop := 268165 ) ], count := [ 6, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 268180, root := ~, start := 266104, stop := 268179 ), rec( content := "\n\n ", count := [ 6, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsStronglyConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 6, 2 ], mathmode := "Text", name := "Prop", next := 268256, root := ~, start := 268201, stop := 268255 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 268279, root := ~, start := 268268, stop := 268278 ), rec( content := " or ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 268295, root := ~, start := 268283, stop := 268294 ), rec( content := ".", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 268306, root := ~, start := 268259, stop := 268305 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 268355, root := ~, start := 268344, stop := 268354 ), rec( content := " if the \ digraph ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "A", next := 268385, root := ~, start := 268371, stop := 268384 ), rec( content := " is stro\ ngly\n connected and ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 268428, root := ~, start := 268416, stop := 268427 ), rec( content := " if it i\ s not. ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "P", next := 268447, root := ~, start := 268443, stop := 268446 ), rec( content := "\n\n \ A digraph ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "A", next := 268477, root := ~, start := 268463, stop := 268476 ), rec( content := " is ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ trongly connected", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "E", next := 268506, root := ~, start := 268481, stop := 268505 ), rec( content := " if ther\ e is a directed\n path from every vertex to every other vertex. See Sectio\ n ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "Ref", next := 268643, root := ~, start := 268593, stop := 268642 ), rec( content := " for the\ definition of a directed\n path. ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "P", next := 268694, root := ~, start := 268690, stop := 268693 ), rec( content := "\n\n \ The method used in this operation is based on Gabow's Algorithm ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Gab00" ) , content := 0, count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "Cite", next := 268789, root := ~, start := 268764, stop := 268788 ), rec( content := " and has\ complexity ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "M", next := 268822, root := ~, start := 268809, stop := 268821 ), rec( content := ", where \ ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "M", next := 268838, root := ~, start := 268830, stop := 268837 ), rec( content := " is\n \ the number of edges (counting multiple edges as one) and ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "M", next := 268911, root := ~, start := 268903, stop := 268910 ), rec( content := " is the\\ n number of vertices in the digraph.\n ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "P", next := 268966, root := ~, start := 268962, stop := 268965 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "A", next := 269008, root := ~, start := 268994, stop := 269007 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(250000);\n\ngap> IsStronglyConnectedDigraph(D);\ntrue\ngap> D := DigraphRemoveEdges(D,\ [[250000, 1]]);\n\ngap>\ IsStronglyConnectedDigraph(D);\nfalse\ngap> D := CycleDigraph(IsMutableDigrap\ h, 250000);\n\ngap> IsStro\ nglyConnectedDigraph(D);\ntrue\ngap> DigraphRemoveEdge(D, [250000, 1]);\n\ngap> IsStronglyConnectedDigra\ ph(D);\nfalse\n", count := [ 6, 3, 6, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 7 ], mathmode := "Text", name := "Example", next := 269685, root := ~, start := 269109, stop := 269684 ), rec( content := "\n ", count := [ 6, 3, 6, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 269702, root := ~, start := 268309, stop := 269701 ) ], count := [ 6, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 269716, root := ~, start := 268186, stop := 269715 ), rec( content := "\n\n ", count := [ 6, 3, 6, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAperiodicDigr\ aph" ), content := 0, count := [ 6, 3, 7, 2 ], mathmode := "Text", name := "Prop", next := 269784, root := ~, start := 269737, stop := 269783 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 269807, root := ~, start := 269796, stop := 269806 ), rec( content := " or ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 269823, root := ~, start := 269811, stop := 269822 ), rec( content := ".", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 269834, root := ~, start := 269787, stop := 269833 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 269883, root := ~, start := 269872, stop := 269882 ), rec( content := " if the \ digraph ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "A", next := 269913, root := ~, start := 269899, stop := 269912 ), rec( content := "\n is\ aperiodic, i.e. if its ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Period" ), content := 0, count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Ref", next := 269973, root := ~, start := 269944, stop := 269972 ), rec( content := " is equa\ l to 1.\n Otherwise, the property is ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 270032, root := ~, start := 270020, stop := 270031 ), rec( content := ".\n " , count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "P", next := 270042, root := ~, start := 270038, stop := 270041 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "A", next := 270084, root := ~, start := 270070, stop := 270083 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> IsAperiodicDigraph(D);\nfalse\ngap> D := Dig\ raph([[2], [3, 5], [4], [5], [1, 2]]);\n\ngap> IsAperiodicDigraph(D);\ntrue\ngap> D := Digraph(IsMutableDigraph,\ [[2], [3, 5], [4], [5], [1, 2]]);\n\ \ngap> IsAperiodicDigraph(D);\ntrue", count := [ 6, 3, 7, 5 ], name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 7, 5 ], mathmode := "Text", name := "Example", next := 270629, root := ~, start := 270184, stop := 270628 ), rec( content := "\n ", count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 270646, root := ~, start := 269837, stop := 270645 ) ], count := [ 6, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 270660, root := ~, start := 269722, stop := 270659 ), rec( content := "\n\n ", count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDirectedTree"\ ), content := 0, count := [ 6, 3, 8, 2 ], mathmode := "Text", name := "Prop", next := 270724, root := ~, start := 270681, stop := 270723 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 270747, root := ~, start := 270736, stop := 270746 ), rec( content := " or ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 270763, root := ~, start := 270751, stop := 270762 ), rec( content := ".", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 270774, root := ~, start := 270727, stop := 270773 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 270814, root := ~, start := 270803, stop := 270813 ), rec( content := " if the \ digraph ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "A", next := 270844, root := ~, start := 270830, stop := 270843 ), rec( content := " is a di\ rected tree, and\n ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 270885, root := ~, start := 270873, stop := 270884 ), rec( content := " if it i\ s not. ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "P", next := 270904, root := ~, start := 270900, stop := 270903 ), rec( content := "\n\n \ A ", count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ irected tree", count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "E", next := 270932, root := ~, start := 270912, stop := 270931 ), rec( content := " is an a\ cyclic digraph with precisely 1 source,\n such that no two vertices share a\ n out-neighbour.\n Note the empty digraph is not considered a directed\n \ tree as it has no source. ", count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "P", next := 271124, root := ~, start := 271120, stop := 271123 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Sources" ), content := 0, count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "Ref", next := 271167, root := ~, start := 271139, stop := 271166 ), rec( content := ".\n " , count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "P", next := 271177, root := ~, start := 271173, stop := 271176 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "A", next := 271219, root := ~, start := 271205, stop := 271218 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], [2]]);\n\ng\ ap> IsDirectedTree(D);\nfalse\ngap> D := Digraph([[3], [3], []]);\n\ngap> IsDirectedTree(D);\nfalse\ngap> D := D\ igraph([[2], [3], []]);\n\ngap> Is\ DirectedTree(D);\ntrue\ngap> D := Digraph([[2, 3], [6], [4, 5], [], [], []]);\ \n\ngap> IsDirectedTree(D);\ntrue\ \n", count := [ 6, 3, 8, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "Example", next := 271803, root := ~, start := 271320, stop := 271802 ), rec( content := "\n ", count := [ 6, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 271820, root := ~, start := 270777, stop := 271819 ) ], count := [ 6, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 271834, root := ~, start := 270666, stop := 271833 ), rec( content := "\n\n ", count := [ 6, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsUndirectedTre\ e" ), content := 0, count := [ 6, 3, 9, 2 ], mathmode := "Text", name := "Prop", next := 271900, root := ~, start := 271855, stop := 271899 ), rec( attributes := rec( Arg := "digraph", Name := "IsUndirectedFor\ est" ), content := 0, count := [ 6, 3, 9, 4 ], mathmode := "Text", name := "Prop", next := 271950, root := ~, start := 271903, stop := 271949 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 271973, root := ~, start := 271962, stop := 271972 ), rec( content := " or ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 271989, root := ~, start := 271977, stop := 271988 ), rec( content := ".", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 272000, root := ~, start := 271953, stop := 271999 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ property ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedTree", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "C", next := 272057, root := ~, start := 272034, stop := 272056 ), rec( content := " returns\ ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 272077, root := ~, start := 272066, stop := 272076 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "A", next := 272111, root := ~, start := 272097, stop := 272110 ), rec( content := " is an u\ ndirected tree, and the property\n ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedForest", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "C", next := 272181, root := ~, start := 272156, stop := 272180 ), rec( content := " returns\ ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 272201, root := ~, start := 272190, stop := 272200 ), rec( content := " if ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "A", next := 272219, root := ~, start := 272205, stop := 272218 ), rec( content := " is an\n\ undirected forest; otherwise, these properties return ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 272296, root := ~, start := 272284, stop := 272295 ), rec( content := ". ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "P", next := 272302, root := ~, start := 272298, stop := 272301 ), rec( content := "\n\n \ An ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected tree", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "E", next := 272333, root := ~, start := 272311, stop := 272332 ), rec( content := " is a sy\ mmetric digraph without loops, in which for\n any pair of distinct vertices\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 272427, root := ~, start := 272419, stop := 272426 ), rec( content := " and ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 272440, root := ~, start := 272432, stop := 272439 ), rec( content := ", there \ is exactly one\n directed path from ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 272494, root := ~, start := 272486, stop := 272493 ), rec( content := " to ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 272506, root := ~, start := 272498, stop := 272505 ), rec( content := ". See ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 272550, root := ~, start := 272512, stop := 272549 ), rec( content := " and ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 272584, root := ~, start := 272555, stop := 272583 ), rec( content := ", and se\ e\n Section ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 272650, root := ~, start := 272606, stop := 272649 ), rec( content := " for the\ definition of\n directed path. This definition implies that an undirected \ tree has\n no multiple edges. ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "P", next := 272771, root := ~, start := 272767, stop := 272770 ), rec( content := "\n\n \ An ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected forest", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "E", next := 272804, root := ~, start := 272780, stop := 272803 ), rec( content := " is a di\ graph, each of whose connected components\n is an undirected tree. In other\ words, an undirected forest is isomorphic to\n a disjoint union of undirec\ ted trees. See ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "Ref", next := 273028, root := ~, start := 272981, stop := 273027 ), rec( content := " and ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DisjointUnion", Label := "for a list of digraphs" ), content := 0, count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "Ref", next := 273105, root := ~, start := 273033, stop := 273104 ), rec( content := ". In par\ ticular, every\n undirected tree is an undirected forest. ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 8 ], mathmode := "Text", name := "P", next := 273177, root := ~, start := 273173, stop := 273176 ), rec( content := "\n\n \ Please note that the digraph with zero vertices is considered to be neither\n \ an undirected tree nor an undirected forest.\n ", count := [ 6, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "P", next := 273316, root := ~, start := 273312, stop := 273315 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "A", next := 273358, root := ~, start := 273344, stop := 273357 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [3], [1, 2]]);\n\ngap> IsUndirectedTree(D);\ntrue\ngap> IsSymmetricDigraph(D) and not D\ igraphHasLoops(D);\ntrue\ngap> D := Digraph([[3], [5], [1, 4], [3], [2]]);\n\ngap> IsConnectedDigraph(D);\nfalse\ \ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedForest(D);\ntrue\ngap> D \ := Digraph([[1, 2], [1], [2]]);\n\ \ngap> IsUndirectedTree(D) or IsUndirectedForest(D);\nfalse\ngap> IsSymmetricD\ igraph(D) or not DigraphHasLoops(D);\nfalse", count := [ 6, 3, 9, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 9, 10 ], mathmode := "Text", name := "Example", next := 274059, root := ~, start := 273459, stop := 274058 ), rec( content := "\n ", count := [ 6, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 274076, root := ~, start := 272003, stop := 274075 ) ], count := [ 6, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 274090, root := ~, start := 271840, stop := 274089 ), rec( content := "\n\n ", count := [ 6, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEulerianDigra\ ph" ), content := 0, count := [ 6, 3, 10, 2 ], mathmode := "Text", name := "Prop", next := 274157, root := ~, start := 274111, stop := 274156 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "K", next := 274180, root := ~, start := 274169, stop := 274179 ), rec( content := " or ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "K", next := 274196, root := ~, start := 274184, stop := 274195 ), rec( content := ".", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Returns", next := 274207, root := ~, start := 274160, stop := 274206 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns true if the digraph ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "A", next := 274284, root := ~, start := 274270, stop := 274283 ), rec( content := " is Eule\ rian.\n ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "P", next := 274306, root := ~, start := 274302, stop := 274305 ), rec( content := "\n\n \ A connected digraph is called ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ ulerian", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "E", next := 274357, root := ~, start := 274342, stop := 274356 ), rec( content := " if ther\ e exists a directed\n circuit on the digraph which includes every edge exac\ tly once. See\n Section ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "Ref", next := 274513, root := ~, start := 274469, stop := 274512 ), rec( content := " for the\ definition of\n a directed circuit. Note that the empty digraph with at mo\ st one vertex is\n considered to be Eulerian.\n ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "P", next := 274654, root := ~, start := 274650, stop := 274653 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "A", next := 274696, root := ~, start := 274682, stop := 274695 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[]]);\n\ngap> IsEul\ erianDigraph(D);\ntrue\ngap> D := Digraph([[2], []]);\n\ngap> IsEulerianDigraph(D);\nfalse\ngap> D := Digraph([[3\ ], [], [2]]);\n\ngap> IsEulerianDi\ graph(D);\nfalse\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> IsEulerianDigraph(D);\ntrue\n", count := [ 6, 3, 10, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 10, 6 ], mathmode := "Text", name := "Example", next := 275259, root := ~, start := 274797, stop := 275258 ), rec( content := "\n ", count := [ 6, 3, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 275276, root := ~, start := 274210, stop := 275275 ) ], count := [ 6, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 275290, root := ~, start := 274096, stop := 275289 ), rec( content := "\n\n ", count := [ 6, 3, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsHamiltonianDi\ graph" ), content := 0, count := [ 6, 3, 11, 2 ], mathmode := "Text", name := "Prop", next := 275360, root := ~, start := 275311, stop := 275359 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 275383, root := ~, start := 275372, stop := 275382 ), rec( content := " or ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 275399, root := ~, start := 275387, stop := 275398 ), rec( content := ".", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Returns", next := 275410, root := ~, start := 275363, stop := 275409 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "A", next := 275448, root := ~, start := 275434, stop := 275447 ), rec( content := " is Hami\ ltonian, then this property returns\n ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 275507, root := ~, start := 275496, stop := 275506 ), rec( content := ", and ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 275525, root := ~, start := 275513, stop := 275524 ), rec( content := " if it i\ s not. ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "P", next := 275544, root := ~, start := 275540, stop := 275543 ), rec( content := "\n\n \ A digraph with ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "C", next := 275573, root := ~, start := 275565, stop := 275572 ), rec( content := " vertice\ s is ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonian", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "E", next := 275604, root := ~, start := 275586, stop := 275603 ), rec( content := " if it h\ as a\n directed cycle of length ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "C", next := 275654, root := ~, start := 275646, stop := 275653 ), rec( content := ". See Se\ ction ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "Ref", next := 275717, root := ~, start := 275668, stop := 275716 ), rec( content := " for the\ definition of a directed cycle.\n Note the empty digraphs on 0 and 1 verti\ ces are considered to be\n Hamiltonian.", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "P", next := 275847, root := ~, start := 275843, stop := 275846 ), rec( content := "\n\n \ The method used in this operation has the worst case complexity as\n ", count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "Ref", next := 275957, root := ~, start := 275924, stop := 275956 ), rec( content := ".\n " , count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "P", next := 275967, root := ~, start := 275963, stop := 275966 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "A", next := 276009, root := ~, start := 275995, stop := 276008 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> g := Digraph([[]]);\n\ngap> IsHam\ iltonianDigraph(g);\ntrue\ngap> g := Digraph([[2], [1]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\ngap> g := Digr\ aph([[3], [], [2]]);\n\ngap> IsHam\ iltonianDigraph(g);\nfalse\ngap> g := Digraph([[2], [3], [1]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\n", count := [ 6, 3, 11, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 11, 7 ], mathmode := "Text", name := "Example", next := 276585, root := ~, start := 276110, stop := 276584 ), rec( content := "\n ", count := [ 6, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Description", next := 276602, root := ~, start := 275413, stop := 276601 ) ], count := [ 6, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 276616, root := ~, start := 275296, stop := 276615 ), rec( content := "\n\n ", count := [ 6, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCycleDigraph"\ ), content := 0, count := [ 6, 3, 12, 2 ], mathmode := "Text", name := "Prop", next := 276680, root := ~, start := 276637, stop := 276679 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 276703, root := ~, start := 276692, stop := 276702 ), rec( content := " or ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 276719, root := ~, start := 276707, stop := 276718 ), rec( content := ".", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 276730, root := ~, start := 276683, stop := 276729 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCycleDigraph", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "C", next := 276772, root := ~, start := 276751, stop := 276771 ), rec( content := " returns\ ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 276792, root := ~, start := 276781, stop := 276791 ), rec( content := " if the \ digraph ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "A", next := 276822, root := ~, start := 276808, stop := 276821 ), rec( content := " is\n \ isomorphic to the cycle digraph with the same number of vertices as\n ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "A", next := 276916, root := ~, start := 276902, stop := 276915 ), rec( content := ", and ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 276934, root := ~, start := 276922, stop := 276933 ), rec( content := " if it i\ s not; see ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CycleDi\ graph" ), content := 0, count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Ref", next := 276985, root := ~, start := 276953, stop := 276984 ), rec( content := ".", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "P", next := 276990, root := ~, start := 276986, stop := 276989 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ycle", count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "E", next := 277023, root := ~, start := 277011, stop := 277022 ), rec( content := " if and \ only if it is strongly connected and has\n the same number of edges as vert\ ices. \n ", count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "P", next := 277123, root := ~, start := 277119, stop := 277122 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "A", next := 277165, root := ~, start := 277151, stop := 277164 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsCycleDigraph(D);\nfalse\ngap> D := CycleDigraph(5);\n\ngap> IsCycleDigraph(D);\ntrue\ngap> D := \ OnDigraphs(D, (1, 2, 3));\n\ngap> \ D = CycleDigraph(5);\nfalse\ngap> IsCycleDigraph(D);\ntrue", count := [ 6, 3, 12, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 12, 6 ], mathmode := "Text", name := "Example", next := 277650, root := ~, start := 277262, stop := 277649 ), rec( content := "\n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 277667, root := ~, start := 276733, stop := 277666 ) ], count := [ 6, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 277681, root := ~, start := 276622, stop := 277680 ), rec( content := "\n\n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 277695, root := ~, start := 260256, stop := 277694 ) , rec( content := "\n \n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Planarity", count := [ 6, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 277738, root := ~, start := 277710, stop := 277737 ), rec( content := "\n ", count := [ 6, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsPlanarDigraph\ " ), content := 0, count := [ 6, 4, 1, 2 ], mathmode := "Text", name := "Prop", next := 277802, root := ~, start := 277758, stop := 277801 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "K", next := 277825, root := ~, start := 277814, stop := 277824 ), rec( content := " or ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "K", next := 277841, root := ~, start := 277829, stop := 277840 ), rec( content := ".", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 277852, root := ~, start := 277805, stop := 277851 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ lanar", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "E", next := 277888, root := ~, start := 277875, stop := 277887 ), rec( content := " digraph\ is a digraph that can be embedded in the plane in\n such a way that its ed\ ges do not intersect. A digraph is planar if and only\n if it does not have\ a subdigraph that is homeomorphic to either the\n complete graph on ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "5\ ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "C", next := 278129, root := ~, start := 278121, stop := 278128 ), rec( content := " vertice\ s or the complete bipartite graph with\n vertex sets of sizes ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "C", next := 278209, root := ~, start := 278201, stop := 278208 ), rec( content := " and ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "C", next := 278222, root := ~, start := 278214, stop := 278221 ), rec( content := ".\n " , count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "P", next := 278232, root := ~, start := 278228, stop := 278231 ), rec( content := "\n \n\ ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPlanarDigraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "C", next := 278264, root := ~, start := 278242, stop := 278263 ), rec( content := " returns\ ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "K", next := 278284, root := ~, start := 278273, stop := 278283 ), rec( content := " if the \ digraph ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "A", next := 278314, root := ~, start := 278300, stop := 278313 ), rec( content := " is\n \ planar and ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "K", next := 278345, root := ~, start := 278333, stop := 278344 ), rec( content := " if it i\ s not. The directions and multiplicities of\n any edges in ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "A", next := 278428, root := ~, start := 278414, stop := 278427 ), rec( content := " are ign\ ored by ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPlanarDigraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "C", next := 278466, root := ~, start := 278444, stop := 278465 ), rec( content := ".\n " , count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "P", next := 278476, root := ~, start := 278472, stop := 278475 ), rec( content := " \n\n \ See also ", count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "Ref", next := 278526, root := ~, start := 278492, stop := 278525 ), rec( content := ".\n " , count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "P", next := 278536, root := ~, start := 278532, stop := 278535 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 6, 4, 1, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 6, 4, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "Cite", next := 278694, root := ~, start := 278676, stop := 278693 ), rec( content := ".\n\n \ ", count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsPlanarDigraph(CompleteDigraph(4));\ntrue\ngap> IsPlanarDigraph(Complet\ eDigraph(5));\nfalse\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(2, 3));\nt\ rue\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(3, 3));\nfalse\ngap> IsPlan\ arDigraph(CompleteDigraph(IsMutableDigraph, 4));\ntrue\ngap> IsPlanarDigraph(C\ ompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsPlanarDigraph(CompleteBip\ artiteDigraph(IsMutableDigraph, 2, 3));\ntrue\ngap> IsPlanarDigraph(CompleteBi\ partiteDigraph(IsMutableDigraph, 3, 3));\nfalse\n", count := [ 6, 4, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 4, 1, 7 ], mathmode := "Text", name := "Example", next := 279233, root := ~, start := 278701, stop := 279232 ), rec( content := "\n ", count := [ 6, 4, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 279250, root := ~, start := 277855, stop := 279249 ) ], count := [ 6, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 279264, root := ~, start := 277743, stop := 279263 ), rec( content := "\n\n ", count := [ 6, 4, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsOuterPlanarDi\ graph" ), content := 0, count := [ 6, 4, 2, 2 ], mathmode := "Text", name := "Prop", next := 279334, root := ~, start := 279285, stop := 279333 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "K", next := 279357, root := ~, start := 279346, stop := 279356 ), rec( content := " or ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "K", next := 279373, root := ~, start := 279361, stop := 279372 ), rec( content := ".", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 279384, root := ~, start := 279337, stop := 279383 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ uter planar", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "E", next := 279427, root := ~, start := 279408, stop := 279426 ), rec( content := " digraph\ is a digraph that can be embedded in the\n plane in such a way that its ed\ ges do not intersect, and all vertices\n belong to the unbounded face of th\ e embedding. A digraph is outer planar\n if and only if it does not have a\ subdigraph that is homeomorphic to either\n the complete graph on ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "4\ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "C", next := 279743, root := ~, start := 279735, stop := 279742 ), rec( content := " vertice\ s or the complete bipartite graph\n with vertex sets of sizes ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "C", next := 279823, root := ~, start := 279815, stop := 279822 ), rec( content := " and ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "C", next := 279836, root := ~, start := 279828, stop := 279835 ), rec( content := ".\n " , count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "P", next := 279846, root := ~, start := 279842, stop := 279845 ), rec( content := "\n \n\ ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sOuterPlanarDigraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "C", next := 279883, root := ~, start := 279856, stop := 279882 ), rec( content := " returns\ ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "K", next := 279903, root := ~, start := 279892, stop := 279902 ), rec( content := " if the \ digraph\n ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "A", next := 279937, root := ~, start := 279923, stop := 279936 ), rec( content := " is oute\ r planar and ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "K", next := 279970, root := ~, start := 279958, stop := 279969 ), rec( content := " if it i\ s not. The\n directions and multiplicities of any edges in ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "A", next := 280053, root := ~, start := 280039, stop := 280052 ), rec( content := " are ign\ ored by\n ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPlanarDigraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "C", next := 280095, root := ~, start := 280073, stop := 280094 ), rec( content := ". ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "P", next := 280102, root := ~, start := 280098, stop := 280101 ), rec( content := "\n\n \ See also ", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "Ref", next := 280146, root := ~, start := 280117, stop := 280145 ), rec( content := ".\n\n \ This method uses the reference implementation in\n ", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 5 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 6, 4, 2, 5 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 6, 4, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 5 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "Cite", next := 280305, root := ~, start := 280287, stop := 280304 ), rec( content := ".\n\n", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsOuterPlanarDigraph(CompleteDigraph(4));\nfalse\ngap> IsOuterPlanarDigr\ aph(CompleteDigraph(5));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteDi\ graph(2, 3));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(3, 3)\ );\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(10));\ntrue\ngap> IsOuterPla\ narDigraph(CompleteDigraph(IsMutableDigraph, 4));\nfalse\ngap> IsOuterPlanarDi\ graph(CompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsOuterPlanarDigraph\ (CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 2, 3));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteD\ igraph(IsMutableDigraph,\n> 3\ , 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(IsMutableDigraph, 10));\ \ntrue\n\n", count := [ 6, 4, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 6 ], mathmode := "Text", name := "Example", next := 281105, root := ~, start := 280308, stop := 281104 ), rec( content := "\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 281122, root := ~, start := 279387, stop := 281121 ) ], count := [ 6, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 281136, root := ~, start := 279270, stop := 281135 ), rec( content := "\n\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 281150, root := ~, start := 277701, stop := 281149 ) , rec( content := "\n\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Homomorphisms and\ transformations", count := [ 6, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 281215, root := ~, start := 281163, stop := 281214 ), rec( content := "\n ", count := [ 6, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDigraphCore" ), content := 0, count := [ 6, 5, 1, 2 ], mathmode := "Text", name := "Prop", next := 281277, root := ~, start := 281235, stop := 281276 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 281300, root := ~, start := 281289, stop := 281299 ), rec( content := " or ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 281316, root := ~, start := 281304, stop := 281315 ), rec( content := ".", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 281327, root := ~, start := 281280, stop := 281326 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 281381, root := ~, start := 281370, stop := 281380 ), rec( content := " if ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "A", next := 281399, root := ~, start := 281385, stop := 281398 ), rec( content := " is a co\ re,\n and ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 281431, root := ~, start := 281419, stop := 281430 ), rec( content := " if it i\ s not.", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "P", next := 281449, root := ~, start := 281445, stop := 281448 ), rec( content := "\n\n \ A digraph ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 281473, root := ~, start := 281465, stop := 281472 ), rec( content := " is a ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ore", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "E", next := 281490, root := ~, start := 281479, stop := 281489 ), rec( content := " if and \ only if it has no proper\n subdigraphs ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 281547, root := ~, start := 281539, stop := 281546 ), rec( content := " such th\ at there exists a homomorphism from ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 281599, root := ~, start := 281591, stop := 281598 ), rec( content := "\n to\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "A\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 281615, root := ~, start := 281607, stop := 281614 ), rec( content := ". In oth\ er words, a digraph ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 281651, root := ~, start := 281643, stop := 281650 ), rec( content := " is a co\ re if and only if\n every endomorphism on ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 281711, root := ~, start := 281703, stop := 281710 ), rec( content := " is an a\ utomorphism on ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 281742, root := ~, start := 281734, stop := 281741 ), rec( content := ".\n " , count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "P", next := 281752, root := ~, start := 281748, stop := 281751 ), rec( content := "\n \ \n If the argument ", count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "A", next := 281794, root := ~, start := 281780, stop := 281793 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(6);\n\n\ gap> IsDigraphCore(D);\ntrue\ngap> D := DigraphSymmetricClosure(CycleDigraph(6\ ));\n\ngap> DigraphHomo\ morphism(D, CompleteDigraph(2));\nTransformation( [ 1, 2, 1, 2, 1, 2 ] )\ngap>\ IsDigraphCore(D);\nfalse\n", count := [ 6, 5, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 6 ], mathmode := "Text", name := "Example", next := 282255, root := ~, start := 281895, stop := 282254 ), rec( content := "\n ", count := [ 6, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 282272, root := ~, start := 281330, stop := 282271 ) ], count := [ 6, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 282286, root := ~, start := 281220, stop := 282285 ), rec( content := "\n\n ", count := [ 6, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEdgeTransitiv\ e" ), content := 0, count := [ 6, 5, 2, 2 ], mathmode := "Text", name := "Prop", next := 282352, root := ~, start := 282307, stop := 282351 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 282375, root := ~, start := 282364, stop := 282374 ), rec( content := " or ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 282391, root := ~, start := 282379, stop := 282390 ), rec( content := ".", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 282402, root := ~, start := 282355, stop := 282401 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "A", next := 282440, root := ~, start := 282426, stop := 282439 ), rec( content := " is a di\ graph without multiple edges, then\n ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sEdgeTransitive", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "C", next := 282510, root := ~, start := 282487, stop := 282509 ), rec( content := " returns\ ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 282530, root := ~, start := 282519, stop := 282529 ), rec( content := " if ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "A", next := 282548, root := ~, start := 282534, stop := 282547 ), rec( content := "\n is\ edge transitive, and ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 282589, root := ~, start := 282577, stop := 282588 ), rec( content := " otherwi\ se. A digraph is\n ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge transitive", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "E", next := 282640, root := ~, start := 282618, stop := 282639 ), rec( content := " if its \ automorphism group acts\n transitively on its edges (via the action\n ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "OnPairs" ), content := 0, count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 282758, root := ~, start := 282722, stop := 282757 ), rec( content := ").\n \ ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "P", next := 282769, root := ~, start := 282765, stop := 282768 ), rec( content := "\n \ \n If the argument ", count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "A", next := 282811, root := ~, start := 282797, stop := 282810 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsEdgeTransitive(CompleteDigraph(2));\ntrue\ngap> IsEdgeTransitive(Chain\ Digraph(3));\nfalse\ngap> IsEdgeTransitive(Digraph([[2], [3, 3, 3], []]));\nEr\ ror, the argument must be a digraph with no multiple edges,\n", count := [ 6, 5, 2, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 5, 2, 5 ], mathmode := "Text", name := "Example", next := 283158, root := ~, start := 282912, stop := 283157 ), rec( content := "\n ", count := [ 6, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 283175, root := ~, start := 282405, stop := 283174 ) ], count := [ 6, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 283189, root := ~, start := 282292, stop := 283188 ), rec( content := "\n\n ", count := [ 6, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsVertexTransit\ ive" ), content := 0, count := [ 6, 5, 3, 2 ], mathmode := "Text", name := "Prop", next := 283257, root := ~, start := 283210, stop := 283256 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 283280, root := ~, start := 283269, stop := 283279 ), rec( content := " or ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 283296, root := ~, start := 283284, stop := 283295 ), rec( content := ".", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 283307, root := ~, start := 283260, stop := 283306 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "A", next := 283345, root := ~, start := 283331, stop := 283344 ), rec( content := " is a di\ graph, then ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sVertexTransitive", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "C", next := 283390, root := ~, start := 283365, stop := 283389 ), rec( content := " returns\ \n ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 283414, root := ~, start := 283403, stop := 283413 ), rec( content := " if ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "A", next := 283432, root := ~, start := 283418, stop := 283431 ), rec( content := " is vert\ ex transitive, and ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 283471, root := ~, start := 283459, stop := 283470 ), rec( content := "\n ot\ herwise. A digraph is ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex transitive", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "E", next := 283524, root := ~, start := 283500, stop := 283523 ), rec( content := " if its \ automorphism group\n acts transitively on its vertices.\n ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "P", next := 283598, root := ~, start := 283594, stop := 283597 ), rec( content := "\n \ \n If the argument ", count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "A", next := 283640, root := ~, start := 283626, stop := 283639 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsVertexTransitive(CompleteDigraph(2));\ntrue\ngap> IsVertexTransitive(C\ hainDigraph(3));\nfalse\n", count := [ 6, 5, 3, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 5 ], mathmode := "Text", name := "Example", next := 283871, root := ~, start := 283741, stop := 283870 ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 283888, root := ~, start := 283310, stop := 283887 ) ], count := [ 6, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 283902, root := ~, start := 283195, stop := 283901 ), rec( content := "\n\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 283916, root := ~, start := 281154, stop := 283915 ) , rec( content := "\n\n \n", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 283931, root := ~, start := 236032, stop := 283930 ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " properties ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Homomorphisms" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Homomorphisms", count := [ 7, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 284018, root := ~, start := 283986, stop := 284017 ) , rec( content := "\n ", count := [ 7, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "Let $E_a=(E_a^0,E_a^1,\\ra_a,\\s\ o_a)$ and $E_b=(E_b^0,E_b^1,\\ra_b,\\so_b)$ be two\ngraphs, and let $\\phi_0 :\ E_a^0 \\to E_b^0$ and $\\phi_1 : E_a^1 \\to E_b^1$ be\nfunctions. Then the pa\ ir $\\phi = (\\phi_0, \\phi_1)$ is a \\emph{graph homomorphism\nfrom} $E_a$ \\\ emph{to} $E_b$ if $\\phi_0(\\so_a(e)) = \\so_b(\\phi_1(e))$ and\n$\\phi_0(\\ra\ _a(e)) = \\ra_b(\\phi_1(e))$ for every $e \\in E_a^1$. If $\\phi_0$ and\n$\\ph\ i_1$ are in addition bijective, then $\\phi$ is a \\emph{graph isomorphism\nfr\ om} $E_a$ \\emph{to} $E_b$. In this case we say that $E_a$ and $E_b$ are\n\\em\ ph{isomorphic} and write $E_a \\cong E_b$.", count := [ 7, 0, 0, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 7, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Acting on digraph\ s", count := [ 7, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 284662, root := ~, start := 284625, stop := 284661 ), rec( content := "\n ", count := [ 7, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, perm", Label := "for a digraph \ and a perm", Name := "OnDigraphs" ), content := 0, count := [ 7, 1, 1, 2 ], mathmode := "Text", name := "Oper", next := 284764, root := ~, start := 284682, stop := 284763 ), rec( attributes := rec( Arg := "digraph, trans", Label := "for a digraph \ and a transformation", Name := "OnDigraphs" ), content := 0, count := [ 7, 1, 1, 4 ], mathmode := "Text", name := "Oper", next := 284860, root := ~, start := 284767, stop := 284859 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "Returns", next := 284892, root := ~, start := 284863, stop := 284891 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 284930, root := ~, start := 284916, stop := 284929 ), rec( content := " is a di\ graph, and the second argument ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 284980, root := ~, start := 284969, stop := 284979 ), rec( content := " is a\n \ ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ermutation", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "E", next := 285008, root := ~, start := 284990, stop := 285007 ), rec( content := " of the \ vertices of ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 285042, root := ~, start := 285028, stop := 285041 ), rec( content := ", then t\ his operation\n returns a digraph constructed by relabelling the vertices o\ f\n ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 285147, root := ~, start := 285133, stop := 285146 ), rec( content := " accordi\ ng to ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 285172, root := ~, start := 285161, stop := 285171 ), rec( content := ". Note \ that for an automorphism\n ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "C", next := 285217, root := ~, start := 285209, stop := 285216 ), rec( content := " of a di\ graph, we have ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 7, 1, 1, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := \ "Text", name := "A", next := 285268, root := ~, start := 285254, stop := 28526\ 7 ), rec( content := ", f) =\n ", count := [ 7, 1, 1, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "C", next := 285283, root := ~, start := 285240, stop := 285282 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 285297, root := ~, start := 285283, stop := 285296 ), rec( content := ".\n " , count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "P", next := 285307, root := ~, start := 285303, stop := 285306 ), rec( content := "\n\n \ If the second argument is a ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ransformation", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "E", next := 285362, root := ~, start := 285341, stop := 285361 ), rec( content := " ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 285375, root := ~, start := 285363, stop := 285374 ), rec( content := " of the\\ n vertices of ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 285413, root := ~, start := 285399, stop := 285412 ), rec( content := ", then t\ his operation returns a digraph\n constructed by transforming the source an\ d range of each edge according to\n ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 285548, root := ~, start := 285536, stop := 285547 ), rec( content := ". Thus a\ vertex which does not appear in the image of\n ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 285618, root := ~, start := 285606, stop := 285617 ), rec( content := " will be\ isolated in the returned digraph, and the returned\n digraph may contain m\ ultiple edges, even if ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 285740, root := ~, start := 285726, stop := 285739 ), rec( content := " does no\ t.\n If ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 285770, root := ~, start := 285758, stop := 285769 ), rec( content := " is math\ ematically a permutation, then the result coincides\n with ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 7, 1, 1, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "A", next := 285867, root := ~, start := 285853, stop := 28586\ 6 ), rec( content := ", AsPermutation(", count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "trans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 2858\ 95, root := ~, start := 285883, stop := 285894 ), rec( content := "))", count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "C", next := 285901, root := ~, start := 285839, stop := 285900 ), rec( content := ".\n " , count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "P", next := 285911, root := ~, start := 285907, stop := 285910 ), rec( content := "\n\n \ The ", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "Ref", next := 285954, root := ~, start := 285921, stop := 285953 ), rec( content := " of ", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "A", next := 285972, root := ~, start := 285958, stop := 285971 ), rec( content := " will no\ t be retained\n in the returned digraph. ", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "P", next := 286027, root := ~, start := 286023, stop := 286026 ), rec( content := "\n\n \ If ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 286050, root := ~, start := 286036, stop := 286049 ), rec( content := " belongs\ to ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "Ref", next := 286092, root := ~, start := 286062, stop := 286091 ), rec( content := ", then\n\ relabelling of the vertices is performed directly on ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 286170, root := ~, start := 286156, stop := 286169 ), rec( content := ".\n I\ f ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 286193, root := ~, start := 286179, stop := 286192 ), rec( content := " belongs\ to ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "Ref", next := 286237, root := ~, start := 286205, stop := 286236 ), rec( content := ", an imm\ utable\n copy of ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 286278, root := ~, start := 286264, stop := 286277 ), rec( content := " with th\ e vertices relabelled is returned.", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 9 ], mathmode := "Text", name := "P", next := 286324, root := ~, start := 286320, stop := 286323 ), rec( content := "\n\n \ ", count := [ 7, 1, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> new := OnDigraphs(D, (1, 2));\n\ngap> OutNeighbours(new);\n[ [ 2, 3, 5 \ ], [ 3 ], [ 2 ], [ 2, 1, 4 ], [ 1, 3, 5 ] ]\ngap> D := Digraph([[2], [], [2]])\ ;\n\ngap> t := Transformation([1, \ 2, 1]);;\ngap> new := OnDigraphs(D, t);\n\ngap> OutNeighbours(new);\n[ [ 2, 2 ], [ ], [ ] ]\ngap> ForAll(\ DigraphEdges(D),\n> e -> IsDigraphEdge(new, [e[1] ^ t, e[2] ^ t]));\ntrue\n", count := [ 7, 1, 1, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 1, 1, 10 ], mathmode := "Text", name := "Example", next := 286967, root := ~, start := 286330, stop := 286966 ), rec( content := "\n ", count := [ 7, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 286984, root := ~, start := 284895, stop := 286983 ) ], count := [ 7, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 286998, root := ~, start := 284667, stop := 286997 ), rec( content := "\n\n ", count := [ 7, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, pair", Name := "OnMultiDigraphs\ " ), content := 0, count := [ 7, 1, 2, 2 ], mathmode := "Text", name := "Oper", next := 287069, root := ~, start := 287019, stop := 287068 ), rec( attributes := rec( Arg := "digraph, perm1, \ perm2", Label := "for a digraph, perm, and perm", Name := "OnMultiDigraphs" ), content := 0, count := [ 7, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 287172, root := ~, start := 287072, stop := 287171 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 287204, root := ~, start := 287175, stop := 287203 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 287242, root := ~, start := 287228, stop := 287241 ), rec( content := " is a di\ graph, and ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ air", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 287272, root := ~, start := 287261, stop := 287271 ), rec( content := " is a pa\ ir consisting of a\n permutation of the vertices and a permutation of the e\ dges of\n ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 287383, root := ~, start := 287369, stop := 287382 ), rec( content := ", then t\ his operation returns a digraph\n constructed by relabelling the vertices a\ nd edges of ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 287494, root := ~, start := 287480, stop := 287493 ), rec( content := "\n ac\ cording to ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm[1]", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 287526, root := ~, start := 287512, stop := 287525 ), rec( content := " and ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm[2]", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 287545, root := ~, start := 287531, stop := 287544 ), rec( content := ", respec\ tively. ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "P", next := 287565, root := ~, start := 287561, stop := 287564 ), rec( content := "\n\n \ In its second form, ", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nMultiDigraphs", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "C", next := 287613, root := ~, start := 287591, stop := 287612 ), rec( content := " returns\ a digraph with vertices\n and edges permuted by ", count := [ 7, 1, 2, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm1", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "A", next := 287684, root := ~, start := 287672, stop := 287683 ), rec( content := " and ", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm2", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "A", next := 287701, root := ~, start := 287689, stop := 287700 ), rec( content := ", respec\ tively. ", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "P", next := 287721, root := ~, start := 287717, stop := 287720 ), rec( content := "\n\n \ Note that ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 7, 1, 2, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := \ "Text", name := "A", next := 287765, root := ~, start := 287751, stop := 28776\ 4 ), rec( content := ",\n perm)=OnMultiDigraphs(", count := [ 7, 1, 2, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 2878\ 09, root := ~, start := 287795, stop := 287808 ), rec( content := ", [perm, ()\ ])", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 287826, root := ~, start := 287737, stop := 287825 ), rec( content := " where " , count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 287844, root := ~, start := 287833, stop := 287843 ), rec( content := " is\n \ a permutation of the vertices of ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 287899, root := ~, start := 287885, stop := 287898 ), rec( content := ". If you\ are only interested\n in the action of a permutation on the vertices of a \ digraph, then you can\n use ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 288031, root := ~, start := 288014, stop := 288030 ), rec( content := " instead\ of ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nMultiDigraphs", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 288065, root := ~, start := 288043, stop := 288064 ), rec( content := ".\n\n \ If ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 288089, root := ~, start := 288075, stop := 288088 ), rec( content := " belongs\ to ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 288131, root := ~, start := 288101, stop := 288130 ), rec( content := ", then\n\ relabelling of the vertices is performed directly on ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 288209, root := ~, start := 288195, stop := 288208 ), rec( content := ".\n I\ f ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 288232, root := ~, start := 288218, stop := 288231 ), rec( content := " belongs\ to ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 288276, root := ~, start := 288244, stop := 288275 ), rec( content := ", an imm\ utable\n copy of ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 288317, root := ~, start := 288303, stop := 288316 ), rec( content := " with th\ e vertices relabelled is returned.", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 2, 8 ], mathmode := "Text", name := "P", next := 288363, root := ~, start := 288359, stop := 288362 ), rec( content := "\n\n \ ", count := [ 7, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := Digraph([\n> [3, 6, 3], [], [3], [9, 10], [9], [], [], [10, 4, 10\ ], [], []]);\n\ngap> p := B\ lissCanonicalLabelling(D1);\n[ (1,7)(3,6)(4,10)(5,9), () ]\ngap> D2 := OnMulti\ Digraphs(D1, p);\n\ngap> Ou\ tNeighbours(D2);\n[ [ ], [ ], [ ], [ ], [ ], [ 6 ], [ 6, 3, 6 ], [ 4, 10,\ 4 ], \n [ 5 ], [ 5, 4 ] ]", count := [ 7, 1, 2, 9 ], name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 9 ], mathmode := "Text", name := "Example", next := 288808, root := ~, start := 288369, stop := 288807 ), rec( content := "\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 288825, root := ~, start := 287207, stop := 288824 ) ], count := [ 7, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 288839, root := ~, start := 287004, stop := 288838 ), rec( content := "\n\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 288853, root := ~, start := 284616, stop := 288852 ) , rec( content := "\n\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Isomorphisms and canonical lab\ ellings" ), content := [ rec( content := "\n ", count := [ 7, 2, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Isomorphisms and \ canonical labellings", count := [ 7, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 288973, root := ~, start := 288917, stop := 288972 ), rec( content := "\n From version 0.11.\ 0 of ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " it is possible to use e\ ither ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " or\n ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "http://pa\ llini.di.uniroma1.it/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 124, root := ~, start := 100, stop := 123 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " (via ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "NautyTracesInte\ rface" ), content := [ rec( content := "https://github.com/gap-packages/NautyT\ racesInterface", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ) , rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "NautyTrac\ esInterface", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := ") to calculate canonical\ labellings and\n automorphism groups of digraphs; see ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "JK07" ), content := 0, count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Cite", next := 289190, root := ~, start := 289172, stop := 289189 ), rec( content := " and ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "MP14" ), content := 0, count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Cite", next := 289219, root := ~, start := 289195, stop := 289218 ), rec( content := " for more details about \ ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " and ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "http://pa\ llini.di.uniroma1.it/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 124, root := ~, start := 100, stop := 123 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := ",\n respectively. \n\\ n ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsUseNaut\ y" ), content := 0, count := [ 7, 2, 1, 2 ], mathmode := "Text", name := "Func", next := 289341, root := ~, start := 289303, stop := 289340 ), rec( attributes := rec( Arg := "", Name := "DigraphsUseBlis\ s" ), content := 0, count := [ 7, 2, 1, 4 ], mathmode := "Text", name := "Func", next := 289382, root := ~, start := 289344, stop := 289381 ), rec( attributes := rec( ), content := [ rec( content := "Nothing." , count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 289412, root := ~, start := 289385, stop := 289411 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se functions can be used to specify whether ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " or ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " should \ be\n used by default by ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ". If ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 1, 5 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 1, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is not \ available,\n then these functions do nothing. Otherwise, by calling\n ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphsUseNauty", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "C", next := 289674, root := ~, start := 289651, stop := 289673 ), rec( content := " subsequ\ ent computations will default to using\n ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " rather \ than ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := ", where \ possible.\n ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "P", next := 289778, root := ~, start := 289774, stop := 289777 ), rec( content := "\n\n \ You can call these functions at any point in a ", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " session\ , as many\n times as you like, it is guaranteed that existing digraphs rema\ in\n valid, and that comparison of existing digraphs and newly created digr\ aphs\n via\n ", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsIsomorphicDigraph" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 290069, root := ~, start := 290015, stop := 290068 ), rec( content := ",\n " , count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsIsomorphicDigraph" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 290157, root := ~, start := 290075, stop := 290156 ), rec( content := ",\n " , count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 290217, root := ~, start := 290163, stop := 290216 ), rec( content := ", and\n \ ", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 290309, root := ~, start := 290227, stop := 290308 ), rec( content := " are als\ o valid.", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "P", next := 290329, root := ~, start := 290325, stop := 290328 ), rec( content := "\n\n \ It is also possible to compute the automorphism group of a specific digraph\n \ using both ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 1, 7 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 1, 7 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 7 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " and ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 1, 7 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 1, 7 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 7 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " using " , count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Ref", next := 290488, root := ~, start := 290452, stop := 290487 ), rec( content := "\n an\ d ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Ref", next := 290555, root := ~, start := 290497, stop := 290554 ), rec( content := ", respec\ tively.\n ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 290587, root := ~, start := 289415, stop := 290586 ) ], count := [ 7, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 290601, root := ~, start := 289288, stop := 290600 ), rec( content := "\n\n ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "AutomorphismGro\ up" ), content := 0, count := [ 7, 2, 2, 2 ], mathmode := "Text", name := "Attr", next := 290690, root := ~, start := 290622, stop := 290689 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 290732, root := ~, start := 290693, stop := 290731 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 290770, root := ~, start := 290756, stop := 290769 ), rec( content := " is a di\ graph, then this attribute contains the group of\n automorphisms of ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 290862, root := ~, start := 290848, stop := 290861 ), rec( content := ". An ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "E", next := 290887, root := ~, start := 290868, stop := 290886 ), rec( content := " of ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 290905, root := ~, start := 290891, stop := 290904 ), rec( content := "\n is\ an isomorphism from ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 290947, root := ~, start := 290933, stop := 290946 ), rec( content := " to itse\ lf. See ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 291024, root := ~, start := 290963, stop := 291023 ), rec( content := " for mor\ e information\n about isomorphisms of digraphs. ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "P", next := 291086, root := ~, start := 291082, stop := 291085 ), rec( content := "\n\n \ If ", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "A", next := 291109, root := ~, start := 291095, stop := 291108 ), rec( content := " is not \ a multidigraph then the automorphism group is\n returned as a group of perm\ utations on the set of vertices of\n ", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "A", next := 291247, root := ~, start := 291233, stop := 291246 ), rec( content := ". ", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "P", next := 291253, root := ~, start := 291249, stop := 291252 ), rec( content := "\n\n \ If ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "A", next := 291276, root := ~, start := 291262, stop := 291275 ), rec( content := " is a mu\ ltidigraph then the automorphism group is returned\n as the direct product \ of a group of permutations on the set of vertices of\n ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "A", next := 291432, root := ~, start := 291418, stop := 291431 ), rec( content := " with a \ group of permutations on the set of edges of\n ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "A", next := 291503, root := ~, start := 291489, stop := 291502 ), rec( content := ". These \ groups can be accessed using ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Label := "for a domain and a positive integer", Oper := "Projection" ), content := 0, count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "Ref", next := 291629, root := ~, start := 291540, stop := 291628 ), rec( content := " on the\\ n returned group.", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "P", next := 291660, root := ~, start := 291656, stop := 291659 ), rec( content := "\n\n \ By default, the automorphism group is found using ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 2, 6 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 2, 6 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 2, 6 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 291945, root := ~, start := 291887, stop := 291944 ), rec( content := ", \n \ ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 291988, root := ~, start := 291952, stop := 291987 ), rec( content := ", ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 292020, root := ~, start := 291990, stop := 292019 ), rec( content := ",\n a\ nd ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 292060, root := ~, start := 292030, stop := 292059 ), rec( content := ".\n " , count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "P", next := 292070, root := ~, start := 292066, stop := 292069 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "A", next := 292106, root := ~, start := 292092, stop := 292105 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> johnson := DigraphFromGraph6String(\"E}lw\");\n\ngap> G := AutomorphismGroup(johnson);\nGroup([ (3,4), (2\ ,3)(4,5), (1,2)(5,6) ])\ngap> cycle := CycleDigraph(9);\n\ngap> G := AutomorphismGroup(cycle);\nGroup([ (1,2,3,4,5,\ 6,7,8,9) ])\ngap> IsCyclic(G) and Size(G) = 9;\ntrue", count := [ 7, 2, 2, 8 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 2, 8 ], mathmode := "Text", name := "Example", next := 292592, root := ~, start := 292208, stop := 292591 ), rec( content := "\n ", count := [ 7, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 292611, root := ~, start := 290735, stop := 292610 ) ], count := [ 7, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 292627, root := ~, start := 290607, stop := 292626 ), rec( content := "\n\n ", count := [ 7, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "BlissAutomorphi\ smGroup" ), content := 0, count := [ 7, 2, 3, 2 ], mathmode := "Text", name := "Attr", next := 292721, root := ~, start := 292648, stop := 292720 ), rec( attributes := rec( Arg := "digraph, vertex_\ colours", Label := "for a digraph and homogenous list", Name := "BlissAutomorphi\ smGroup" ), content := 0, count := [ 7, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 292838, root := ~, start := 292724, stop := 292837 ), rec( attributes := rec( Arg := "digraph, vertex_\ colours, edge_colours", Label := "for a digraph, homogenous list, and list", Name := "BlissAutomorphi\ smGroup" ), content := 0, count := [ 7, 2, 3, 6 ], mathmode := "Text", name := "Oper", next := 292976, root := ~, start := 292841, stop := 292975 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Returns", next := 293018, root := ~, start := 292979, stop := 293017 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "A", next := 293056, root := ~, start := 293042, stop := 293055 ), rec( content := " is a di\ graph, then this attribute contains the group of\n automorphisms of ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "A", next := 293148, root := ~, start := 293134, stop := 293147 ), rec( content := " as calc\ ulated using ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 3, 7 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 3, 7 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "P", next := 293222, root := ~, start := 293218, stop := 293221 ), rec( content := "\n\n \ The attribute ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 3, 8 ] , mathmode := "Text", name := "Ref", next := 293295, root := ~, start := 293242, stop := 293294 ), rec( content := " and\n \ operation ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "Ref", next := 293397, root := ~, start := 293314, stop := 293396 ), rec( content := " returns\ \n the value of either ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissAutomorphismGroup", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "C", next := 293459, root := ~, start := 293430, stop := 293458 ), rec( content := " or ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "Ref", next := 293505, root := ~, start := 293463, stop := 293504 ), rec( content := ". These \ groups are, of course, equal but\n their generating sets may differ. ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "P", next := 293588, root := ~, start := 293584, stop := 293587 ), rec( content := "\n Th\ e attribute ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph, homogeneous list, and list" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 293695, root := ~, start := 293607, stop := 293694 ), rec( content := " returns\ the value of\n ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissAutomorphismGroup", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "C", next := 293750, root := ~, start := 293721, stop := 293749 ), rec( content := " as it i\ s not implemented for ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 3, 9 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 3, 9 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " The \n \ requirements for the optional arguments ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex_colours", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "A", next := 293858, root := ~, start := 293837, stop := 293857 ), rec( content := " and \n \ ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge_colours", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "A", next := 293887, root := ~, start := 293868, stop := 293886 ), rec( content := " are doc\ umented in ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph, homogeneous list, and list" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 293994, root := ~, start := 293906, stop := 293993 ), rec( content := ".\n\n \ See also ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 294040, root := ~, start := 294010, stop := 294039 ), rec( content := ", and ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 294076, root := ~, start := 294046, stop := 294075 ), rec( content := ".\n " , count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "P", next := 294086, root := ~, start := 294082, stop := 294085 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "A", next := 294122, root := ~, start := 294108, stop := 294121 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := BlissAutomorphismGroup(JohnsonDigraph(5, 2));;\ngap> IsSymmetricGro\ up(G);\ntrue\ngap> Size(G);\n120", count := [ 7, 2, 3, 11 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 11 ], mathmode := "Text", name := "Example", next := 294361, root := ~, start := 294224, stop := 294360 ), rec( content := "\n ", count := [ 7, 2, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Description", next := 294380, root := ~, start := 293021, stop := 294379 ) ], count := [ 7, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 294396, root := ~, start := 292633, stop := 294395 ), rec( content := "\n\n ", count := [ 7, 2, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, vert_c\ olours]", Name := "NautyAutomorphismGroup" ), content := 0, count := [ 7, 2, 4, 2 ], mathmode := "Text", name := "Attr", next := 294484, root := ~, start := 294417, stop := 294483 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 294526, root := ~, start := 294487, stop := 294525 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "A", next := 294564, root := ~, start := 294550, stop := 294563 ), rec( content := " is a di\ graph, then this attribute contains the group of\n automorphisms of ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "A", next := 294656, root := ~, start := 294642, stop := 294655 ), rec( content := " as calc\ ulated using ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 4, 3 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 4, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno via ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 4, 3 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 4, 3 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 4, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := ".\n\n \ ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( content := " FIXME b\ e more explicit about what the second arg is ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n\n \ The attribute ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 4, 3 ] , mathmode := "Text", name := "Ref", next := 294891, root := ~, start := 294838, stop := 294890 ), rec( content := " and\n \ operation ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 294993, root := ~, start := 294910, stop := 294992 ), rec( content := " returns\ \n the value of either ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyAutomorphismGroup", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "C", next := 295055, root := ~, start := 295026, stop := 295054 ), rec( content := " or ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 295124, root := ~, start := 295060, stop := 295123 ), rec( content := ".\n T\ hese groups are, of course, equal but their generating sets may differ.", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "P", next := 295206, root := ~, start := 295202, stop := 295205 ), rec( content := "\n\n \ See also ", count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "Ref", next := 295251, root := ~, start := 295221, stop := 295250 ), rec( content := ", and ", count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "Ref", next := 295287, root := ~, start := 295257, stop := 295286 ), rec( content := ".\n " , count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "P", next := 295297, root := ~, start := 295293, stop := 295296 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "A", next := 295333, root := ~, start := 295319, stop := 295332 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> NautyAutomorphismGroup(JohnsonDigraph(5, 2));\nGroup([ (3,4)(6,7)(8,9), \ (2,3)(5,6)(9,10), (2,5)(3,6)(4,7),\n (1,2)(6,8)(7,9) ])", count := [ 7, 2, 4, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 4, 6 ], mathmode := "Text", name := "Log", next := 295589, root := ~, start := 295435, stop := 295588 ), rec( content := "\n ", count := [ 7, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 295608, root := ~, start := 294529, stop := 295607 ) ], count := [ 7, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 295624, root := ~, start := 294402, stop := 295623 ), rec( content := "\n\n ", count := [ 7, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vert_co\ lours", Label := "for a digraph and a homogeneous list", Name := "AutomorphismGro\ up" ), content := 0, count := [ 7, 2, 5, 2 ], mathmode := "Text", name := "Oper", next := 295754, root := ~, start := 295645, stop := 295753 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 295796, root := ~, start := 295757, stop := 295795 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation computes the automorphism group of a vertex-coloured digraph.\n \ A vertex-coloured digraph can be specified by its underlying digraph\n " , count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 295986, root := ~, start := 295972, stop := 295985 ), rec( content := " and its\ colouring ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 296024, root := ~, start := 296005, stop := 296023 ), rec( content := ". Let ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "C", next := 296038, root := ~, start := 296030, stop := 296037 ), rec( content := " be the\\ n number of vertices of ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 296086, root := ~, start := 296072, stop := 296085 ), rec( content := ". The co\ louring ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 296121, root := ~, start := 296102, stop := 296120 ), rec( content := " may\n \ have one of the following two forms:\n\n ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 7, 2, 5, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode\ := "Text", name := "C", next := 296218, root := ~, start := 296210, stop := 2\ 96217 ), rec( content := " integers, where ", count := [ 7, 2, 5, 5 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "vert_colours", count := [ 7, 2, 5, 5 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode := "T\ ext", name := "A", next := 296254, root := ~, start := 296235, stop := 296253 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 5, 5 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ] , math\ mode := "Text", name := "C", next := 296264, root := ~, start := 296254, stop \ := 296263 ), rec( content := " is the\n colour of vertex ", count := [ \ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "i", count := [ 7, 2, 5, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], math\ mode := "Text", name := "C", next := 296305, root := ~, start := 296297, stop \ := 296304 ), rec( content := ", using the colours ", count := [ 7, 2, 5, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 5, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode := "Tex\ t", name := "C", next := 296340, root := ~, start := 296325, stop := 296339 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 5, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "C", next\ := 296374, root := ~, start := 296358, stop := 296373 ), rec( content := "; o\ r\n ", count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 5, 5 ], name := "Item", next := 296392, root := ~, start := 296\ 185, stop := 296391 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 5, 7 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 \ ], mathmode := "Text", name := "A", next := 296505, root := ~, start := 296491\ , stop := 296504 ), rec( content := ")", count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "C", next := 296510, root := ~, start := 296472, stop := 29650\ 9 ), rec( content := ", such that\n ", count := [ 7, 2, 5, 7 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "vert_colours", count := [ 7, 2, 5, 7 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], mathmode := "T\ ext", name := "A", next := 296549, root := ~, start := 296530, stop := 296548 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 5, 7 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ] , math\ mode := "Text", name := "C", next := 296559, root := ~, start := 296549, stop \ := 296558 ), rec( content := " is the list of all vertices with colour\n \ ", count := [ 7, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 5, 7 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], math\ mode := "Text", name := "C", next := 296616, root := ~, start := 296608, stop \ := 296615 ), rec( content := ".\n ", count := [ 7, 2, 5, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], name := "Ite\ m", next := 296631, root := ~, start := 296399, stop := 296630 ) ], count := [ 7, 2, 5, 4 ], mathmode := "Text", name := "List", next := 296643, root := ~, start := 296172, stop := 296642 ), rec( content := "\n\n \ The ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism group", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "E", next := 296678, root := ~, start := 296653, stop := 296677 ), rec( content := " of a co\ loured digraph ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 296715, root := ~, start := 296701, stop := 296714 ), rec( content := " with\n \ colouring ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 296754, root := ~, start := 296735, stop := 296753 ), rec( content := " is the \ group consisting of its automorphisms; an\n ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "E", next := 296827, root := ~, start := 296808, stop := 296826 ), rec( content := " of ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 296845, root := ~, start := 296831, stop := 296844 ), rec( content := " is an i\ somorphism of coloured\n digraphs from ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 296908, root := ~, start := 296894, stop := 296907 ), rec( content := " to itse\ lf. This group is equal to the\n subgroup of ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ utomorphismGroup(", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "di\ graph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 2969\ 99, root := ~, start := 296985, stop := 296998 ), rec( content := ")", count := [ 7, 2, 5, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "C", next := 297004, root := ~, start := 296964, stop := 297003 ), rec( content := " consist\ ing of those\n automorphisms that preserve the colouring specified by ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 297103, root := ~, start := 297084, stop := 297102 ), rec( content := ". See\n \ ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 5, 9 ] , mathmode := "Text", name := "Ref", next := 297167, root := ~, start := 297113, stop := 297166 ), rec( content := ", and se\ e ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "Ref", next := 297260, root := ~, start := 297177, stop := 297259 ), rec( content := "\n fo\ r more information about isomorphisms of coloured digraphs. ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "P", next := 297331, root := ~, start := 297327, stop := 297330 ), rec( content := "\n\n \ If ", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "A", next := 297354, root := ~, start := 297340, stop := 297353 ), rec( content := " is not \ a multidigraph then the automorphism group is\n returned as a group of perm\ utations on the set of vertices of\n ", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "A", next := 297492, root := ~, start := 297478, stop := 297491 ), rec( content := ". ", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "P", next := 297498, root := ~, start := 297494, stop := 297497 ), rec( content := "\n\n \ If ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "A", next := 297521, root := ~, start := 297507, stop := 297520 ), rec( content := " is a mu\ ltidigraph then the automorphism group is returned\n as the direct product \ of a group of permutations on the set of vertices of\n ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "A", next := 297677, root := ~, start := 297663, stop := 297676 ), rec( content := " with a \ group of permutations on the set of edges of\n ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "A", next := 297748, root := ~, start := 297734, stop := 297747 ), rec( content := ". These \ groups can be accessed using ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Label := "for a domain and a positive integer", Oper := "Projection" ), content := 0, count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "Ref", next := 297874, root := ~, start := 297785, stop := 297873 ), rec( content := " on the\\ n returned group.", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "P", next := 297905, root := ~, start := 297901, stop := 297904 ), rec( content := "\n\n \ By default, the automorphism group is found using ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 5, 12 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 5, 12 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 5, 12 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 5, 12 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph and homogenous list" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 298216, root := ~, start := 298132, stop := 298215 ), rec( content := ",\n " , count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 298258, root := ~, start := 298222, stop := 298257 ), rec( content := ", ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 298290, root := ~, start := 298260, stop := 298289 ), rec( content := ",\n a\ nd ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 298330, root := ~, start := 298300, stop := 298329 ), rec( content := ".\n\n " , count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> cycle := CycleDigraph(9);\n\nga\ p> G := AutomorphismGroup(cycle);;\ngap> IsCyclic(G) and Size(G) = 9;\ntrue\ng\ ap> colours := [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;\ngap> H := AutomorphismGrou\ p(cycle, colours);;\ngap> Size(H);\n3\ngap> H = AutomorphismGroup(cycle, [1, 2\ , 3, 1, 2, 3, 1, 2, 3]);\ntrue\ngap> H = SubgroupByProperty(G, p -> OnTuplesSe\ ts(colours, p) = colours);\ntrue\ngap> IsTrivial(AutomorphismGroup(cycle, [1, \ 1, 2, 2, 2, 2, 2, 2, 2]));\ntrue", count := [ 7, 2, 5, 13 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 13 ], mathmode := "Text", name := "Example", next := 298852, root := ~, start := 298335, stop := 298851 ), rec( content := "\n ", count := [ 7, 2, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 298869, root := ~, start := 295799, stop := 298868 ) ], count := [ 7, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 298883, root := ~, start := 295630, stop := 298882 ), rec( content := "\n\n ", count := [ 7, 2, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vert_co\ lours, edge_colours", Label := "for a digraph, homogeneous list, and list", Name := "AutomorphismGro\ up" ), content := 0, count := [ 7, 2, 6, 2 ], mathmode := "Text", name := "Oper", next := 299036, root := ~, start := 298904, stop := 299035 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "Returns", next := 299078, root := ~, start := 299039, stop := 299077 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation computes the automorphism group of a vertex- and/or\n edge-co\ loured digraph.\n A coloured digraph can be specified by its underlying dig\ raph\n ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 299278, root := ~, start := 299264, stop := 299277 ), rec( content := " and col\ ourings ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 299313, root := ~, start := 299294, stop := 299312 ), rec( content := ", ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge_colours", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 299334, root := ~, start := 299315, stop := 299333 ), rec( content := ".\n L\ et ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "C", next := 299352, root := ~, start := 299344, stop := 299351 ), rec( content := " be the \ number of vertices of ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 299396, root := ~, start := 299382, stop := 299395 ), rec( content := ". The co\ lourings\n must have the following forms:\n\n ", count := [ 7, 2, 6, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 6, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "vert_colours", count := [ 7, 2, 6, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], mathmode := \ "Text", name := "A", next := 299500, root := ~, start := 299481, stop := 29949\ 9 ), rec( content := " must be ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "fail", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "K", next :\ = 299520, root := ~, start := 299509, stop := 299519 ), rec( content := " or a\ list of ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "n", count :\ = [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next := 299542, root := ~, star\ t := 299534, stop := 299541 ), rec( content := " integers,\n where ", coun\ t := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "vert_colours", count := [ 7, \ 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, \ 2, 6, 5 ], mathmode := "Text", name := "A", next := 299586, root := ~, start :\ = 299567, stop := 299585 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "[i]", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next :\ = 299596, root := ~, start := 299586, stop := 299595 ), rec( content := " is t\ he colour of vertex ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "i", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next := 299629, root\ := ~, start := 299621, stop := 299628 ), rec( content := ",\n using th\ e colours ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[1 .. m]", \ count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next := 299672, root := \ ~, start := 299657, stop := 299671 ), rec( content := " for some ", count := [ \ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "m <= n", count := [ 7, 2, 6, 5 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], math\ mode := "Text", name := "C", next := 299698, root := ~, start := 299682, stop \ := 299697 ), rec( content := ";\n ", count := [ 7, 2, 6, 5 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], name := "Ite\ m", next := 299713, root := ~, start := 299466, stop := 299712 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 6, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "edge_colours", count := [ 7, 2, 6, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := \ "Text", name := "A", next := 299754, root := ~, start := 299735, stop := 29975\ 3 ), rec( content := " must be ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "fail", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "K", next :\ = 299774, root := ~, start := 299763, stop := 299773 ), rec( content := " or a\ list of ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "n", count :\ = [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 6, 7 ], mathmode := "Text", name := "C", next := 299796, root := ~, star\ t := 299788, stop := 299795 ), rec( content := " lists of\n integers of\ the same shape as ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\ OutNeighbours(digraph)", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := "Text", name := \ "C", next := 299873, root := ~, start := 299844, stop := 299872 ), rec( conten\ t := ", where\n ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "edge_colours", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "A", next\ := 299908, root := ~, start := 299889, stop := 299907 ), rec( attributes := re\ c( ), content := [ rec( content := "[i][j]", count := [ 7, 2, 6, 7 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmod\ e := "Text", name := "C", next := 299921, root := ~, start := 299908, stop := \ 299920 ), rec( content := " is the colour of the edge\n ", count := [ 7\ , 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "OutNeighbours(digraph)[i][j]", cou\ nt := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 6, 7 ], mathmode := "Text", name := "C", next := 299991, root := \ ~, start := 299956, stop := 299990 ), rec( content := ", using the colours ", coun\ t := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[1 .. k]", count := [ 7, 2, 6\ , 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6\ , 7 ], mathmode := "Text", name := "C", next := 300026, root := ~, start := 30\ 0011, stop := 300025 ), rec( content := "\n for some ", count := [ 7, 2\ , 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "k <= n", count := [ 7, 2, 6, 7 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], math\ mode := "Text", name := "C", next := 300060, root := ~, start := 300044, stop \ := 300059 ), rec( content := ";\n ", count := [ 7, 2, 6, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], name := "Ite\ m", next := 300075, root := ~, start := 299720, stop := 300074 ) ], count := [ 7, 2, 6, 4 ], mathmode := "Text", name := "List", next := 300087, root := ~, start := 299453, stop := 300086 ), rec( content := "\n\n \ Giving ", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "A", next := 300119, root := ~, start := 300100, stop := 300118 ), rec( content := " [", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge_colours", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "A", next := 300140, root := ~, start := 300121, stop := 300139 ), rec( content := "] as ", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "C", next := 300156, root := ~, start := 300145, stop := 300155 ), rec( content := " is\n \ equivalent to setting all vertices [edges] to be the same colour. ", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "P", next := 300234, root := ~, start := 300230, stop := 300233 ), rec( content := "\n\n \ Unlike ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 300300, root := ~, start := 300247, stop := 300299 ), rec( content := ", it is\\ n possible to obtain the automorphism group of an edge-coloured multidigrap\ h\n (see ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 300425, root := ~, start := 300396, stop := 300424 ), rec( content := ") when n\ o two edges share the same\n source, range, and colour.\n\n The ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism group", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "E", next := 300525, root := ~, start := 300500, stop := 300524 ), rec( content := " of a ve\ rtex/edge-coloured digraph\n ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 300578, root := ~, start := 300564, stop := 300577 ), rec( content := " with co\ louring ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 300602, root := ~, start := 300594, stop := 300601 ), rec( content := " is the \ group consisting of its\n vertex/edge-colour preserving automorphisms; an\n\ ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "E", next := 300709, root := ~, start := 300690, stop := 300708 ), rec( content := " of ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 300727, root := ~, start := 300713, stop := 300726 ), rec( content := " is an i\ somorphism of vertex/edge-coloured\n digraphs from ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 300802, root := ~, start := 300788, stop := 300801 ), rec( content := " to itse\ lf. This group is equal to the\n subgroup of ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ utomorphismGroup(", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 3008\ 93, root := ~, start := 300879, stop := 300892 ), rec( content := ")", count := [ 7, 2, 6, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "C", next := 300898, root := ~, start := 300858, stop := 300897 ), rec( content := " consist\ ing of those\n automorphisms that preserve the colouring specified by ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 300992, root := ~, start := 300978, stop := 300991 ), rec( content := ". See\n \ ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 301056, root := ~, start := 301002, stop := 301055 ), rec( content := ", and se\ e ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 301149, root := ~, start := 301066, stop := 301148 ), rec( content := "\n fo\ r more information about isomorphisms of coloured digraphs. ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "P", next := 301220, root := ~, start := 301216, stop := 301219 ), rec( content := "\n\n \ If ", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "A", next := 301243, root := ~, start := 301229, stop := 301242 ), rec( content := " is not \ a multidigraph then the automorphism group is\n returned as a group of perm\ utations on the set of vertices of\n ", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "A", next := 301381, root := ~, start := 301367, stop := 301380 ), rec( content := ". ", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "P", next := 301387, root := ~, start := 301383, stop := 301386 ), rec( content := "\n\n \ If ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "A", next := 301410, root := ~, start := 301396, stop := 301409 ), rec( content := " is a mu\ ltidigraph then the automorphism group is returned\n as the direct product \ of a group of permutations on the set of vertices of\n ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "A", next := 301566, root := ~, start := 301552, stop := 301565 ), rec( content := " with a \ group of permutations on the set of edges of\n ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "A", next := 301637, root := ~, start := 301623, stop := 301636 ), rec( content := ". These \ groups can be accessed using ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Label := "for a domain and a positive integer", Oper := "Projection" ), content := 0, count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "Ref", next := 301763, root := ~, start := 301674, stop := 301762 ), rec( content := " on the\\ n returned group.", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "P", next := 301794, root := ~, start := 301790, stop := 301793 ), rec( content := "\n\n \ By default, the automorphism group is found using ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 6, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 6, 13 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 6, 13 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 6, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph, homogenous list, and list" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 302114, root := ~, start := 302022, stop := 302113 ), rec( content := ",\n \ ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 302157, root := ~, start := 302121, stop := 302156 ), rec( content := ", ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 302189, root := ~, start := 302159, stop := 302188 ), rec( content := ", and\n \ ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 302230, root := ~, start := 302200, stop := 302229 ), rec( content := ".\n\n " , count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> cycle := CycleDigraph(12);\n\n\ gap> vert_colours := List([1 .. 12], x -> x mod 3 + 1);;\ngap> edge_colours :=\ List([1 .. 12], x -> [x mod 2 + 1]);;\ngap> Size(AutomorphismGroup(cycle));\n\ 12\ngap> Size(AutomorphismGroup(cycle, vert_colours));\n4\ngap> Size(Automorph\ ismGroup(cycle, fail, edge_colours));\n6\ngap> Size(AutomorphismGroup(cycle, v\ ert_colours, edge_colours));\n2\ngap> IsTrivial(AutomorphismGroup(cycle,\n> ve\ rt_colours, List([1 .. 12], x -> [x mod 4 + 1])));\ntrue\n", count := [ 7, 2, 6, 14 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 6, 14 ], mathmode := "Text", name := "Example", next := 302777, root := ~, start := 302235, stop := 302776 ), rec( content := "\n ", count := [ 7, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 302794, root := ~, start := 299081, stop := 302793 ) ], count := [ 7, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 302808, root := ~, start := 298889, stop := 302807 ), rec( content := "\n\n ", count := [ 7, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "BlissCanonicalL\ abelling" ), content := 0, count := [ 7, 2, 7, 2 ], mathmode := "Text", name := "Attr", next := 302903, root := ~, start := 302829, stop := 302902 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "NautyCanonicalL\ abelling" ), content := 0, count := [ 7, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 302980, root := ~, start := 302906, stop := 302979 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion, or a list of two permutations.", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "Returns", next := 303047, root := ~, start := 302983, stop := 303046 ), rec( attributes := rec( ), content := [ rec( content := "\n A f\ unction ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "M", next := 303090, root := ~, start := 303079, stop := 303089 ), rec( content := " that ma\ ps a digraph to a digraph is a ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical\n representative map", count := [ 7, 2, 7, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "E", next := 303170, root := ~, start := 303129, stop := 303169 ), rec( content := " if the \ following two conditions hold for all\n digraphs ", count := [ 7, 2, 7, 5 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "M", next := 303237, root := ~, start := 303229, stop := 303236 ), rec( content := " and ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "M", next := 303250, root := ~, start := 303242, stop := 303249 ), rec( content := ": ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 6 ], mathmode := "Text", name := "P", next := 303256, root := ~, start := 303252, stop := 303255 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 7, 8 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "\\rho(G)", count := [ 7, 2, 7, 8 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 8 ], mathmode := "Tex\ t", name := "M", next := 303304, root := ~, start := 303290, stop := 303303 ), rec( cont\ ent := " and ", count := [ 7, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count :\ = [ 7, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 7, 8 ], mathmode := "Text", name := "M", next := 303317, root := ~, star\ t := 303309, stop := 303316 ), rec( content := " are isomorphic as digraphs; a\ nd\n ", count := [ 7, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 8 ], name := "Item", next := 303363, root := ~, start := 303\ 275, stop := 303362 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\rho(G)=\\\ rho(H)", count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "M", next := 3\ 03407, root := ~, start := 303385, stop := 303406 ), rec( content := " if and \ only if ", count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count :\ = [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 7, 10 ], mathmode := "Text", name := "M", next := 303431, root := ~ , star\ t := 303423, stop := 303430 ), rec( content := " and ", count := [ 7, 2, 7, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "H", count := [ 7, 2, 7, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 10 ], mathmode := \ "Text", name := "M", next := 303444, root := ~, start := 303436, stop := 30344\ 3 ), rec( content := " are\n isomorphic as digraphs.\n ", count :=\ [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 10 ], name := "Ite\ m", next := 303494, root := ~, start := 303370, stop := 303493 ) ], count := [ 7, 2, 7, 7 ], mathmode := "Text", name := "List", next := 303506, root := ~, start := 303262, stop := 303505 ), rec( content := "\n\n \ A ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical labelling", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "E", next := 303540, root := ~, start := 303514, stop := 303539 ), rec( content := " of a di\ graph ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 303562, root := ~, start := 303554, stop := 303561 ), rec( content := " (under \ ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 303581, root := ~, start := 303570, stop := 303580 ), rec( content := ") is an\\ n isomorphism of ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 303616, root := ~, start := 303608, stop := 303615 ), rec( content := " onto it\ s ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical representative", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "E", next := 303657, root := ~, start := 303626, stop := 303656 ), rec( content := ",\n " , count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho(G)", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 303677, root := ~, start := 303663, stop := 303676 ), rec( content := ". See " , count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "Ref", next := 303739, root := ~, start := 303684, stop := 303738 ), rec( content := "\n fo\ r more information about isomorphisms of digraphs. ", count := [ 7, 2, 7, 12 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "P", next := 303801, root := ~, start := 303797, stop := 303800 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "C", next := 303837, root := ~, start := 303807, stop := 303836 ), rec( content := " returns\ a canonical labelling of the digraph\n ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "A", next := 303901, root := ~, start := 303887, stop := 303900 ), rec( content := " found u\ sing ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila and Petteri Kaski.\n ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyCanonicalLabelling", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "C", next := 303993, root := ~, start := 303963, stop := 303992 ), rec( content := " returns\ a canonical labelling of the digraph\n ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "A", next := 304057, root := ~, start := 304043, stop := 304056 ), rec( content := " found u\ sing ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by Bren\ dan McKay and Adolfo Piperno.\n Note that the canonical labellings returned\ by ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " and ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " are not\ \n usually the same (and may depend of the version used).", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "P", next := 304256, root := ~, start := 304252, stop := 304255 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "C", next := 304292, root := ~, start := 304262, stop := 304291 ), rec( content := " can onl\ y be computed if ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "A", next := 304331, root := ~, start := 304317, stop := 304330 ), rec( content := " has\n \ no multiple edges; see ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "Ref", next := 304392, root := ~, start := 304363, stop := 304391 ), rec( content := ". ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 15 ], mathmode := "Text", name := "P", next := 304398, root := ~, start := 304394, stop := 304397 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := DigraphFromDiSparse6String(\".ImNS_AiB?qRN\");\n\ngap> BlissCanonicalLabelling(digraph1);\n(1\ ,9,5,7)(3,6,4,10)\ngap> p := (1, 2, 7, 5)(3, 9)(6, 10, 8);;\ngap> digraph2 := \ OnDigraphs(digraph1, p);\n\ngap> \ digraph1 = digraph2;\nfalse\ngap> OnDigraphs(digraph1, BlissCanonicalLabelling\ (digraph1)) =\n> OnDigraphs(digraph2, BlissCanonicalLabelling(digraph2));\n\ true", count := [ 7, 2, 7, 16 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 16 ], mathmode := "Text", name := "Example", next := 304894, root := ~, start := 304404, stop := 304893 ), rec( content := "\n ", count := [ 7, 2, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "Description", next := 304911, root := ~, start := 303050, stop := 304910 ) ], count := [ 7, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 304925, root := ~, start := 302814, stop := 304924 ), rec( content := "\n\n ", count := [ 7, 2, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, colours\ ", Label := "for a digraph and a list", Name := "BlissCanonicalLabelling" ), content := 0, count := [ 7, 2, 8, 2 ], mathmode := "Text", name := "Oper", next := 305044, root := ~, start := 304946, stop := 305043 ), rec( attributes := rec( Arg := "digraph, colours\ ", Label := "for a digraph and a list", Name := "NautyCanonicalLabelling" ), content := 0, count := [ 7, 2, 8, 4 ], mathmode := "Text", name := "Oper", next := 305145, root := ~, start := 305047, stop := 305144 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion.", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "Returns", next := 305181, root := ~, start := 305148, stop := 305180 ), rec( attributes := rec( ), content := [ rec( content := "\n A f\ unction ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "M", next := 305224, root := ~, start := 305213, stop := 305223 ), rec( content := " that ma\ ps a coloured digraph to a coloured digraph is\n a ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical representative map", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "E", next := 305320, root := ~, start := 305285, stop := 305319 ), rec( content := " if the \ following two conditions hold\n for all coloured digraphs ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "M", next := 305396, root := ~, start := 305388, stop := 305395 ), rec( content := " and ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "M", next := 305409, root := ~, start := 305401, stop := 305408 ), rec( content := ":\n\n \ ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 8, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "\\rho(G)", count := [ 7, 2, 8, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 7 ], mathmode := "Tex\ t", name := "M", next := 305458, root := ~, start := 305444, stop := 305457 ), rec( cont\ ent := " and ", count := [ 7, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count :\ = [ 7, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 8, 7 ], mathmode := "Text", name := "M", next := 305471, root := ~, star\ t := 305463, stop := 305470 ), rec( content := " are isomorphic as coloured di\ graphs; and\n ", count := [ 7, 2, 8, 7 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 7, 2, 8, 7 ], name := "Item", next := 305526, root := ~, start := 305\ 429, stop := 305525 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\rho(G)=\\\ rho(H)", count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "M", next := 305\ 570, root := ~, start := 305548, stop := 305569 ), rec( content := " if and on\ ly if ", count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count := \ [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 2, 8, 9 ], mathmode := "Text", name := "M", next := 305594, root := ~, star\ t := 305586, stop := 305593 ), rec( content := " and ", count := [ 7, 2, 8, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "H", count := [ 7, 2, 8, 9 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 9 ], mathmode := "Text", name\ := "M", next := 305607, root := ~, start := 305599, stop := 305606 ), rec( cont\ ent := " are\n isomorphic as coloured digraphs.\n ", count := [ 7,\ 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 9 ], name := "Ite\ m", next := 305666, root := ~, start := 305533, stop := 305665 ) ], count := [ 7, 2, 8, 6 ], mathmode := "Text", name := "List", next := 305678, root := ~, start := 305416, stop := 305677 ), rec( content := "\n\n \ A ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical labelling", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "E", next := 305712, root := ~, start := 305686, stop := 305711 ), rec( content := " of a co\ loured digraph ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 305743, root := ~, start := 305735, stop := 305742 ), rec( content := " (under\\ n ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 305766, root := ~, start := 305755, stop := 305765 ), rec( content := ") is an \ isomorphism of ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 305797, root := ~, start := 305789, stop := 305796 ), rec( content := " onto it\ s ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical\n representative", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "E", next := 305844, root := ~, start := 305807, stop := 305843 ), rec( content := ", ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho(G)", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 305860, root := ~, start := 305846, stop := 305859 ), rec( content := ". See " , count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "Ref", next := 305950, root := ~, start := 305867, stop := 305949 ), rec( content := " for mor\ e information about\n isomorphisms of coloured digraphs. ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "P", next := 306021, root := ~, start := 306017, stop := 306020 ), rec( content := "\n\n \ A coloured digraph can be specified by its underlying digraph ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 306103, root := ~, start := 306089, stop := 306102 ), rec( content := "\n an\ d its colouring ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 306140, root := ~, start := 306126, stop := 306139 ), rec( content := ". Let " , count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "C", next := 306155, root := ~, start := 306147, stop := 306154 ), rec( content := " be the \ number of vertices of\n ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 306203, root := ~, start := 306189, stop := 306202 ), rec( content := ". The co\ louring ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 306233, root := ~, start := 306219, stop := 306232 ), rec( content := " may hav\ e one of the following\n two forms: ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 13 ], mathmode := "Text", name := "P", next := 306283, root := ~, start := 306279, stop := 306282 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 8, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "n", count := [ 7, 2, 8, 15 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ], mathm\ ode := "Text", name := "C", next := 306335, root := ~, start := 306327, stop :\ = 306334 ), rec( content := " integers, where ", count := [ 7, 2, 8, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 8, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ], mathmode := "Te\ xt", name := "A", next := 306366, root := ~, start := 306352, stop := 306365 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 8, 15 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15\ ], mathmode := "Text", name := "C", next := 306376, root := ~, start := 30636\ 6, stop := 306375 ), rec( content := " is the\n colour of vertex ", coun\ t := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 8, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ] , math\ mode := "Text", name := "C", next := 306417, root := ~, start := 306409, stop \ := 306416 ), rec( content := ", using the colours ", count := [ 7, 2, 8, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 8, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ], mathmode := "T\ ext", name := "C", next := 306452, root := ~, start := 306437, stop := 306451 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 8, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "C", next\ := 306486, root := ~, start := 306470, stop := 306485 ), rec( content := "; o\ r\n ", count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 15 ], name := "Item", next := 306504, root := ~, start := 306\ 302, stop := 306503 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 8, 1\ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, \ 17 ], mathmode := "Text", name := "A", next := 306617, root := ~, start := 306\ 603, stop := 306616 ), rec( content := ")", count := [ 7, 2, 8, 17 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], mathmod\ e := "Text", name := "C", next := 306622, root := ~, start := 306584, stop := \ 306621 ), rec( content := ", such that\n ", count := [ 7, 2, 8, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 8, 17 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], mathmode := "Te\ xt", name := "A", next := 306656, root := ~, start := 306642, stop := 306655 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 8, 17 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17\ ], mathmode := "Text", name := "C", next := 306666, root := ~, start := 30665\ 6, stop := 306665 ), rec( content := " is the list of all vertices with colour\ \n ", count := [ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 8, 17 ], mathmode := "Text", name := "C", next := 306723, root := ~ , star\ t := 306715, stop := 306722 ), rec( content := ".\n ", count := [ 7, 2, 8\ , 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], name := "Ite\ m", next := 306738, root := ~, start := 306511, stop := 306737 ) ], count := [ 7, 2, 8, 14 ], mathmode := "Text", name := "List", next := 306750, root := ~, start := 306289, stop := 306749 ), rec( content := "\n\n \ If ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 306773, root := ~, start := 306759, stop := 306772 ), rec( content := " and ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 306792, root := ~, start := 306778, stop := 306791 ), rec( content := " togethe\ r form a coloured digraph,\n ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "C", next := 306861, root := ~, start := 306831, stop := 306860 ), rec( content := " returns\ a canonical labelling of the digraph\n ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 306925, root := ~, start := 306911, stop := 306924 ), rec( content := " found u\ sing ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila and Petteri Kaski.\n Similarly, ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyCanonicalLabelling", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "C", next := 307028, root := ~, start := 306998, stop := 307027 ), rec( content := " returns\ a canonical labelling of\n the digraph ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 307092, root := ~, start := 307078, stop := 307091 ), rec( content := " found u\ sing ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by Bren\ dan McKay and Adolfo\n Piperno. Note that the canonical labellings returne\ d by ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " and\n \ ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " are not\ usually the same (and may depend of the version used).", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "P", next := 307292, root := ~, start := 307288, stop := 307291 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "C", next := 307328, root := ~, start := 307298, stop := 307327 ), rec( content := " can onl\ y be computed if ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 307367, root := ~, start := 307353, stop := 307366 ), rec( content := " has\n \ no multiple edges; see ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "Ref", next := 307428, root := ~, start := 307399, stop := 307427 ), rec( content := ". The ca\ nonical\n labelling of ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 307475, root := ~, start := 307461, stop := 307474 ), rec( content := " is give\ n as a permutation of its vertices.\n The canonical representative of ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 307569, root := ~, start := 307555, stop := 307568 ), rec( content := " can be \ created from\n ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 307608, root := ~, start := 307594, stop := 307607 ), rec( content := " and its\ canonical labelling ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "C", next := 307645, root := ~, start := 307637, stop := 307644 ), rec( content := " by usin\ g the\n operation ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a perm", Oper := "OnDigraphs" ), content := 0, count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "Ref", next := 307731, root := ~, start := 307673, stop := 307730 ), rec( content := ":\n " , count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap> OnDigraphs(digraph, p);", count := [ 7, 2, 8, 21 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 21 ], mathmode := "Text", name := "Log", next := 307776, root := ~, start := 307737, stop := 307775 ), rec( content := "\n\n \ The colouring of the canonical representative can easily be\n constructed. \ A vertex ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 307876, root := ~, start := 307868, stop := 307875 ), rec( content := " (in ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "A", next := 307895, root := ~, start := 307881, stop := 307894 ), rec( content := ") has co\ lour ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 307916, root := ~, start := 307908, stop := 307915 ), rec( content := " if\n \ and only if the vertex ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ^ p", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 307959, root := ~, start := 307947, stop := 307958 ), rec( content := " (in the\ canonical representative) has\n colour ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 308017, root := ~, start := 308009, stop := 308016 ), rec( content := ", where \ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 308033, root := ~, start := 308025, stop := 308032 ), rec( content := " is the \ permutation of the canonical\n labelling that acts on the vertices of ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "A", next := 308127, root := ~, start := 308113, stop := 308126 ), rec( content := ". In par\ ticular, if\n ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "A", next := 308165, root := ~, start := 308151, stop := 308164 ), rec( content := " has the\ first form that is described above, then the\n colouring of the canonical \ representative is given by:\n\n ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap> List(DigraphVertices(digraph), i -> colours[i / p]);", count := [ 7, 2, 8, 23 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 23 ], mathmode := "Text", name := "Log", next := 308351, root := ~, start := 308283, stop := 308350 ), rec( content := "\n\n \ On the other hand, if ", count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "A", next := 308393, root := ~, start := 308379, stop := 308392 ), rec( content := " has the\ second form above, then the\n canonical representative has colouring:\n\n \ ", count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap> OnTuplesSets(colours, p);", count := [ 7, 2, 8, 25 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 25 ], mathmode := "Text", name := "Log", next := 308520, root := ~, start := 308479, stop := 308519 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 27 ], mathmode := "Text", name := "P", next := 308530, root := ~, start := 308526, stop := 308529 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := DigraphFromDiSparse6String(\".ImNS_AiB?qRN\");\n\ngap> colours := [[1, 2, 8, 9, 10], [3, 4, 5,\ 6, 7]];;\ngap> p := BlissCanonicalLabelling(digraph, colours);\n(1,5,8,4,10,3\ ,9)(6,7)\ngap> OnDigraphs(digraph, p);\n\ngap> OnTuplesSets(colours, p);\n[ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10\ ] ]\ngap> colours := [1, 1, 1, 1, 2, 3, 1, 3, 2, 1];;\ngap> p := BlissCanonic\ alLabelling(digraph, colours);\n(1,6,9,7)(3,4,5,8,10)\ngap> OnDigraphs(digraph\ , p);\n\ngap> List(DigraphVertice\ s(digraph), i -> colours[i / p]);\n[ 1, 1, 1, 1, 1, 1, 2, 2, 3, 3 ]", count := [ 7, 2, 8, 28 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 28 ], mathmode := "Text", name := "Example", next := 309241, root := ~, start := 308536, stop := 309240 ), rec( content := "\n ", count := [ 7, 2, 8, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "Description", next := 309258, root := ~, start := 305184, stop := 309257 ) ], count := [ 7, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 309272, root := ~, start := 304931, stop := 309271 ), rec( content := "\n\n ", count := [ 7, 2, 8, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "BlissCanonicalD\ igraph" ), content := 0, count := [ 7, 2, 9, 2 ], mathmode := "Text", name := "Attr", next := 309343, root := ~, start := 309293, stop := 309342 ), rec( attributes := rec( Arg := "digraph", Name := "NautyCanonicalD\ igraph" ), content := 0, count := [ 7, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 309396, root := ~, start := 309346, stop := 309395 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Returns", next := 309428, root := ~, start := 309399, stop := 309427 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ attribute ", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "C", next := 309493, root := ~, start := 309463, stop := 309492 ), rec( content := " returns\ the canonical\n representative found by applying ", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissCa\ nonicalLabelling", Label := "for a digraph" ), content := 0, count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Ref", next := 309618, root := ~, start := 309553, stop := 309617 ), rec( content := ". The di\ graph returned is canonical in the sense\n that\n\n ", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 9, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "BlissCanonicalDigraph(", count := [ 7, 2, 9, 7 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), c\ ontent := [ rec( content := "digraph", count := [ 7, 2, 9, 7 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], mathmode := "T\ ext", name := "A", next := 309748, root := ~, start := 309734, stop := 309747 \ ), rec( content := ")", count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ C", next := 309753, root := ~, start := 309709, stop := 309752 ), rec( content\ := " and ", count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 7, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 9, 7 ], mathmode := "Text", name := "A", next := 309772, root := \ ~, start := 309758, stop := 309771 ), rec( content := " are\n isomorphi\ c as digraphs; and\n ", count := [ 7, 2, 9, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], name := "Item", next := 3098\ 26, root := ~, start := 309694, stop := 309825 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n If ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "gr", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "C", next := 30986\ 0, root := ~, start := 309851, stop := 309859 ), rec( content := " is any digr\ aph then ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "BlissCanoni\ calDigraph(gr)", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "C", next\ := 309913, root := ~, start := 309881, stop := 309912 ), rec( content := "\n \ and ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "BlissCanoni\ calDigraph(", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph"\ , count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ]\ , count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "A", next := 309965, ro\ ot := ~, start := 309951, stop := 309964 ), rec( content := ")", count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "C", next := 309970, root := ~, start \ := 309926, stop := 309969 ), rec( content := " are equal if and only\n \ if ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "gr", count := [ \ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 2, 9, 9 ], mathmode := "Text", name := "C", next := 310013, root := ~, star\ t := 310004, stop := 310012 ), rec( content := " and ", count := [ 7, 2, 9, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "digraph", count := [ 7, 2, 9, 9 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 9 ], mathmode := "Text\ ", name := "A", next := 310032, root := ~, start := 310018, stop := 310031 ), rec( cont\ ent := " are isomorphic as digraphs.\n ", count := [ 7, 2, 9, 9 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 9 ], name := "Ite\ m", next := 310074, root := ~, start := 309833, stop := 310073 ) ], count := [ 7, 2, 9, 6 ], mathmode := "Text", name := "List", next := 310086, root := ~, start := 309681, stop := 310085 ), rec( content := "\n\n \ Analogously, the attribute ", count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyCanonicalLabelling", count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "C", next := 310149, root := ~, start := 310119, stop := 310148 ), rec( content := " returns\ the\n canonical representative found by applying ", count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyCa\ nonicalLabelling", Label := "for a digraph" ), content := 0, count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "Ref", next := 310274, root := ~, start := 310209, stop := 310273 ), rec( content := ".\n " , count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "P", next := 310284, root := ~, start := 310280, stop := 310283 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "A", next := 310320, root := ~, start := 310306, stop := 310319 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := Digraph([[1], [2, 3], [3], [1, 2, 3]]);\n\ngap> canon := BlissCanonicalDigraph(digraph);\n\ngap> OutNeighbours(canon);\n[ [ 1 ], \ [ 2 ], [ 3, 2 ], [ 1, 3, 2 ] ]\n", count := [ 7, 2, 9, 13 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 13 ], mathmode := "Text", name := "Example", next := 310713, root := ~, start := 310422, stop := 310712 ), rec( content := "\n ", count := [ 7, 2, 9, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Description", next := 310730, root := ~, start := 309431, stop := 310729 ) ], count := [ 7, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 310744, root := ~, start := 309278, stop := 310743 ), rec( content := "\n\n ", count := [ 7, 2, 9, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphGroup" ) , content := 0, count := [ 7, 2, 10, 2 ], mathmode := "Text", name := "Attr", next := 310806, root := ~, start := 310765, stop := 310805 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "Returns", next := 310848, root := ~, start := 310809, stop := 310847 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 310886, root := ~, start := 310872, stop := 310885 ), rec( content := " is immu\ table and was created knowing a subgroup of\n its automorphism group, then \ this group is stored in the attribute\n ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGroup", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "C", next := 311032, root := ~, start := 311013, stop := 311031 ), rec( content := ". If ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 311051, root := ~, start := 311037, stop := 311050 ), rec( content := " is muta\ ble, or was not created\n knowing a subgroup of its automorphism group, the\ n ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGroup", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "C", next := 311157, root := ~, start := 311138, stop := 311156 ), rec( content := "\n re\ turns the entire automorphism group of ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 311217, root := ~, start := 311203, stop := 311216 ), rec( content := ". \n \ Note that if ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 311254, root := ~, start := 311240, stop := 311253 ), rec( content := " is muta\ ble, then the automorphism group is\n recomputed every time this function i\ s called.\n ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "P", next := 311357, root := ~, start := 311353, stop := 311356 ), rec( content := "\n\n \ Note that certain other constructor operations such as ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CayleyD\ igraph" ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "Ref", next := 311452, root := ~, start := 311419, stop := 311451 ), rec( content := ", ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Biparti\ teDoubleDigraph" ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "Ref", next := 311490, root := ~, start := 311454, stop := 311489 ), rec( content := ", and ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DoubleD\ igraph" ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "Ref", next := 311529, root := ~, start := 311496, stop := 311528 ), rec( content := ", may no\ t require a group as one of the arguments,\n but use the standard construct\ or method using a group, and hence set the\n ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGroup", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "C", next := 311680, root := ~, start := 311661, stop := 311679 ), rec( content := " attribu\ te for the resulting digraph.\n\n ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> n := 4;;\ngap> adj := function(x, y)\n> return (((x - y) mod n) = 1\ ) or (((x - y) mod n) = n - 1);\n> end;;\ngap> group := CyclicGroup(IsPermG\ roup, n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph, group, [1\ .. n], \\^, adj);\n\ngap> HasDigrap\ hGroup(D);\nfalse\ngap> DigraphGroup(D);\nGroup([ (2,4), (1,2)(3,4) ])\ngap> A\ utomorphismGroup(D);\nGroup([ (2,4), (1,2)(3,4) ])\ngap> D := Digraph(group, [\ 1 .. n], \\^, adj);\n\ngap> HasDig\ raphGroup(D);\ntrue\ngap> DigraphGroup(D);\nGroup([ (1,2,3,4) ])\ngap> D := Do\ ubleDigraph(D);\n\ngap> HasDigrap\ hGroup(D);\ntrue\ngap> DigraphGroup(D);\nGroup([ (1,2,3,4)(5,6,7,8), (1,5)(2,6\ )(3,7)(4,8) ])\ngap> AutomorphismGroup(D) =\n> Group([(6, 8), (5, 7), (4, 6), \ (3, 5), (2, 4),\n> (1, 2)(3, 4)(5, 6)(7, 8)]);\ntrue\ngap> D := Digraph\ ([[2, 3], [], []]);\n\ngap> HasDig\ raphGroup(D);\nfalse\ngap> HasAutomorphismGroup(D);\nfalse\ngap> DigraphGroup(\ D);\nGroup([ (2,3) ])\ngap> HasAutomorphismGroup(D);\ntrue\ngap> group := Dihe\ dralGroup(8);\n\ngap> D := CayleyDigraph\ (group);\n\ngap> HasDigraphGroup(\ D);\ntrue\ngap> GeneratorsOfGroup(DigraphGroup(D));\n[ (1,2)(3,8)(4,6)(5,7), (\ 1,3,4,7)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) ]", count := [ 7, 2, 10, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 10, 5 ], mathmode := "Text", name := "Example", next := 313150, root := ~, start := 311723, stop := 313149 ), rec( content := "\n ", count := [ 7, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 313167, root := ~, start := 310851, stop := 313166 ) ], count := [ 7, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 313181, root := ~, start := 310750, stop := 313180 ), rec( content := "\n\n ", count := [ 7, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphOrbits" ), content := 0, count := [ 7, 2, 11, 2 ], mathmode := "Text", name := "Attr", next := 313244, root := ~, start := 313202, stop := 313243 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable list of lists of integers.\n ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Returns", next := 313313, root := ~, start := 313247, stop := 313312 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOrbits", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "C", next := 313354, root := ~, start := 313334, stop := 313353 ), rec( content := " returns\ the orbits of the action of the\n ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Ref", next := 313425, root := ~, start := 313399, stop := 313424 ), rec( content := " on the \ set of vertices of ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "A", next := 313466, root := ~, start := 313452, stop := 313465 ), rec( content := ".\n\n \ ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := Group([(2, 3)(7, 8, 9), (1, 2, 3)(4, 5, 6)(8, 9)]);;\ngap> D := Edg\ eOrbitsDigraph(G, [1, 2]);\n\ngap>\ DigraphOrbits(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]\ngap> D := Digra\ phMutableCopy(D);\n\ngap> DigraphOrb\ its(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6, 7, 8, 9 ] ]\n", count := [ 7, 2, 11, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 11, 4 ], mathmode := "Text", name := "Example", next := 313855, root := ~, start := 313473, stop := 313854 ), rec( content := "\n ", count := [ 7, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Description", next := 313872, root := ~, start := 313316, stop := 313871 ) ], count := [ 7, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 313886, root := ~, start := 313187, stop := 313885 ), rec( content := "\n\n ", count := [ 7, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphOrbitRep\ s" ), content := 0, count := [ 7, 2, 12, 2 ], mathmode := "Text", name := "Attr", next := 313952, root := ~, start := 313907, stop := 313951 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable list of integers.\n ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Returns", next := 314012, root := ~, start := 313955, stop := 314011 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOrbitReps", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "C", next := 314056, root := ~, start := 314033, stop := 314055 ), rec( content := " returns\ a list of orbit representatives\n of the action of the ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Ref", next := 314148, root := ~, start := 314122, stop := 314147 ), rec( content := " on the \ set of vertices of\n ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "A", next := 314193, root := ~, start := 314179, stop := 314192 ), rec( content := ".\n\n \ ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CayleyDigraph(AlternatingGroup(4));\n\ngap> DigraphOrbitReps(D);\n[ 1 ]\ngap> D := DigraphMutableCo\ py(D);\n\ngap> DigraphOrbitReps(D)\ ;\n[ 1 ]\ngap> D := DigraphFromDigraph6String(\"&IGO??S?`?_@?a?CK?O\");\n\ngap> DigraphOrbitReps(D);\n[ 1, 2, \ 3, 4, 5, 6, 7, 8, 9, 10 ]\ngap> DigraphOrbitReps(DigraphMutableCopy(D));\n[ 1,\ 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\n", count := [ 7, 2, 12, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 4 ], mathmode := "Text", name := "Example", next := 314714, root := ~, start := 314200, stop := 314713 ), rec( content := "\n ", count := [ 7, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 314731, root := ~, start := 314015, stop := 314730 ) ], count := [ 7, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 314745, root := ~, start := 313892, stop := 314744 ), rec( content := "\n\n ", count := [ 7, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSchreier\ Vector" ), content := 0, count := [ 7, 2, 13, 2 ], mathmode := "Text", name := "Attr", next := 314816, root := ~, start := 314766, stop := 314815 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable list of integers.\n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Returns", next := 314876, root := ~, start := 314819, stop := 314875 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphSchreierVector", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 314925, root := ~, start := 314897, stop := 314924 ), rec( content := " returns\ the so-called ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ chreier vector", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "E", next := 314970, root := ~, start := 314948, stop := 314969 ), rec( content := "\n of\ the action of the ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Ref", next := 315022, root := ~, start := 314996, stop := 315021 ), rec( content := " on the \ set of vertices of\n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "A", next := 315067, root := ~, start := 315053, stop := 315066 ), rec( content := ". The Sc\ hreier vector is a list ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ch", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 315109, root := ~, start := 315099, stop := 315108 ), rec( content := " of inte\ gers with\n length ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "A", next := 3151\ 73, root := ~, start := 315159, stop := 315172 ), rec( content := ")", count := [ 7, 2, 13, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 315178, root := ~, start := 315138, stop := 315177 ), rec( content := " where:\\ n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "sch[i] < 0:", count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "C", next\ := 315230, root := ~, start := 315209, stop := 315229 ) ], count := [ 7, 2, 13, 4 ], name := "Mar\ k", next := 315237, root := ~, start := 315203, stop := 315236 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n implies that ", count := [ \ 7, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 13, 5 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 5 ], math\ mode := "Text", name := "C", next := 315280, root := ~, start := 315272, stop \ := 315279 ), rec( content := " is an orbit representative and\n ", coun\ t := [ 7, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphOrbitReps(", count := \ [ 7, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attri\ butes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 13,\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13\ , 5 ], mathmode := "Text", name := "A", next := 315354, root := ~, start := 31\ 5340, stop := 315353 ), rec( content := ")[-sch[i]] = i", count := [ 7, 2, 13,\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13\ , 5 ], mathmode := "Text", name := "C", next := 315372, root := ~, start := 31\ 5320, stop := 315371 ), rec( content := ".\n ", count := [ 7, 2, 13, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 5 ], name := "Ite\ m", next := 315387, root := ~, start := 315244, stop := 315386 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "sch[i] > 0:", count := [ 7, 2, 13, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 13, 6 ], mathmode := "Text", name := "C", next\ := 315422, root := ~, start := 315401, stop := 315421 ) ], count := [ 7, 2, 13, 6 ], name := "Mar\ k", next := 315429, root := ~, start := 315395, stop := 315428 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n implies that ", count := [ \ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "i / gens[sch[i]]", count := [ 7,\ 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7\ , 2, 13, 7 ], mathmode := "Text", name := "C", next := 315487, root := ~, star\ t := 315464, stop := 315486 ), rec( content := "\n is one step closer t\ o the root (or representative) of the tree, where\n ", count := [ 7, 2,\ 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "gens", count := [ 7, 2, 13, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 7 ], math\ mode := "Text", name := "C", next := 315585, root := ~, start := 315574, stop \ := 315584 ), rec( content := " is the generators of ", count := [ 7, 2, 13, 7 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphGroup(", count := [ 7, 2, 13, 7 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "digraph", count := [ 7, 2, 13, 7 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 7 ], mathmode := "Text", \ name := "A", next := 315637, root := ~, start := 315623, stop := 315636 ), rec\ ( content := ")", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "C", next\ := 315642, root := ~, start := 315607, stop := 315641 ), rec( content := ".\n\ ", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 13, 7 ], name := "Item", next := 315657, root := ~, start := 315\ 436, stop := 315656 ) ], count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "List", next := 315669, root := ~, start := 315190, stop := 315668 ), rec( content := "\n\n \ ", count := [ 7, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> n := 4;;\ngap> adj := function(x, y)\n> return (((x - y) mod n) = 1\ ) or (((x - y) mod n) = n - 1);\n> end;;\ngap> group := CyclicGroup(IsPermG\ roup, n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph, group, [1\ .. n], \\^, adj);\n\ngap> sch := Di\ graphSchreierVector(D);\n[ -1, 2, 2, 1 ]\ngap> D := CayleyDigraph(AlternatingG\ roup(4));\n\ngap> sch := Digraph\ SchreierVector(D);\n[ -1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1 ]\ngap> DigraphOrbit\ Reps(D);\n[ 1 ]\ngap> gens := GeneratorsOfGroup(DigraphGroup(D));\n[ (1,5,7)(2\ ,4,8)(3,6,9)(10,11,12), (1,2,3)(4,7,10)(5,9,11)(6,8,12) ]\ngap> 10 / gens[sch[\ 10]];\n7\ngap> 7 / gens[sch[7]];\n5\ngap> 5 / gens[sch[5]];\n1", count := [ 7, 2, 13, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 13, 10 ], mathmode := "Text", name := "Example", next := 316446, root := ~, start := 315675, stop := 316445 ), rec( content := "\n ", count := [ 7, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Description", next := 316463, root := ~, start := 314879, stop := 316462 ) ], count := [ 7, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 316477, root := ~, start := 314751, stop := 316476 ), rec( content := "\n\n ", count := [ 7, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, v", Name := "DigraphStabiliz\ er" ), content := 0, count := [ 7, 2, 14, 2 ], mathmode := "Text", name := "Oper", next := 316547, root := ~, start := 316498, stop := 316546 ), rec( attributes := rec( ), content := [ rec( content := "\n A p\ ermutation group.\n ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Returns", next := 316597, root := ~, start := 316550, stop := 316596 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphStabilizer", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "C", next := 316642, root := ~, start := 316618, stop := 316641 ), rec( content := " returns\ the stabilizer of the vertex ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "A", next := 316688, root := ~, start := 316680, stop := 316687 ), rec( content := "\n un\ der of the action of the ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Ref", next := 316746, root := ~, start := 316720, stop := 316745 ), rec( content := " on the \ set of\n vertices of ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "A", next := 316791, root := ~, start := 316777, stop := 316790 ), rec( content := ".\n\n \ ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> Digr\ aphStabilizer(D, 2);\nGroup(())\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\nga\ p> DigraphStabilizer(D, 2);\nGroup(())\n", count := [ 7, 2, 14, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 14, 4 ], mathmode := "Text", name := "Example", next := 317167, root := ~, start := 316798, stop := 317166 ), rec( content := "\n ", count := [ 7, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 317184, root := ~, start := 316600, stop := 317183 ) ], count := [ 7, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 317198, root := ~, start := 316483, stop := 317197 ), rec( content := "\n\n ", count := [ 7, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Label := "for digraphs", Name := "IsIsomorphicDigraph" ), content := 0, count := [ 7, 2, 15, 2 ], mathmode := "Text", name := "Oper", next := 317299, root := ~, start := 317219, stop := 317298 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "K", next := 317322, root := ~, start := 317311, stop := 317321 ), rec( content := " or ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "K", next := 317338, root := ~, start := 317326, stop := 317337 ), rec( content := ".", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Returns", next := 317349, root := ~, start := 317302, stop := 317348 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "K", next := 317404, root := ~, start := 317393, stop := 317403 ), rec( content := " if ther\ e exists an isomorphism from the\n digraph ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "A", next := 317472, root := ~, start := 317457, stop := 317471 ), rec( content := " to the \ digraph ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "A", next := 317503, root := ~, start := 317488, stop := 317502 ), rec( content := ". See " , count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Ref", next := 317571, root := ~, start := 317510, stop := 317570 ), rec( content := " for mor\ e information\n about isomorphisms of digraphs. ", count := [ 7, 2, 15, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "P", next := 317633, root := ~, start := 317629, stop := 317632 ), rec( content := "\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 15, 4 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 15, 4 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 15, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 15, 4 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Ref", next := 317941, root := ~, start := 317911, stop := 317940 ), rec( content := ", and ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Ref", next := 317977, root := ~, start := 317947, stop := 317976 ), rec( content := ".\n\n \ ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := CycleDigraph(4);\n\ \ngap> digraph2 := CycleDigraph(5);\n\ \ngap> IsIsomorphicDigraph(digraph1, digraph2);\nfalse\ngap> digraph2 := Digra\ phReverse(digraph1);\n\ngap> IsIso\ morphicDigraph(digraph1, digraph2);\ntrue\ngap> digraph1 := Digraph([[3], [], \ []]);\n\ngap> digraph2 := Digraph([\ [], [], [2]]);\n\ngap> IsIsomorphic\ Digraph(digraph1, digraph2);\ntrue", count := [ 7, 2, 15, 5 ], name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 15, 5 ], mathmode := "Text", name := "Example", next := 318579, root := ~, start := 317984, stop := 318578 ), rec( content := "\n ", count := [ 7, 2, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Description", next := 318596, root := ~, start := 317352, stop := 318595 ) ], count := [ 7, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 318610, root := ~, start := 317204, stop := 318609 ), rec( content := "\n\n ", count := [ 7, 2, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2, colours1, colours2", Label := "for digraphs and homogeneous lists", Name := "IsIsomorphicDig\ raph" ), content := 0, count := [ 7, 2, 16, 2 ], mathmode := "Text", name := "Oper", next := 318757, root := ~, start := 318631, stop := 318756 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "K", next := 318780, root := ~, start := 318769, stop := 318779 ), rec( content := " or ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "K", next := 318796, root := ~, start := 318784, stop := 318795 ), rec( content := ".", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Returns", next := 318807, root := ~, start := 318760, stop := 318806 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation tests for isomorphism of coloured digraphs. A coloured\n digra\ ph can be specified by its underlying digraph ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 318969, root := ~, start := 318954, stop := 318968 ), rec( content := " and its\ \n colouring ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 319007, root := ~, start := 318992, stop := 319006 ), rec( content := ". Let " , count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "C", next := 319022, root := ~, start := 319014, stop := 319021 ), rec( content := " be the \ number of vertices of\n ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 319071, root := ~, start := 319056, stop := 319070 ), rec( content := ". The co\ louring ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 319102, root := ~, start := 319087, stop := 319101 ), rec( content := " may hav\ e one of the following\n two forms:\n\n ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "n", count := [ 7, 2, 16, 5 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ], mathm\ ode := "Text", name := "C", next := 319199, root := ~, start := 319191, stop :\ = 319198 ), rec( content := " integers, where ", count := [ 7, 2, 16, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 16, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ], mathmode := "Te\ xt", name := "A", next := 319230, root := ~, start := 319216, stop := 319229 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 16, 5 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5\ ], mathmode := "Text", name := "C", next := 319240, root := ~, start := 31923\ 0, stop := 319239 ), rec( content := " is the\n colour of vertex ", coun\ t := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 16, 5 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ] , math\ mode := "Text", name := "C", next := 319281, root := ~, start := 319273, stop \ := 319280 ), rec( content := ", using the colours ", count := [ 7, 2, 16, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 16, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ], mathmode := "T\ ext", name := "C", next := 319316, root := ~, start := 319301, stop := 319315 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 16, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "C", next\ := 319350, root := ~, start := 319334, stop := 319349 ), rec( content := "; o\ r\n ", count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 16, 5 ], name := "Item", next := 319368, root := ~, start := 319\ 166, stop := 319367 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 16, \ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16,\ 7 ], mathmode := "Text", name := "A", next := 319481, root := ~, start := 319\ 467, stop := 319480 ), rec( content := ")", count := [ 7, 2, 16, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], mathmod\ e := "Text", name := "C", next := 319486, root := ~, start := 319448, stop := \ 319485 ), rec( content := ", such that\n ", count := [ 7, 2, 16, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 16, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], mathmode := "Te\ xt", name := "A", next := 319520, root := ~, start := 319506, stop := 319519 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 16, 7 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7\ ], mathmode := "Text", name := "C", next := 319530, root := ~, start := 31952\ 0, stop := 319529 ), rec( content := " is the list of all vertices with colour\ \n ", count := [ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 16, 7 ], mathmode := "Text", name := "C", next := 319587, root := ~ , star\ t := 319579, stop := 319586 ), rec( content := ".\n ", count := [ 7, 2, 1\ 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], name := "Ite\ m", next := 319602, root := ~, start := 319375, stop := 319601 ) ], count := [ 7, 2, 16, 4 ], mathmode := "Text", name := "List", next := 319614, root := ~, start := 319153, stop := 319613 ), rec( content := "\n\n \ If ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 319638, root := ~, start := 319623, stop := 319637 ), rec( content := " and ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 319658, root := ~, start := 319643, stop := 319657 ), rec( content := " are dig\ raphs without multiple edges,\n and ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 319719, root := ~, start := 319704, stop := 319718 ), rec( content := " and ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours2", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 319739, root := ~, start := 319724, stop := 319738 ), rec( content := " are col\ ourings of ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 319773, root := ~, start := 319758, stop := 319772 ), rec( content := "\n an\ d ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 319797, root := ~, start := 319782, stop := 319796 ), rec( content := ", respec\ tively, then this operation returns ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "K", next := 319852, root := ~, start := 319841, stop := 319851 ), rec( content := "\n if\ there exists an isomorphism between these two coloured digraphs. See\n ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "Ref", next := 320017, root := ~, start := 319934, stop := 320016 ), rec( content := " for mor\ e information about isomorphisms of coloured digraphs. ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "P", next := 320084, root := ~, start := 320080, stop := 320083 ), rec( content := "\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "URL\ ", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 16, 10 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 16, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 16, 10 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 16, 10 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Ref", next := 320392, root := ~, start := 320362, stop := 320391 ), rec( content := ", and ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Ref", next := 320428, root := ~, start := 320398, stop := 320427 ), rec( content := ".\n\n \ ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := ChainDigraph(4);\n\ \ngap> digraph2 := ChainDigraph(3);\n\ \ngap> IsIsomorphicDigraph(digraph1, digraph2,\n> [[1, 4], [2, 3]], [[1, 2], \ [3]]);\nfalse\ngap> digraph2 := DigraphReverse(digraph1);\n\ngap> IsIsomorphicDigraph(digraph1, digraph2,\n> [1\ , 1, 1, 1], [1, 1, 1, 1]);\ntrue\ngap> IsIsomorphicDigraph(digraph1, digraph2,\ \n> [1, 2, 2, 1], [1, 2, 2, 1]);\ntrue\ngap> IsIsomorphicDigraph(digraph1, di\ graph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfalse", count := [ 7, 2, 16, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 16, 11 ], mathmode := "Text", name := "Example", next := 321041, root := ~, start := 320435, stop := 321040 ), rec( content := "\n ", count := [ 7, 2, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 321058, root := ~, start := 318810, stop := 321057 ) ], count := [ 7, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 321072, root := ~, start := 318616, stop := 321071 ), rec( content := "\n\n ", count := [ 7, 2, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Label := "for digraphs", Name := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 17, 2 ], mathmode := "Text", name := "Oper", next := 321173, root := ~, start := 321093, stop := 321172 ), rec( attributes := rec( ), content := [ rec( content := " A permut\ ation, or a pair of permutations, or ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "K", next := 321242, root := ~, start := 321231, stop := 321241 ), rec( content := ".", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Returns", next := 321253, root := ~, start := 321176, stop := 321252 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns an isomorphism between the digraphs ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "A", next := 321348, root := ~, start := 321333, stop := 321347 ), rec( content := "\n an\ d ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "A", next := 321372, root := ~, start := 321357, stop := 321371 ), rec( content := " if one \ exists, else this operation returns ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "K", next := 321427, root := ~, start := 321416, stop := 321426 ), rec( content := ".\n " , count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "P", next := 321437, root := ~, start := 321433, stop := 321436 ), rec( content := "\n\n \ An ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ somorphism", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "E", next := 321464, root := ~, start := 321446, stop := 321463 ), rec( content := " from a \ digraph ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 321495, root := ~, start := 321480, stop := 321494 ), rec( content := " to a di\ graph\n ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 321528, root := ~, start := 321513, stop := 321527 ), rec( content := " is a bi\ jection ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 321552, root := ~, start := 321544, stop := 321551 ), rec( content := " from th\ e vertices of\n ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 321593, root := ~, start := 321578, stop := 321592 ), rec( content := " to the \ vertices of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 321628, root := ~, start := 321613, stop := 321627 ), rec( content := " with th\ e following\n property: for all vertices ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 321687, root := ~, start := 321679, stop := 321686 ), rec( content := " and ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 321700, root := ~, start := 321692, stop := 321699 ), rec( content := " of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 321719, root := ~, start := 321704, stop := 321718 ), rec( content := ",\n " , count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 321738, root := ~, start := 321725, stop := 321737 ), rec( content := " is an e\ dge of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 321768, root := ~, start := 321753, stop := 321767 ), rec( content := " if and \ only if ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i ^ p, j\n ^ p]", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 321811, root := ~, start := 321784, stop := 321810 ), rec( content := " is an e\ dge of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 321841, root := ~, start := 321826, stop := 321840 ), rec( content := ". ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "P", next := 321847, root := ~, start := 321843, stop := 321846 ), rec( content := "\n\n \ If there exists such an isomorphism, then this operation returns one.\n The\ form of this isomorphism is a permutation ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "C", next := 321981, root := ~, start := 321973, stop := 321980 ), rec( content := " of the \ vertices of\n ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "A", next := 322020, root := ~, start := 322005, stop := 322019 ), rec( content := " such th\ at ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "P", next := 322035, root := ~, start := 322031, stop := 322034 ), rec( content := "\n\n \ ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph1", count := [ 7, 2, 17, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "A", next := 322070, root := ~, start := 322055, stop := 32206\ 9 ), rec( content := ", p) = digraph2", count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "C", next := 322089, root := ~, start := 322041, stop := 322088 ), rec( content := ".\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 17, 6 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 17, 6 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 17, 6 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 17, 6 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Ref", next := 322398, root := ~, start := 322368, stop := 322397 ), rec( content := ", and ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Ref", next := 322434, root := ~, start := 322404, stop := 322433 ), rec( content := ".\n\n \ ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := CycleDigraph(4);\n\ \ngap> digraph2 := CycleDigraph(5);\n\ \ngap> IsomorphismDigraphs(digraph1, digraph2);\nfail\ngap> digraph1 := Comple\ teBipartiteDigraph(10, 5);\n\ngap> digraph2 := CompleteBipartiteDigraph(5, 10);\n\ngap> p := I\ somorphismDigraphs(digraph1, digraph2);\n(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10\ ,15)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\n", count := [ 7, 2, 17, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 17, 7 ], mathmode := "Text", name := "Example", next := 323057, root := ~, start := 322441, stop := 323056 ), rec( content := "\n ", count := [ 7, 2, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 323074, root := ~, start := 321256, stop := 323073 ) ], count := [ 7, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 323088, root := ~, start := 321078, stop := 323087 ), rec( content := "\n\n ", count := [ 7, 2, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2, colours1, colours2", Label := "for digraphs and homogeneous lists", Name := "IsomorphismDigr\ aphs" ), content := 0, count := [ 7, 2, 18, 2 ], mathmode := "Text", name := "Oper", next := 323235, root := ~, start := 323109, stop := 323234 ), rec( attributes := rec( ), content := [ rec( content := " A permut\ ation, or ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "K", next := 323277, root := ~, start := 323266, stop := 323276 ), rec( content := ".", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Returns", next := 323288, root := ~, start := 323238, stop := 323287 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation searches for an isomorphism between coloured digraphs. A\n col\ oured digraph can be specified by its underlying digraph ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 323461, root := ~, start := 323446, stop := 323460 ), rec( content := "\n an\ d its colouring ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 323499, root := ~, start := 323484, stop := 323498 ), rec( content := ". Let " , count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "C", next := 323514, root := ~, start := 323506, stop := 323513 ), rec( content := " be the \ number of vertices\n of ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 323563, root := ~, start := 323548, stop := 323562 ), rec( content := ". The co\ louring ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 323594, root := ~, start := 323579, stop := 323593 ), rec( content := " may hav\ e one of the\n following two forms:\n\n ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "n", count := [ 7, 2, 18, 5 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ], mathm\ ode := "Text", name := "C", next := 323691, root := ~, start := 323683, stop :\ = 323690 ), rec( content := " integers, where ", count := [ 7, 2, 18, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 18, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ], mathmode := "Te\ xt", name := "A", next := 323722, root := ~, start := 323708, stop := 323721 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 18, 5 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5\ ], mathmode := "Text", name := "C", next := 323732, root := ~, start := 32372\ 2, stop := 323731 ), rec( content := " is the\n colour of vertex ", coun\ t := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 18, 5 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ] , math\ mode := "Text", name := "C", next := 323773, root := ~, start := 323765, stop \ := 323772 ), rec( content := ", using the colours ", count := [ 7, 2, 18, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 18, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ], mathmode := "T\ ext", name := "C", next := 323808, root := ~, start := 323793, stop := 323807 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 18, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "C", next\ := 323842, root := ~, start := 323826, stop := 323841 ), rec( content := "; o\ r\n ", count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 18, 5 ], name := "Item", next := 323860, root := ~, start := 323\ 658, stop := 323859 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 18, \ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18,\ 7 ], mathmode := "Text", name := "A", next := 323973, root := ~, start := 323\ 959, stop := 323972 ), rec( content := ")", count := [ 7, 2, 18, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], mathmod\ e := "Text", name := "C", next := 323978, root := ~, start := 323940, stop := \ 323977 ), rec( content := ", such that\n ", count := [ 7, 2, 18, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 18, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], mathmode := "Te\ xt", name := "A", next := 324012, root := ~, start := 323998, stop := 324011 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 18, 7 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7\ ], mathmode := "Text", name := "C", next := 324022, root := ~, start := 32401\ 2, stop := 324021 ), rec( content := " is the list of all vertices with colour\ \n ", count := [ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 18, 7 ], mathmode := "Text", name := "C", next := 324079, root := ~ , star\ t := 324071, stop := 324078 ), rec( content := ".\n ", count := [ 7, 2, 1\ 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], name := "Ite\ m", next := 324094, root := ~, start := 323867, stop := 324093 ) ], count := [ 7, 2, 18, 4 ], mathmode := "Text", name := "List", next := 324106, root := ~, start := 323645, stop := 324105 ), rec( content := "\n\n \ An ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ somorphism", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "E", next := 324133, root := ~, start := 324115, stop := 324132 ), rec( content := " between\ coloured digraphs is an isomorphism between\n the underlying digraphs that\ preserves the colourings. See ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "Ref", next := 324310, root := ~, start := 324250, stop := 324309 ), rec( content := " for mor\ e information\n about isomorphisms of digraphs. More precisely, let ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 324396, root := ~, start := 324388, stop := 324395 ), rec( content := " be an\n\ isomorphism of digraphs from the digraph ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 324463, root := ~, start := 324448, stop := 324462 ), rec( content := " (with c\ olouring\n ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 324499, root := ~, start := 324484, stop := 324498 ), rec( content := ") to the\ digraph ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 324531, root := ~, start := 324516, stop := 324530 ), rec( content := " (with c\ olouring\n ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 324567, root := ~, start := 324552, stop := 324566 ), rec( content := "), and l\ et ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 324586, root := ~, start := 324578, stop := 324585 ), rec( content := " be the \ permutation of the vertices of\n ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 324644, root := ~, start := 324629, stop := 324643 ), rec( content := " that co\ rresponds to ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 324673, root := ~, start := 324665, stop := 324672 ), rec( content := ". Then \ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 324689, root := ~, start := 324681, stop := 324688 ), rec( content := " preserv\ es the\n colourings of ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 324737, root := ~, start := 324722, stop := 324736 ), rec( content := " and ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 324757, root := ~, start := 324742, stop := 324756 ), rec( content := " – and\ hence is an\n isomorphism of coloured digraphs – if ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "colours1", count := [ 7, 2, 18, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := \ "Text", name := "A", next := 324840, root := ~, start := 324825, stop := 32483\ 9 ), rec( content := "[i] =\n ", count := [ 7, 2, 18, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "colours2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 3248\ 67, root := ~, start := 324852, stop := 324866 ), rec( content := "[i ^ p]", count := [ 7, 2, 18, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 324878, root := ~, start := 324822, stop := 324877 ), rec( content := " for all\ vertices ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 324904, root := ~, start := 324896, stop := 324903 ), rec( content := " in ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 324923, root := ~, start := 324908, stop := 324922 ), rec( content := ".\n " , count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "P", next := 324933, root := ~, start := 324929, stop := 324932 ), rec( content := "\n\n \ This operation returns such an isomorphism if one exists, else it returns\n \ ", count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "K", next := 325028, root := ~, start := 325017, stop := 325027 ), rec( content := ". ", count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "P", next := 325034, root := ~, start := 325030, stop := 325033 ), rec( content := "\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "URL\ ", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 18, 11 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 18, 11 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 18, 11 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 18, 11 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Ref", next := 325342, root := ~, start := 325312, stop := 325341 ), rec( content := ", and ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Ref", next := 325378, root := ~, start := 325348, stop := 325377 ), rec( content := ".\n\n \ ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := ChainDigraph(4);\n\ \ngap> digraph2 := ChainDigraph(3);\n\ \ngap> IsomorphismDigraphs(digraph1, digraph2,\n> [[1, 4], [2, 3]], [[1, 2], \ [3]]);\nfail\ngap> digraph2 := DigraphReverse(digraph1);\n\ngap> colours1 := [1, 1, 1, 1];;\ngap> colours2 := [1\ , 1, 1, 1];;\ngap> p := IsomorphismDigraphs(digraph1, digraph2, colours1, colo\ urs2);\n(1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\ngap> List(\ DigraphVertices(digraph1), i -> colours1[i ^ p]) = colours2;\ntrue\ngap> colou\ rs1 := [1, 1, 2, 2];;\ngap> colours2 := [2, 2, 1, 1];;\ngap> p := IsomorphismD\ igraphs(digraph1, digraph2, colours1, colours2);\n(1,4)(2,3)\ngap> OnDigraphs(\ digraph1, p) = digraph2;\ntrue\ngap> List(DigraphVertices(digraph1), i -> colo\ urs1[i ^ p]) = colours2;\ntrue\ngap> IsomorphismDigraphs(digraph1, digraph2,\n\ > [1, 1, 2, 2], [1, 1, 2, 2]);\nfail", count := [ 7, 2, 18, 12 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 18, 12 ], mathmode := "Text", name := "Example", next := 326361, root := ~, start := 325385, stop := 326360 ), rec( content := "\n ", count := [ 7, 2, 18, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Description", next := 326378, root := ~, start := 323291, stop := 326377 ) ], count := [ 7, 2, 18, 1 ], mathmode := "Text", name := "ManSection", next := 326392, root := ~, start := 323094, stop := 326391 ), rec( content := "\n\n ", count := [ 7, 2, 18, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "RepresentativeO\ utNeighbours" ), content := 0, count := [ 7, 2, 19, 2 ], mathmode := "Text", name := "Attr", next := 326469, root := ~, start := 326413, stop := 326468 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of lists.", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Returns", next := 326518, root := ~, start := 326472, stop := 326517 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the list ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "C", next := 326580, root := ~, start := 326570, stop := 326579 ), rec( content := " of ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut-neighbours", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "E", next := 326605, root := ~, start := 326584, stop := 326604 ), rec( content := " of each\ \n representative of the orbits of the action of ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Ref", next := 326690, root := ~, start := 326664, stop := 326689 ), rec( content := " on\n \ the vertex set of the digraph ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "A", next := 326742, root := ~, start := 326728, stop := 326741 ), rec( content := ". ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "P", next := 326748, root := ~, start := 326744, stop := 326747 ), rec( content := "\n\n \ More specifically, if ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eps", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 326787, root := ~, start := 326776, stop := 326786 ), rec( content := " is the \ list of orbit representatives,\n then a vertex ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 326852, root := ~, start := 326844, stop := 326851 ), rec( content := " appears\ in ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut[i]", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 326877, root := ~, start := 326864, stop := 326876 ), rec( content := " each ti\ me\n there exists an edge with source ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eps[i]", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 326939, root := ~, start := 326925, stop := 326938 ), rec( content := " and ran\ ge ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 326958, root := ~, start := 326950, stop := 326957 ), rec( content := " in\n \ ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "A", next := 326980, root := ~, start := 326966, stop := 326979 ), rec( content := ".\n " , count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "P", next := 326990, root := ~, start := 326986, stop := 326989 ), rec( content := "\n\n \ If ", count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "Ref", next := 327025, root := ~, start := 326999, stop := 327024 ), rec( content := " is triv\ ial, then ", count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "Ref", next := 327070, root := ~, start := 327043, stop := 327069 ), rec( content := "\n is\ returned.\n\n ", count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3,\ 4]]);\n\ngap> DigraphGroup(D);\n\ Group(())\ngap> RepresentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ]\ , [ 2 ], [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := Digraph(IsMutableDigraph,\ [\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3, 4]]);\n\ngap> DigraphGroup(D);\nGroup(())\ngap> Repr\ esentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ], [ 2 ], [ 1, 2, 3, \ 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\ \n\ngap> G := DigraphGroup(D);;\n\ gap> GeneratorsOfGroup(G);\n[ (1,2)(3,4)(5,6)(7,8), (1,3,2,4)(5,7,6,8), (1,5)(\ 2,6)(3,8)(4,7) ]\ngap> Set(RepresentativeOutNeighbours(D), Set);\n[ [ 2, 3, 5 \ ] ]", count := [ 7, 2, 19, 6 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 6 ], mathmode := "Text", name := "Example", next := 327963, root := ~, start := 327093, stop := 327962 ), rec( content := "\n ", count := [ 7, 2, 19, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Description", next := 327980, root := ~, start := 326521, stop := 327979 ) ], count := [ 7, 2, 19, 1 ], mathmode := "Text", name := "ManSection", next := 327994, root := ~, start := 326398, stop := 327993 ), rec( content := "\n\n ", count := [ 7, 2, 19, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd transformation or permutation", Name := "IsDigraphIsomorphism" ), content := 0, count := [ 7, 2, 20, 2 ], mathmode := "Text", name := "Oper", next := 328132, root := ~, start := 328015, stop := 328131 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphIsomorphism" ), content := 0, count := [ 7, 2, 20, 4 ], mathmode := "Text", name := "Oper", next := 328200, root := ~, start := 328135, stop := 328199 ), rec( attributes := rec( Arg := "digraph, x", Label := "for a digraph \ and a transformation or permutation", Name := "IsDigraphAutomorphism" ), content := 0, count := [ 7, 2, 20, 6 ], mathmode := "Text", name := "Oper", next := 328323, root := ~, start := 328203, stop := 328322 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphAutomo\ rphism" ), content := 0, count := [ 7, 2, 20, 8 ], mathmode := "Text", name := "Oper", next := 328384, root := ~, start := 328326, stop := 328383 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "K", next := 328407, root := ~, start := 328396, stop := 328406 ), rec( content := " or ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "K", next := 328423, root := ~, start := 328411, stop := 328422 ), rec( content := ".", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Returns", next := 328434, root := ~, start := 328387, stop := 328433 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphIsomorphism", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "C", next := 328482, root := ~, start := 328455, stop := 328481 ), rec( content := " returns\ ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "K", next := 328502, root := ~, start := 328491, stop := 328501 ), rec( content := " if the \ permutation or\n transformation ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "A", next := 328552, root := ~, start := 328544, stop := 328551 ), rec( content := " is an i\ somorphism from the digraph ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "A", next := 328598, root := ~, start := 328588, stop := 328597 ), rec( content := " to\n \ the digraph ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "A", next := 328628, root := ~, start := 328618, stop := 328627 ), rec( content := ".\n " , count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "P", next := 328638, root := ~, start := 328634, stop := 328637 ), rec( content := "\n\n \ ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphAutomorphism", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "C", next := 328672, root := ~, start := 328644, stop := 328671 ), rec( content := " returns\ ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "K", next := 328692, root := ~, start := 328681, stop := 328691 ), rec( content := " if the \ permutation or\n transformation ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "A", next := 328742, root := ~, start := 328734, stop := 328741 ), rec( content := " is an a\ utomorphism of the digraph ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "A", next := 328791, root := ~, start := 328777, stop := 328790 ), rec( content := ".\n " , count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "P", next := 328801, root := ~, start := 328797, stop := 328800 ), rec( content := "\n\n \ A permutation or transformation ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "A", next := 328847, root := ~, start := 328839, stop := 328846 ), rec( content := " is an " , count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ somorphism", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "E", next := 328872, root := ~, start := 328854, stop := 328871 ), rec( content := " from a\\ n digraph ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "A", next := 328902, root := ~, start := 328892, stop := 328901 ), rec( content := " to a di\ graph ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "A", next := 328926, root := ~, start := 328916, stop := 328925 ), rec( content := " if the \ following hold:\n ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 13 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "x", count := [ 7, 2, 20, 13 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 13 ], mathmode := "Text", name\ := "A", next := 328990, root := ~, start := 328982, stop := 328989 ), rec( cont\ ent := " is a bijection from the vertices of ", count := [ 7, 2, 20, 13 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "src", count := [ 7, 2, 20, 13 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 13 ], mathmode := "Text\ ", name := "A", next := 329037, root := ~, start := 329027, stop := 329036 ), rec( cont\ ent := " to those of\n ", count := [ 7, 2, 20, 13 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "ran", count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "A", next\ := 329068, root := ~, start := 329058, stop := 329067 ), rec( content := ";\n\ ", count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 13 ], name := "Item", next := 329083, root := ~, start := 328\ 967, stop := 329082 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[u ^\ ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "x", count := [ 7\ , 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [\ 7, 2, 20, 15 ], mathmode := "Text", name := "A", next := 329121, root := ~, s\ tart := 329113, stop := 329120 ), rec( content := ", v ^ ", count := [ 7, 2, 2\ 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "x", count := [ 7, 2, 20, 15 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathm\ ode := "Text", name := "A", next := 329135, root := ~, start := 329127, stop :\ = 329134 ), rec( content := "]", count := [ 7, 2, 20, 15 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Tex\ t", name := "C", next := 329140, root := ~, start := 329105, stop := 329139 ), rec( cont\ ent := " is an edge of\n ", count := [ 7, 2, 20, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "ran", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Text", name := \ "A", next := 329173, root := ~, start := 329163, stop := 329172 ), rec( conten\ t := " if and only if ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content\ := "[u, v]", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "C", next\ := 329202, root := ~, start := 329189, stop := 329201 ), rec( content := " is\ an\n edge of ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "src", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "A", next :\ = 329235, root := ~, start := 329225, stop := 329234 ), rec( content := "; and\ \n ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 15 ], name := "Item", next := 329254, root := ~, start := 329\ 090, stop := 329253 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x", \ count := [ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 7, 2, 20, 17 ], mathmode := "Text", name := "A", next := 329284, root :\ = ~, start := 329276, stop := 329283 ), rec( content := " fixes every ", count\ := [ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 20, 17 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17\ ], mathmode := "Text", name := "C", next := 329305, root := ~, start := 32929\ 7, stop := 329304 ), rec( content := " which is not a vertex of ", count := [ \ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "src", count := [ 7, 2, 20, 17 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17\ ], mathmode := "Text", name := "A", next := 329341, root := ~, start := 32933\ 1, stop := 329340 ), rec( content := ".\n ", count := [ 7, 2, 20, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17 ], name := "Ite\ m", next := 329356, root := ~, start := 329261, stop := 329355 ) ], count := [ 7, 2, 20, 12 ], mathmode := "Text", name := "List", next := 329368, root := ~, start := 328954, stop := 329367 ), rec( content := "\n Se\ e also ", count := [ 7, 2, 20, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 20, 19 ], mathmode := "Text", name := "Ref", next := 329442, root := ~, start := 329382, stop := 329441 ), rec( content := ".", count := [ 7, 2, 20, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "P", next := 329447, root := ~, start := 329443, stop := 329446 ), rec( content := "\n \n\ If ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "A", next := 329471, root := ~, start := 329460, stop := 329470 ), rec( content := " and ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "A", next := 329487, root := ~, start := 329476, stop := 329486 ), rec( content := ", or ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "A", next := 329502, root := ~, start := 329492, stop := 329501 ), rec( content := ", are gi\ ven, then they must\n represent vertex colourings; see \n ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "Ref", next := 329648, root := ~, start := 329572, stop := 329647 ), rec( content := " \n f\ or details of the permissible values for\n these arguments. The homomorphis\ m must then also have the property:\n \n ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 22 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col1[i] = col2[i ^ x]", count := [ 7, 2, 20, 22 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ], math\ mode := "Text", name := "C", next := 329831, root := ~, start := 329803, stop \ := 329830 ), rec( content := " for all vertices ", count := [ 7, 2, 20, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 2, 20, 22 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ], mathmode := "Text", name\ := "C", next := 329857, root := ~, start := 329849, stop := 329856 ), rec( cont\ ent := " of ", count := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "src", count\ := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 22 ], mathmode := "Text", name := "A", next := 329871, root :\ = ~, start := 329861, stop := 329870 ), rec( content := ",\n for ", coun\ t := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "IsDigraphIsomorphism", count \ := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 22 ], mathmode := "Text", name := "C", next := 329912, root :\ = ~, start := 329885, stop := 329911 ), rec( content := ". ", count := [ 7, 2,\ 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ] , name := "Ite\ m", next := 329921, root := ~, start := 329788, stop := 329920 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 24 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col[i] = col[i ^ x]", count := [ 7, 2, 20, 24 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ], math\ mode := "Text", name := "C", next := 329969, root := ~, start := 329943, stop \ := 329968 ), rec( content := " for all vertices ", count := [ 7, 2, 20, 24 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 2, 20, 24 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ], mathmode := "Text", name\ := "C", next := 329995, root := ~, start := 329987, stop := 329994 ), rec( cont\ ent := " of ", count := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 24 ], mathmode := "Text", name := "A", next := 330013, root :\ = ~, start := 329999, stop := 330012 ), rec( content := ",\n for ", coun\ t := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "IsDigraphAutomorphism", count\ := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 24 ], mathmode := "Text", name := "C", next := 330055, root :\ = ~, start := 330027, stop := 330054 ), rec( content := ". ", count := [ 7, 2,\ 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ] , name := "Ite\ m", next := 330064, root := ~, start := 329928, stop := 330063 ) ], count := [ 7, 2, 20, 21 ], mathmode := "Text", name := "List", next := 330076, root := ~, start := 329775, stop := 330075 ), rec( content := "\n\n \ For some digraphs, it can be faster to use ", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphAutomorphism", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "C", next := 330153, root := ~, start := 330125, stop := 330152 ), rec( content := "\n th\ an to test membership in the automorphism group of ", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "A", next := 330225, root := ~, start := 330211, stop := 330224 ), rec( content := ".\n\n \ ", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> IsDigraphAutomorphism(src, (1, 2, 3));\nfalse\ngap> IsDigra\ phAutomorphism(src, (2, 3));\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2\ , 1, 1]);\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2, 2, 1]);\nfalse\ng\ ap> IsDigraphAutomorphism(src, (2, 3)(4, 5));\nfalse\ngap> IsDigraphAutomorphi\ sm(src, (1, 4));\nfalse\ngap> IsDigraphAutomorphism(src, ());\ntrue\ngap> ran \ := Digraph([[2, 1], [2], [2, 3]]);\n\ngap> IsDigraphIsomorphism(src, ran, (1, 2));\ntrue\ngap> IsDigraphIsomorph\ ism(ran, src, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ntr\ ue\ngap> IsDigraphIsomorphism(src, Digraph([[3], [1, 3], [2]]), (1, 2, 3));\nf\ alse\ngap> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 3], [2, 1, 3]);\ntrue\ \ngap> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 2], [2, 1, 3]);\nfalse\n" , count := [ 7, 2, 20, 27 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 27 ], mathmode := "Text", name := "Example", next := 331168, root := ~, start := 330232, stop := 331167 ), rec( content := "\n ", count := [ 7, 2, 20, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Description", next := 331185, root := ~, start := 328437, stop := 331184 ) ], count := [ 7, 2, 20, 1 ], mathmode := "Text", name := "ManSection", next := 331199, root := ~, start := 328000, stop := 331198 ), rec( content := "\n\n ", count := [ 7, 2, 20, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Name := "IsDigraphColour\ ing" ), content := 0, count := [ 7, 2, 21, 2 ], mathmode := "Text", name := "Oper", next := 331277, root := ~, start := 331224, stop := 331276 ), rec( attributes := rec( Arg := "digraph, t", Label := "for a transfor\ mation", Name := "IsDigraphColouring" ), content := 0, count := [ 7, 2, 21, 4 ], mathmode := "Text", name := "Oper", next := 331368, root := ~, start := 331282, stop := 331367 ), rec( attributes := rec( ), content := [ rec( content := " ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "K", next := 331394, root := ~, start := 331383, stop := 331393 ), rec( content := " or ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "K", next := 331410, root := ~, start := 331398, stop := 331409 ), rec( content := ". ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Returns", next := 331422, root := ~, start := 331373, stop := 331421 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphColouring", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "C", next := 331486, root := ~, start := 331461, stop := 331485 ), rec( content := " verifie\ s whether or not \n the list ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "A", next := 331538, root := ~, start := 331527, stop := 331537 ), rec( content := " describ\ es a proper colouring of the digraph \n ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "A", next := 331604, root := ~, start := 331590, stop := 331603 ), rec( content := ". \n \ ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "P", next := 331617, root := ~, start := 331613, stop := 331616 ), rec( content := "\n\n \ A list ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 331643, root := ~, start := 331632, stop := 331642 ), rec( content := " describ\ es a ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "E", next := 331679, root := ~, start := 331656, stop := 331678 ), rec( content := " of a di\ graph\n ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 331713, root := ~, start := 331699, stop := 331712 ), rec( content := " if ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 331728, root := ~, start := 331717, stop := 331727 ), rec( content := " consist\ s of positive integers, the length of\n ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 331791, root := ~, start := 331780, stop := 331790 ), rec( content := " equals \ the number of vertices in ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 331839, root := ~, start := 331825, stop := 331838 ), rec( content := ", and fo\ r any\n vertices ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ , v", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 331879, root := ~, start := 331868, stop := 331878 ), rec( content := " of ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 331897, root := ~, start := 331883, stop := 331896 ), rec( content := " if ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 331909, root := ~, start := 331901, stop := 331908 ), rec( content := " and ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 331922, root := ~, start := 331914, stop := 331921 ), rec( content := " are adj\ acent,\n then ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "list", count := [ 7, 2, 21, 6 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := \ "Text", name := "A", next := 331962, root := ~, start := 331951, stop := 33196\ 1 ), rec( content := "[u] >< ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 3319\ 86, root := ~, start := 331975, stop := 331985 ), rec( content := "[v]", count := [ 7, 2, 21, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 331993, root := ~, start := 331948, stop := 331992 ), rec( content := ".\n \ ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "P", next := 332005, root := ~, start := 332001, stop := 332004 ), rec( content := "\n\n \ A transformation ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 332038, root := ~, start := 332030, stop := 332037 ), rec( content := " describ\ es a proper colouring of a digraph\n ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 332101, root := ~, start := 332087, stop := 332100 ), rec( content := ", if ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ mageListOfTransformation(", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "A", next := 332143, root := ~, start := 332135, stop := 33214\ 2 ), rec( content := ",\n DigraphNrVertices(", count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 3321\ 83, root := ~, start := 332169, stop := 332182 ), rec( content := "))", count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "C", next := 332189, root := ~, start := 332106, stop := 332188 ), rec( content := " is a pr\ oper colouring of \n ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 332236, root := ~, start := 332222, stop := 332235 ), rec( content := ". ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "P", next := 332242, root := ~, start := 332238, stop := 332241 ), rec( content := "\n\n \ See also ", count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsDigra\ phHomomorphism" ), content := 0, count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "Ref", next := 332294, root := ~, start := 332259, stop := 332293 ), rec( content := ".\n \ \n ", count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := JohnsonDigraph(5, 3);\n\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 4, 5, 6, 7]);\ntru\ e\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, 7]);\nfalse\ngap> Is\ DigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, -1]);\nfalse\ngap> IsDigraphCo\ louring(D, [1, 2, 3]);\nfalse\ngap> IsDigraphColouring(D, IdentityTransformati\ on);\ntrue\n", count := [ 7, 2, 21, 9 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 9 ], mathmode := "Text", name := "Example", next := 332730, root := ~, start := 332309, stop := 332729 ), rec( content := "\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Description", next := 332749, root := ~, start := 331427, stop := 332748 ) ], count := [ 7, 2, 21, 1 ], mathmode := "Text", name := "ManSection", next := 332765, root := ~, start := 331207, stop := 332764 ), rec( content := "\n\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 332779, root := ~, start := 288857, stop := 332778 ) , rec( content := "\n\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Homomorphisms of \ digraphs", count := [ 7, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 332836, root := ~, start := 332792, stop := 332835 ), rec( content := "\n\n The following me\ thods exist to find homomorphisms between digraphs.\n If an argument to one\ of these methods is a digraph with multiple edges,\n then the multiplicity\ of edges will be ignored in order to perform the\n calculation; the digrap\ h will be treated as if it has no multiple edges.\n\n ", count := [ 7, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, hook, us\ er_param, max_results, hint, injective, image, partial_map, colors1, colors2[,\ order, aut_grp]", Name := "HomomorphismDigraphsFinder" ), content := 0, count := [ 7, 3, 1, 2 ], mathmode := "Text", name := "Func", next := 333317, root := ~, start := 333159, stop := 333316 ), rec( attributes := rec( ), content := [ rec( content := "The argum\ ent ", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ser_param", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "A", next := 333359, root := ~, start := 333342, stop := 333358 ), rec( content := ".", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 333370, root := ~, start := 333320, stop := 333369 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function finds homomorphisms from the digraph ", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "A", next := 333451, root := ~, start := 333442, stop := 333450 ), rec( content := " to the \ digraph\n ", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "A", next := 333480, root := ~, start := 333471, stop := 333479 ), rec( content := " subject\ to the conditions imposed by the other arguments as\n described below.", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "P", next := 333565, root := ~, start := 333561, stop := 333564 ), rec( content := "\n\n \ If ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 333582, root := ~, start := 333574, stop := 333581 ), rec( content := " and ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 333595, root := ~, start := 333587, stop := 333594 ), rec( content := " are hom\ omorphisms found by\n ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismDigraphsFinder", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 333660, root := ~, start := 333627, stop := 333659 ), rec( content := ", then " , count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 333675, root := ~, start := 333667, stop := 333674 ), rec( content := " cannot \ be obtained from\n ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 333712, root := ~, start := 333704, stop := 333711 ), rec( content := " by righ\ t multiplying by an automorphism of ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "A", next := 333765, root := ~, start := 333756, stop := 333764 ), rec( content := " in\n \ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "a\ ut_grp", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "A", next := 333787, root := ~, start := 333773, stop := 333786 ), rec( content := ".\n\n \ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "hook", count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "A", next := 33\ 3824, root := ~, start := 333813, stop := 333823 ) ], count := [ 7, 3, 1, 5 ], name := "Mar\ k", next := 333831, root := ~, start := 333807, stop := 333830 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a f\ unction or ", count := [ 7, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 7, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 6 ], mathmode := "Text", name := "K", next := 333902, root := \ ~, start := 333891, stop := 333901 ), rec( content := ".", count := [ 7, 3, 1,\ 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := 0, count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "P", next\ := 333907, root := ~, start := 333903, stop := 333906 ), rec( content := "\n\ \n If ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "hook", coun\ t := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 7 ], mathmode := "Text", name := "A", next := 333931, root := \ ~, start := 333920, stop := 333930 ), rec( content := " is a function, then it\ must have two arguments\n ", count := [ 7, 3, 1, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "user_param", count := [ 7, 3, 1, 7 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name \ := "A", next := 334004, root := ~, start := 333987, stop := 334003 ), rec( cont\ ent := " (see below) and a transformation ", count := [ 7, 3, 1, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "t", count := [ 7, 3, 1, 7 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name :\ = "C", next := 334046, root := ~, start := 334038, stop := 334045 ), rec( cont\ ent := ". The\n function ", count := [ 7, 3, 1, 7 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attri\ butes := rec( ), content := [ rec( content := "hook", count := [ 7, 3, 1, 7 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ]\ , mathmode := "Text", name := "A", next := 334083, root := ~, start := 334072,\ stop := 334082 ), rec( content := "(", count := [ 7, 3, 1, 7 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [\ rec( content := "user_param", count := [ 7, 3, 1, 7 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", na\ me := "A", next := 334101, root := ~, start := 334084, stop := 334100 ), rec( \ content := ", t)", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next\ := 334109, root := ~, start := 334069, stop := 334108 ), rec( content := " is\ called every time\n a new homomorphism ", count := [ 7, 3, 1, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "t", count := [ 7, 3, 1, 7 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name\ := "C", next := 334166, root := ~, start := 334158, stop := 334165 ), rec( cont\ ent := " is found by\n ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "HomomorphismDigraphsFinder", count := [ 7, 3, 1, 7 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text\ ", name := "C", next := 334220, root := ~, start := 334187, stop := 334219 ), rec( cont\ ent := ". If the function returns ", count := [ 7, 3, 1, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "true", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "K" , next\ := 334257, root := ~, start := 334246, stop := 334256 ), rec( content := ",\n\ then ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Homomorphis\ mDigraphsFinder", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next\ := 334305, root := ~, start := 334272, stop := 334304 ), rec( content := " st\ ops and does not find any\n further homomorphisms. This feature might b\ e useful if you are\n searching for a homomorphism that satisfies some \ condition that you\n cannot specify via the other arguments to\n \ ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "HomomorphismDigraphsFinder", \ count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next := 334572, root := \ ~, start := 334539, stop := 334571 ), rec( content := ".\n ", count := [ \ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := 0, count := [ 7, 3, 1, 8 ], mathmode := "Text", name\ := "P", next := 334586, root := ~, start := 334582, stop := 334585 ), rec( cont\ ent := "\n\n If ", count := [ 7, 3, 1, 8 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "hook", count := [ 7, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 1, 8 ], mathmode := "Text", name := "A", next := 3\ 34610, root := ~, start := 334599, stop := 334609 ), rec( content := " is ", coun\ t := [ 7, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 7, 3, 1, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 8 ] , math\ mode := "Text", name := "K", next := 334625, root := ~, start := 334614, stop \ := 334624 ), rec( content := ", then a default function is used which\n \ simply adds every new homomorphism found by\n ", count := [ 7, 3, 1, 8\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "HomomorphismDigraphsFinder", count := [ 7, \ 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, \ 3, 1, 8 ], mathmode := "Text", name := "C", next := 334758, root := ~, start :\ = 334725, stop := 334757 ), rec( content := " to ", count := [ 7, 3, 1, 8 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "user_param", count := [ 7, 3, 1, 8 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 8 ], mathmode := "T\ ext", name := "A", next := 334779, root := ~, start := 334762, stop := 334778 ), rec( cont\ ent := ", which must be a\n mutable list in this case.\n ", count \ := [ 7, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 6 ], name := "Ite\ m", next := 334845, root := ~, start := 333838, stop := 334844 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "user_param", count := [ 7, 3, 1, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 3, 1, 9 ], mathmode := "Text", name := "A", next\ := 334876, root := ~, start := 334859, stop := 334875 ) ], count := [ 7, 3, 1, 9 ], name := "Mar\ k", next := 334883, root := ~, start := 334853, stop := 334882 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If ", count := [ 7, 3, 1, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "hook", count := [ 7, 3, 1, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode\ := "Text", name := "A", next := 334919, root := ~, start := 334908, stop := 3\ 34918 ), rec( content := " is a function, then ", count := [ 7, 3, 1, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "user_param", count := [ 7, 3, 1, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := \ "Text", name := "A", next := 334957, root := ~, start := 334940, stop := 33495\ 6 ), rec( content := " can be any ", count := [ 7, 3, 1, 10 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "GAP", count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "\ Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := "\n\ object. The object ", count := [ 7, 3, 1, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "user_param", count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "A" , next\ := 335019, root := ~, start := 335002, stop := 335018 ), rec( content := " is\ used as the first argument of\n the function ", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "hook", count := [ 7, 3, 1, 10 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode \ := "Text", name := "A", next := 335085, root := ~, start := 335074, stop := 33\ 5084 ), rec( content := ". For example, ", count := [ 7, 3, 1, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "user_param", count := [ 7, 3, 1, 10 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Tex\ t", name := "A", next := 335117, root := ~, start := 335100, stop := 335116 ), rec( cont\ ent := " might be a\n transformation semigroup, and ", count := [ 7, 3,\ 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "hook", count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "A", next :=\ 335181, root := ~, start := 335170, stop := 335180 ), rec( content := "(", co\ unt := [ 7, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec\ ( attributes := rec( ), content := [ rec( content := "user_param", count := [\ 7, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := \ [ 7, 3, 1, 10 ], mathmode := "Text", name := "A", next := 335199, root := ~, s\ tart := 335182, stop := 335198 ), rec( content := ", t)", count := [ 7, 3, 1, \ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1,\ 10 ], mathmode := "Text", name := "C", next := 335207, root := ~, start := 33\ 5167, stop := 335206 ), rec( content := "\n might set ", count := [ 7, \ 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "user_param", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 0 ], mathmode := "Text", name := "A", next := 335243, root := ~, start := 3352\ 26, stop := 335242 ), rec( content := " to be the closure of ", count := [ 7, \ 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "user_param", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 0 ], mathmode := "Text", name := "A", next := 335282, root := ~, start := 3352\ 65, stop := 335281 ), rec( content := " and\n ", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "t", count := [ 7, 3, 1, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := \ "Text", name := "C", next := 335303, root := ~, start := 335295, stop := 33530\ 2 ), rec( content := ". ", count := [ 7, 3, 1, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 7, 3, 1, 11 ], mathmode := "Text", name := "P", next := 335309, root := ~, star\ t := 335305, stop := 335308 ), rec( content := "\n\n If the value of ", coun\ t := [ 7, 3, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "hook", count := [ 7, 3, 1, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 1 ], mathmode := "Text", name := "A", next := 335346, root := ~, start := 3353\ 35, stop := 335345 ), rec( content := " is ", count := [ 7, 3, 1, 11 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "fail", count := [ 7, 3, 1, 11 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 11 ], mathmode := "Text", name\ := "K", next := 335361, root := ~, start := 335350, stop := 335360 ), rec( cont\ ent := ", then the value of\n ", count := [ 7, 3, 1, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "user_param", count := [ 7, 3, 1, 11 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 11 ], mathmode := "Text", name\ := "A", next := 335406, root := ~, start := 335389, stop := 335405 ), rec( cont\ ent := " must be a mutable list.\n ", count := [ 7, 3, 1, 11 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], name := "Ite\ m", next := 335444, root := ~, start := 334890, stop := 335443 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "max_results", count := [ 7, 3, 1, 12 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 12 ], mathmode := "Text", name := "A", next\ := 335476, root := ~, start := 335458, stop := 335475 ) ], count := [ 7, 3, 1, 12 ], name := "Mar\ k", next := 335483, root := ~, start := 335452, stop := 335482 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a p\ ositive integer or ", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "infinity", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "K", next :\ = 335566, root := ~, start := 335551, stop := 335565 ), rec( content := ".\n \ ", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Homomorphism\ DigraphsFinder", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "C", next\ := 335609, root := ~, start := 335576, stop := 335608 ), rec( content := " wi\ ll return after it has found\n ", count := [ 7, 3, 1, 13 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "max_results", count := [ 7, 3, 1, 13 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 13 ], mathmode := "Text\ ", name := "A", next := 335667, root := ~, start := 335649, stop := 335666 ), rec( cont\ ent := " homomorphisms or the search is complete, whichever\n happens f\ irst.\n ", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 3, 1, 13 ], name := "Item", next := 335755, root := ~, start := 335\ 490, stop := 335754 ), rec( attributes := rec( ), content := [ rec( attribute\ s := rec( ), content := [ rec( content := "hint", count := [ 7, 3, 1, 14 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 14 ], math\ mode := "Text", name := "A", next := 335780, root := ~, start := 335769, stop \ := 335779 ) ], count := [ 7, 3, 1, 14 ], name := "Mark", next := 335787, root := ~, start := 335\ 763, stop := 335786 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n This argument should be a positive integer or ", count := [ 7, 3,\ 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "fail", count := [ 7, 3, 1, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 15 ], math\ mode := "Text", name := "K", next := 335866, root := ~, start := 335855, stop \ := 335865 ), rec( content := ". ", count := [ 7, 3, 1, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, coun\ t := [ 7, 3, 1, 16 ], mathmode := "Text", name := "P", next := 335872, root :=\ ~, start := 335868, stop := 335871 ), rec( content := "\n\n If ", coun\ t := [ 7, 3, 1, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "hint", count := [ 7, 3, 1, 16\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 6 ], mathmode := "Text", name := "A", next := 335896, root := ~, start := 3358\ 85, stop := 335895 ), rec( content := " is a positive integer, then only homor\ phisms of rank\n ", count := [ 7, 3, 1, 16 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "hint", count := [ 7, 3, 1, 16 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 3, 1, 16 ], mathmode := "Text", name := "A", next \ := 335969, root := ~, start := 335958, stop := 335968 ), rec( content := " are\ found.", count := [ 7, 3, 1, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 1, 17 ], math\ mode := "Text", name := "P", next := 335984, root := ~, start := 335980, stop \ := 335983 ), rec( content := "\n\n If ", count := [ 7, 3, 1, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "hint", count := [ 7, 3, 1, 17 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 17 ], mathmode := "Text" , name\ := "A", next := 336008, root := ~, start := 335997, stop := 336007 ), rec( cont\ ent := " is ", count := [ 7, 3, 1, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 7, 3, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 17 ], mathmode := "Text", name := "K", next := 336023, root :=\ ~, start := 336012, stop := 336022 ), rec( content := ", then no restriction \ is put on the rank\n of homomorphisms found.\n ", count := [ 7, 3,\ 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 15 ], name := "Ite\ m", next := 336109, root := ~, start := 335794, stop := 336108 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "injective", count := [ 7, 3, 1, 18 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 3, 1, 18 ], mathmode := "Text", name := "A", next\ := 336139, root := ~, start := 336123, stop := 336138 ) ], count := [ 7, 3, 1, 18 ], name := "Mar\ k", next := 336146, root := ~, start := 336117, stop := 336145 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be ", coun\ t := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "0", count := [ 7, 3, 1, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ] , math\ mode := "Text", name := "C", next := 336200, root := ~, start := 336192, stop \ := 336199 ), rec( content := ", ", count := [ 7, 3, 1, 19 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "1", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "C", next\ := 336210, root := ~, start := 336202, stop := 336209 ), rec( content := ", o\ r ", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "2", count := [ 7\ , 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 3, 1, 19 ], mathmode := "Text", name := "C", next := 336223, root := ~, star\ t := 336215, stop := 336222 ), rec( content := ". If it is\n ", count \ := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "2", count := [ 7, 3, 1, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ] , math\ mode := "Text", name := "C", next := 336251, root := ~, start := 336243, stop \ := 336250 ), rec( content := ", then only embeddings are found, if it is ", coun\ t := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "1", count := [ 7, 3, 1, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ] , math\ mode := "Text", name := "C", next := 336302, root := ~, start := 336294, stop \ := 336301 ), rec( content := ", then only\n injective homomorphisms are\ found, and if it is ", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "0", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "C", next := 336\ 378, root := ~, start := 336370, stop := 336377 ), rec( content := " there are\ no\n restrictions imposed by this argument.\n ", count := [ 7, \ 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := 0, count := [ 7, 3, 1, 20 ], mathmode := "Text", name \ := "P", next := 336451, root := ~, start := 336447, stop := 336450 ), rec( cont\ ent := "\n\n For backwards compatibility, ", count := [ 7, 3, 1, 20 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "injective", count := [ 7, 3, 1, 20 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "\ Text", name := "A", next := 336506, root := ~, start := 336490, stop := 336505\ ), rec( content := " can also be ", count := [ 7, 3, 1, 20 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "false", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "Text", name :=\ "K", next := 336531, root := ~, start := 336519, stop := 336530 ), rec( conte\ nt := "\n or ", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "true", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "K", next := 3\ 36554, root := ~, start := 336543, stop := 336553 ), rec( content := " which c\ orrespond to the values ", count := [ 7, 3, 1, 20 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "0", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "C", next := \ 336594, root := ~, start := 336586, stop := 336593 ), rec( content := " and ", coun\ t := [ 7, 3, 1, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "1", count := [ 7, 3, 1, 20 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 20 ] , math\ mode := "Text", name := "C", next := 336607, root := ~, start := 336599, stop \ := 336606 ), rec( content := "\n described in the previous paragraph, r\ espectively. \n ", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ], name := "Item", next := 3366\ 81, root := ~, start := 336153, stop := 336680 ), rec( attributes := rec( ), content := [ rec( attr\ ibutes := rec( ), content := [ rec( content := "image", count := [ 7, 3, 1, 2\ 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, \ 21 ], mathmode := "Text", name := "A", next := 336707, root := ~, start := 336\ 695, stop := 336706 ) ], count := [ 7, 3, 1, 21 ], name := "Mark", next := 3367\ 14, root := ~, start := 336689, stop := 336713 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n This argument should be a subset of the vertices of the grap\ h ", count := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "D2", count := [ \ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 3, 1, 22 ], mathmode := "Text", name := "A", next := 336807, root := ~, star\ t := 336798, stop := 336806 ), rec( content := ".\n ", count := [ 7, 3,\ 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "HomomorphismDigraphsFinder", count :=\ [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 3, 1, 22 ], mathmode := "Text", name := "C", next := 336850, root := ~, star\ t := 336817, stop := 336849 ), rec( content := " only finds homomorphisms from\ \n ", count := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "D1", count \ := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 7, 3, 1, 22 ], mathmode := "Text", name := "A", next := 336898, root := \ ~, start := 336889, stop := 336897 ), rec( content := " to the subgraph of ", coun\ t := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "D2", count := [ 7, 3, 1, 22 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ] , math\ mode := "Text", name := "A", next := 336927, root := ~, start := 336918, stop \ := 336926 ), rec( content := " induced by the vertices\n ", count := [ \ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "image", count := [ 7, 3, 1, 22 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ] , math\ mode := "Text", name := "A", next := 336972, root := ~, start := 336960, stop \ := 336971 ), rec( content := ".\n ", count := [ 7, 3, 1, 22 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ], name := "Ite\ m", next := 336987, root := ~, start := 336721, stop := 336986 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "partial_map", count := [ 7, 3, 1, 23 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 23 ], mathmode := "Text", name := "A", next\ := 337019, root := ~, start := 337001, stop := 337018 ) ], count := [ 7, 3, 1, 23 ], name := "Mar\ k", next := 337026, root := ~, start := 336995, stop := 337025 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a p\ artial map from ", count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D1\ ", count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "A", next := 337100, root\ := ~, start := 337091, stop := 337099 ), rec( content := " to ", count := [ 7\ , 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "D2", count := [ 7, 3, 1, 24 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], math\ mode := "Text", name := "A", next := 337113, root := ~, start := 337104, stop \ := 337112 ), rec( content := ",\n that is, a (not necessarily dense) li\ st of vertices of the digraph\n ", count := [ 7, 3, 1, 24 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "D2", count := [ 7, 3, 1, 24 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], mathmode := "Text", name \ := "A", next := 337207, root := ~, start := 337198, stop := 337206 ), rec( cont\ ent := " of length no greater than the number vertices in the digraph\n \ ", count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "D1", count := [ 7, 3, 1, 24 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ] , math\ mode := "Text", name := "A", next := 337286, root := ~, start := 337277, stop \ := 337285 ), rec( content := ". ", count := [ 7, 3, 1, 24 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "HomomorphismDigraphsFinder", count := [ 7, 3, 1, 24 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], mathmode\ := "Text", name := "C", next := 337321, root := ~, start := 337288, stop := 3\ 37320 ), rec( content := " only finds homomorphisms\n extending ", coun\ t := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "partial_map", count := [ 7, 3\ , 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, \ 3, 1, 24 ], mathmode := "Text", name := "A", next := 337383, root := ~, start \ := 337365, stop := 337382 ), rec( content := " (if any).\n ", count := [ \ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], name := "Ite\ m", next := 337407, root := ~, start := 337033, stop := 337406 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "colors1", count := [ 7, 3, 1, 25 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 7, 3, 1, 25 ], mathmode := "Text", name := "A", next\ := 337441, root := ~, start := 337427, stop := 337440 ) ], count := [ 7, 3, 1, 25 ], name := "Mar\ k", next := 337448, root := ~, start := 337421, stop := 337447 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This should be a list repre\ senting possible colours of vertices in the\n digraph ", count := [ 7, \ 3, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "D1", count := [ 7, 3, 1, 26 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 26 ], math\ mode := "Text", name := "A", next := 337566, root := ~, start := 337557, stop \ := 337565 ), rec( content := "; see \n ", count := [ 7, 3, 1, 26 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label :\ = "for a digraph and a homogeneous list", Oper := "AutomorphismGroup" ), conte\ nt := 0, count := [ 7, 3, 1, 26 ], mathmode := "Text", name := "Ref", next := \ 337668, root := ~, start := 337581, stop := 337667 ), rec( content := " \n \ for details of the permissible values for this argument.\n ", count :\ = [ 7, 3, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 26 ], name := "Ite\ m", next := 337748, root := ~, start := 337455, stop := 337747 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "colors2", count := [ 7, 3, 1, 27 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 7, 3, 1, 27 ], mathmode := "Text", name := "A", next\ := 337782, root := ~, start := 337768, stop := 337781 ) ], count := [ 7, 3, 1, 27 ], name := "Mar\ k", next := 337789, root := ~, start := 337762, stop := 337788 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This should be a list repre\ senting possible colours of vertices in the\n digraph ", count := [ 7, \ 3, 1, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "D2", count := [ 7, 3, 1, 28 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 28 ], math\ mode := "Text", name := "A", next := 337907, root := ~, start := 337898, stop \ := 337906 ), rec( content := "; see \n ", count := [ 7, 3, 1, 28 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label :\ = "for a digraph and a homogeneous list", Oper := "AutomorphismGroup" ), conte\ nt := 0, count := [ 7, 3, 1, 28 ], mathmode := "Text", name := "Ref", next := \ 338009, root := ~, start := 337922, stop := 338008 ), rec( content := " \n \ for details of the permissible values for this argument.\n ", count :\ = [ 7, 3, 1, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 28 ], name := "Ite\ m", next := 338089, root := ~, start := 337796, stop := 338088 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "order", count := [ 7, 3, 1, 29 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 7, 3, 1, 29 ], mathmode := "Text", name := "A", next :\ = 338114, root := ~, start := 338102, stop := 338113 ) ], count := [ 7, 3, 1, 29 ], name := "Mar\ k", next := 338121, root := ~, start := 338096, stop := 338120 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n The optional argument ", coun\ t := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "order", count := [ 7, 3, 1, 3\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, \ 30 ], mathmode := "Text", name := "A", next := 338177, root := ~, start := 338\ 165, stop := 338176 ), rec( content := " specifies the order the\n vert\ ices in ", count := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "D1", count \ := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 7, 3, 1, 30 ], mathmode := "Text", name := "A", next := 338231, root := \ ~, start := 338222, stop := 338230 ), rec( content := " appear in the search f\ or homomorphisms. \n The value of this parameter can have a large impac\ t\n on the runtime of the function. It seems in many cases to be a good\ \n idea for this to be the ", count := [ 7, 3, 1, 30 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphWe\ lshPowellOrder" ), content := 0, count := [ 7, 3, 1, 30 ], mathmode := "Text", name\ := "Ref", next := 338478, root := ~, start := 338441, stop := 338477 ), rec( cont\ ent := ", i.e.\n vertices ordered from highest to lowest degree.\n \ ", count := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 30 ], name := "Ite\ m", next := 338554, root := ~, start := 338128, stop := 338553 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n The optional argument ", coun\ t := [ 7, 3, 1, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "aut_grp", count := [ 7, 3, 1,\ 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1\ , 32 ], mathmode := "Text", name := "A", next := 338612, root := ~, start := 3\ 38598, stop := 338611 ), rec( content := " should be a subgroup of the\n \ automorphism group of ", count := [ 7, 3, 1, 32 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "D2", count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "A", next :=\ 338680, root := ~, start := 338671, stop := 338679 ), rec( content := ". This\ function returns unique\n representatives of the homomorphisms found u\ p to right multiplication\n by ", count := [ 7, 3, 1, 32 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "aut_grp", count := [ 7, 3, 1, 32 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 32 ], mathmode := "Text", name\ := "A", next := 338814, root := ~, start := 338800, stop := 338813 ), rec( cont\ ent := ". If this argument is not specific, it defaults to the\n full a\ utomorphism group of ", count := [ 7, 3, 1, 32 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "D2", count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "A", next := 33\ 8913, root := ~, start := 338904, stop := 338912 ), rec( content := ", which m\ ay be costly to calculate.\n ", count := [ 7, 3, 1, 32 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 32 ], name := "Item" , next := 3389\ 62, root := ~, start := 338561, stop := 338961 ) ], count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "List", next := 338974, root := ~, start := 333794, stop := 338973 ), rec( content := "\n\n \ ", count := [ 7, 3, 1, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := ChainDigraph(10);\n\ngap>\ D := DigraphSymmetricClosure(D);\n\ngap> HomomorphismDigraphsFinder(D, D, fail, [], infinity, 2, 0,\ \n> [3, 4], [], fail, fail);\n[ Transformation( [ 3, 4, 3, 4, 3, 4, 3, 4, 3, 4\ ] ), \n Transformation( [ 4, 3, 4, 3, 4, 3, 4, 3, 4, 3 ] ) ]\ngap> D2 := Com\ pleteDigraph(6);;\ngap> HomomorphismDigraphsFinder(D, D2, fail, [], 1, fail, 0\ ,\n> [1 .. 6], [1, 2, 1], fail, fail);\n[ Transformation( [ 1, 2, 1, 3, 4, 5, \ 6, 1, 2, 1 ] ) ]\ngap> func := function(user_param, t)\n> Add(user_param, t * \ user_param[1]);\n> end;;\ngap> HomomorphismDigraphsFinder(D, D2, func, [Transf\ ormation([2, 2])],\n> 3, fail, 0, [1 .. 6], [1, 2, 1], fail, fail);\n[ Transfo\ rmation( [ 2, 2 ] ), \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 2 ] ), \ \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 3 ] ), \n Transformation( [ \ 2, 2, 2, 3, 4, 5, 6, 2, 2, 4 ] ) ]\ngap> HomomorphismDigraphsFinder(NullDigrap\ h(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [1, 2, 3], fail, fail, f\ ail, fail,\n> Group(()));\n[ IdentityTransformation, Transformation( [ 1, 3, 3\ ] ), \n Transformation( [ 2, 1 ] ), Transformation( [ 2, 3, 3 ] ), \n Trans\ formation( [ 3, 1, 3 ] ), Transformation( [ 3, 2, 3 ] ) ]\ngap> HomomorphismDi\ graphsFinder(NullDigraph(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [\ 1, 2, 3], fail, fail, fail, fail,\n> Group((1, 2)));\n[ IdentityTransformation\ , Transformation( [ 1, 3, 3 ] ), \n Transformation( [ 3, 1, 3 ] ) ]", count := [ 7, 3, 1, 35 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 1, 35 ], mathmode := "Text", name := "Example", next := 340529, root := ~, start := 338980, stop := 340528 ), rec( content := "\n ", count := [ 7, 3, 1, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 340546, root := ~, start := 333373, stop := 340545 ) ], count := [ 7, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 340560, root := ~, start := 333144, stop := 340559 ), rec( content := "\n\n ", count := [ 7, 3, 1, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphHomomorphism" ), content := 0, count := [ 7, 3, 2, 2 ], mathmode := "Text", name := "Oper", next := 340640, root := ~, start := 340581, stop := 340639 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "K", next := 340685, root := ~, start := 340674, stop := 340684 ), rec( content := ".", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 340696, root := ~, start := 340643, stop := 340695 ), rec( attributes := rec( ), content := [ rec( content := "\n A h\ omomorphism from ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 340752, root := ~, start := 340737, stop := 340751 ), rec( content := " to ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 340771, root := ~, start := 340756, stop := 340770 ), rec( content := " is a ma\ pping from\n the vertex set of ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 340827, root := ~, start := 340812, stop := 340826 ), rec( content := " to a su\ bset of the vertices of\n ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 340878, root := ~, start := 340863, stop := 340877 ), rec( content := ", such t\ hat every pair of vertices ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 340925, root := ~, start := 340913, stop := 340924 ), rec( content := " which h\ as\n an edge ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ->j", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 340959, root := ~, start := 340948, stop := 340958 ), rec( content := " is mapp\ ed to a pair of vertices ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ a,b]", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 341004, root := ~, start := 340992, stop := 341003 ), rec( content := " which\n\ has an edge ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ ->b", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 341038, root := ~, start := 341027, stop := 341037 ), rec( content := ". Note \ that non-adjacent vertices can still be\n mapped to adjacent vertices. ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "P", next := 341123, root := ~, start := 341119, stop := 341122 ), rec( content := "\n\n \ ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphHomomorphism", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "C", next := 341155, root := ~, start := 341129, stop := 341154 ), rec( content := " returns\ a single homomorphism between\n ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "A", next := 341213, root := ~, start := 341198, stop := 341212 ), rec( content := " and ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "A", next := 341233, root := ~, start := 341218, stop := 341232 ), rec( content := " if it e\ xists, otherwise it returns\n ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "K", next := 341284, root := ~, start := 341273, stop := 341283 ), rec( content := ".\n", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> DigraphHomomorphism\ (gr1, gr1);\nIdentityTransformation\ngap> map := DigraphHomomorphism(gr1, gr2)\ ;\nTransformation( [ 3, 1, 5, 4, 5 ] )\ngap> IsDigraphHomomorphism(gr1, gr2, m\ ap);\ntrue\n", count := [ 7, 3, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 5 ], mathmode := "Text", name := "Example", next := 341632, root := ~, start := 341286, stop := 341631 ), rec( content := "\n ", count := [ 7, 3, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 341649, root := ~, start := 340699, stop := 341648 ) ], count := [ 7, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 341663, root := ~, start := 340566, stop := 341662 ), rec( content := "\n\n ", count := [ 7, 3, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "HomomorphismsDigraphs" ), content := 0, count := [ 7, 3, 3, 2 ], mathmode := "Text", name := "Oper", next := 341745, root := ~, start := 341684, stop := 341744 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "HomomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 3, 4 ], mathmode := "Text", name := "Oper", next := 341824, root := ~, start := 341748, stop := 341823 ), rec( attributes := rec( ), content := [ rec( content := " A list o\ f transformations.", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Returns", next := 341873, root := ~, start := 341827, stop := 341872 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismsDigraphsRepresentatives", count := [ 7, 3, 3, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 341937, root := ~, start := 341894, stop := 341936 ), rec( content := " finds e\ very\n ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Ref", next := 341987, root := ~, start := 341954, stop := 341986 ), rec( content := " between\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 342011, root := ~, start := 341996, stop := 342010 ), rec( content := " and\n \ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 342035, root := ~, start := 342020, stop := 342034 ), rec( content := ", up to \ right multiplication by an element of the\n ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 3, 3, 5 ] , mathmode := "Text", name := "Ref", next := 342142, root := ~, start := 342089, stop := 342141 ), rec( content := " of ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 342161, root := ~, start := 342146, stop := 342160 ), rec( content := ".\n I\ n other words, every homomorphism ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 342210, root := ~, start := 342202, stop := 342209 ), rec( content := " between\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 342234, root := ~, start := 342219, stop := 342233 ), rec( content := " and\n \ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 342258, root := ~, start := 342243, stop := 342257 ), rec( content := " can be \ written as the composition ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ = g * x", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 342309, root := ~, start := 342293, stop := 342308 ), rec( content := ",\n w\ here ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 342329, root := ~, start := 342321, stop := 342328 ), rec( content := " is one \ of the ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismsDigraphsRepresentatives", count := [ 7, 3, 3, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 342387, root := ~, start := 342344, stop := 342386 ), rec( content := "\n an\ d ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 342404, root := ~, start := 342396, stop := 342403 ), rec( content := " is an a\ utomorphism of ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 342442, root := ~, start := 342427, stop := 342441 ), rec( content := ".\n " , count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "P", next := 342452, root := ~, start := 342448, stop := 342451 ), rec( content := "\n\n \ ", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismsDigraphs", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "C", next := 342486, root := ~, start := 342458, stop := 342485 ), rec( content := " returns\ all homomorphisms between\n ", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "A", next := 342540, root := ~, start := 342525, stop := 342539 ), rec( content := " and ", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "A", next := 342560, root := ~, start := 342545, stop := 342559 ), rec( content := ".\n " , count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> HomomorphismsDigrap\ hs(gr1, gr2);\n[ Transformation( [ 1, 3, 1 ] ), Transformation( [ 1, 3, 3 ] ),\ \n Transformation( [ 1, 5, 4, 4, 5 ] ), Transformation( [ 2, 2, 2 ] ), \n T\ ransformation( [ 3, 1, 3 ] ), Transformation( [ 3, 1, 5, 4, 5 ] ), \n Transfo\ rmation( [ 3, 3, 1 ] ), Transformation( [ 3, 3, 3 ] ) ]\ngap> HomomorphismsDig\ raphsRepresentatives(gr1, CompleteDigraph(3));\n[ Transformation( [ 2, 1 ] ), \ Transformation( [ 2, 1, 2 ] ) ]\n", count := [ 7, 3, 3, 7 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 7 ], mathmode := "Text", name := "Example", next := 343166, root := ~, start := 342566, stop := 343165 ), rec( content := "\n ", count := [ 7, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Description", next := 343183, root := ~, start := 341876, stop := 343182 ) ], count := [ 7, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 343197, root := ~, start := 341669, stop := 343196 ), rec( content := "\n\n ", count := [ 7, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphMonomorphism" ), content := 0, count := [ 7, 3, 4, 2 ], mathmode := "Text", name := "Oper", next := 343277, root := ~, start := 343218, stop := 343276 ), rec( attributes := rec( ), content := [ rec( content := "A transfo\ rmation, or ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "K", next := 343321, root := ~, start := 343310, stop := 343320 ), rec( content := ".", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 343332, root := ~, start := 343280, stop := 343331 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphMonomorphism", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "C", next := 343379, root := ~, start := 343353, stop := 343378 ), rec( content := " returns\ a single ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ njective", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "E", next := 343413, root := ~, start := 343397, stop := 343412 ), rec( content := " ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 343453, root := ~, start := 343414, stop := 343452 ), rec( content := " between\ ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "A", next := 343477, root := ~, start := 343462, stop := 343476 ), rec( content := " and ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "A", next := 343497, root := ~, start := 343482, stop := 343496 ), rec( content := "\n if\ one exists, otherwise it returns ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "K", next := 343549, root := ~, start := 343538, stop := 343548 ), rec( content := ".\n " , count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> DigraphMonomorphism\ (gr1, gr1);\nIdentityTransformation\ngap> DigraphMonomorphism(gr1, gr2);\nTran\ sformation( [ 3, 1, 5, 4, 5 ] )\n", count := [ 7, 3, 4, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 4 ], mathmode := "Text", name := "Example", next := 343846, root := ~, start := 343555, stop := 343845 ), rec( content := "\n ", count := [ 7, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 343863, root := ~, start := 343335, stop := 343862 ) ], count := [ 7, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 343877, root := ~, start := 343203, stop := 343876 ), rec( content := "\n\n ", count := [ 7, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "MonomorphismsDigraphs" ), content := 0, count := [ 7, 3, 5, 2 ], mathmode := "Text", name := "Oper", next := 343959, root := ~, start := 343898, stop := 343958 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "MonomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 5, 4 ], mathmode := "Text", name := "Oper", next := 344038, root := ~, start := 343962, stop := 344037 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Returns", next := 344086, root := ~, start := 344041, stop := 344085 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations behave the same as ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphs" ), content := 0, count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Ref", next := 344178, root := ~, start := 344143, stop := 344177 ), rec( content := " and\n \ ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Ref", next := 344237, root := ~, start := 344187, stop := 344236 ), rec( content := ", except\ they only return\n ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ njective", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "E", next := 344283, root := ~, start := 344267, stop := 344282 ), rec( content := " homomor\ phisms.\n ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> MonomorphismsDigrap\ hs(gr1, gr2);\n[ Transformation( [ 1, 5, 4, 4, 5 ] ), \n Transformation( [ 3,\ 1, 5, 4, 5 ] ) ]\ngap> MonomorphismsDigraphsRepresentatives(gr1, CompleteDigr\ aph(3));\n[ Transformation( [ 2, 1 ] ) ]\n", count := [ 7, 3, 5, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 5, 6 ], mathmode := "Text", name := "Example", next := 344680, root := ~, start := 344303, stop := 344679 ), rec( content := "\n ", count := [ 7, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 344697, root := ~, start := 344089, stop := 344696 ) ], count := [ 7, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 344711, root := ~, start := 343883, stop := 344710 ), rec( content := "\n\n ", count := [ 7, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphEpimorphism" ), content := 0, count := [ 7, 3, 6, 2 ], mathmode := "Text", name := "Oper", next := 344790, root := ~, start := 344732, stop := 344789 ), rec( attributes := rec( ), content := [ rec( content := "A transfo\ rmation, or ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "K", next := 344834, root := ~, start := 344823, stop := 344833 ), rec( content := ".", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 344845, root := ~, start := 344793, stop := 344844 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEpimorphism", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "C", next := 344891, root := ~, start := 344866, stop := 344890 ), rec( content := " returns\ a single ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ urjective", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "E", next := 344926, root := ~, start := 344909, stop := 344925 ), rec( content := " ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Ref", next := 344966, root := ~, start := 344927, stop := 344965 ), rec( content := " between\ ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "A", next := 344990, root := ~, start := 344975, stop := 344989 ), rec( content := " and ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "A", next := 345010, root := ~, start := 344995, stop := 345009 ), rec( content := "\n if\ one exists, otherwise it returns ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "K", next := 345062, root := ~, start := 345051, stop := 345061 ), rec( content := ".\n " , count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphRemoveEdge(CompleteDigraph(3), [1, 2]);\n\ngap> DigraphEpimorphism(gr2, gr1);\ \nfail\ngap> DigraphEpimorphism(gr1, gr2);\nTransformation( [ 3, 1, 2, 3 ] )\n\ ", count := [ 7, 3, 6, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 4 ], mathmode := "Text", name := "Example", next := 345402, root := ~, start := 345068, stop := 345401 ), rec( content := "\n ", count := [ 7, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 345419, root := ~, start := 344848, stop := 345418 ) ], count := [ 7, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 345433, root := ~, start := 344717, stop := 345432 ), rec( content := "\n\n ", count := [ 7, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "EpimorphismsDigraphs" ), content := 0, count := [ 7, 3, 7, 2 ], mathmode := "Text", name := "Oper", next := 345514, root := ~, start := 345454, stop := 345513 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "EpimorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 7, 4 ], mathmode := "Text", name := "Oper", next := 345592, root := ~, start := 345517, stop := 345591 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Returns", next := 345640, root := ~, start := 345595, stop := 345639 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations behave the same as ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphs" ), content := 0, count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 345732, root := ~, start := 345697, stop := 345731 ), rec( content := " and\n \ ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 345791, root := ~, start := 345741, stop := 345790 ), rec( content := ", except\ they only return\n ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ urjective", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "E", next := 345838, root := ~, start := 345821, stop := 345837 ), rec( content := " homomor\ phisms.\n ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphSymmetricClosure(CycleDigraph(3));\n\ngap> EpimorphismsDigraphsRepre\ sentatives(gr1, gr2);\n[ Transformation( [ 3, 1, 2, 1 ] ), Transformation( [ 3\ , 1, 2, 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ) ]\ngap> EpimorphismsDigrap\ hs(gr1, gr2);\n[ Transformation( [ 1, 2, 1, 3 ] ), Transformation( [ 1, 2, 3, \ 1 ] ), \n Transformation( [ 1, 2, 3, 2 ] ), Transformation( [ 1, 3, 1, 2 ] ),\ \n Transformation( [ 1, 3, 2, 1 ] ), Transformation( [ 1, 3, 2, 3 ] ), \n T\ ransformation( [ 2, 1, 2, 3 ] ), Transformation( [ 2, 1, 3, 1 ] ), \n Transfo\ rmation( [ 2, 1, 3, 2 ] ), Transformation( [ 2, 3, 1, 2 ] ), \n Transformatio\ n( [ 2, 3, 1, 3 ] ), Transformation( [ 2, 3, 2, 1 ] ), \n Transformation( [ 3\ , 1, 2, 1 ] ), Transformation( [ 3, 1, 2, 3 ] ), \n Transformation( [ 3, 1, 3\ , 2 ] ), Transformation( [ 3, 2, 1, 2 ] ), \n Transformation( [ 3, 2, 1, 3 ] \ ), Transformation( [ 3, 2, 3, 1 ] ) ]\n", count := [ 7, 3, 7, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 7, 6 ], mathmode := "Text", name := "Example", next := 346925, root := ~, start := 345858, stop := 346924 ), rec( content := "\n ", count := [ 7, 3, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Description", next := 346942, root := ~, start := 345643, stop := 346941 ) ], count := [ 7, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 346956, root := ~, start := 345439, stop := 346955 ), rec( content := "\n\n ", count := [ 7, 3, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphEmbedding" ), content := 0, count := [ 7, 3, 8, 2 ], mathmode := "Text", name := "Oper", next := 347033, root := ~, start := 346977, stop := 347032 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "K", next := 347078, root := ~, start := 347067, stop := 347077 ), rec( content := ".", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 347089, root := ~, start := 347036, stop := 347088 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ embedding of a digraph ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 347151, root := ~, start := 347136, stop := 347150 ), rec( content := " into an\ other digraph\n ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 347192, root := ~, start := 347177, stop := 347191 ), rec( content := " is a ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Ref", next := 347231, root := ~, start := 347198, stop := 347230 ), rec( content := " from ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 347252, root := ~, start := 347237, stop := 347251 ), rec( content := "\n to\ ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 347275, root := ~, start := 347260, stop := 347274 ), rec( content := " which h\ as the additional property that a pair of\n vertices ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 347351, root := ~, start := 347338, stop := 347350 ), rec( content := " which h\ ave no edge ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 347384, root := ~, start := 347371, stop := 347383 ), rec( content := " in ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 347403, root := ~, start := 347388, stop := 347402 ), rec( content := " are\n\n\ mapped to a pair of vertices ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ a, b]", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 347455, root := ~, start := 347442, stop := 347454 ), rec( content := " which h\ ave no edge ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ ->b", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 347486, root := ~, start := 347475, stop := 347485 ), rec( content := " in\n \ ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 347509, root := ~, start := 347494, stop := 347508 ), rec( content := ".", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "P", next := 347514, root := ~, start := 347510, stop := 347513 ), rec( content := "\n\n \ In other words, an embedding ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "C", next := 347557, root := ~, start := 347549, stop := 347556 ), rec( content := " is an i\ somorphism from\n ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "A", next := 347600, root := ~, start := 347585, stop := 347599 ), rec( content := " to the \ ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Induced\ Subdigraph" ), content := 0, count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "Ref", next := 347639, root := ~, start := 347608, stop := 347638 ), rec( content := " of ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "A", next := 347658, root := ~, start := 347643, stop := 347657 ), rec( content := "\n on\ the image of ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "C", next := 347687, root := ~, start := 347679, stop := 347686 ), rec( content := ". ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "P", next := 347693, root := ~, start := 347689, stop := 347692 ), rec( content := "\n\n \ ", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEmbedding", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "C", next := 347722, root := ~, start := 347699, stop := 347721 ), rec( content := " returns\ a single embedding if one exists, otherwise\n it returns ", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "K", next := 347801, root := ~, start := 347790, stop := 347800 ), rec( content := ".\n " , count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := ChainDigraph(3);\n\ngap> \ DigraphEmbedding(gr, CompleteDigraph(4));\nfail\ngap> DigraphEmbedding(gr, Dig\ raph([[3], [1, 4], [1], [3]]));\nTransformation( [ 2, 4, 3, 4 ] )\n", count := [ 7, 3, 8, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 8, 6 ], mathmode := "Text", name := "Example", next := 348055, root := ~, start := 347807, stop := 348054 ), rec( content := "\n ", count := [ 7, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 348072, root := ~, start := 347092, stop := 348071 ) ], count := [ 7, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 348086, root := ~, start := 346962, stop := 348085 ), rec( content := "\n\n ", count := [ 7, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2", Name := "EmbeddingsDigra\ phs" ), content := 0, count := [ 7, 3, 9, 2 ], mathmode := "Text", name := "Oper", next := 348153, root := ~, start := 348107, stop := 348152 ), rec( attributes := rec( Arg := "D1, D2", Name := "EmbeddingsDigra\ phsRepresentatives" ), content := 0, count := [ 7, 3, 9, 4 ], mathmode := "Text", name := "Oper", next := 348217, root := ~, start := 348156, stop := 348216 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 348265, root := ~, start := 348220, stop := 348264 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations behave the same as ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphs" ), content := 0, count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 348357, root := ~, start := 348322, stop := 348356 ), rec( content := " and\n \ ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 348416, root := ~, start := 348366, stop := 348415 ), rec( content := ", except\ they only return\n embeddings of ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "A", next := 348469, root := ~, start := 348460, stop := 348468 ), rec( content := " into ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "A", next := 348484, root := ~, start := 348475, stop := 348483 ), rec( content := ".", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "P", next := 348489, root := ~, start := 348485, stop := 348488 ), rec( content := "\n\n \ See also ", count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsDigra\ phEmbedding" ), content := 0, count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 348536, root := ~, start := 348504, stop := 348535 ), rec( content := ".\n " , count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := NullDigraph(2);\n\ngap> D\ 2 := CycleDigraph(5);\n\ngap> Embeddi\ ngsDigraphsRepresentatives(D1, D2);\n[ Transformation( [ 1, 3, 3 ] ), Transfor\ mation( [ 1, 4, 3, 4 ] ) ]\ngap> EmbeddingsDigraphs(D1, D2);\n[ Transformation\ ( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 ] ), \n Transformation( [ 2, 4,\ 4, 5, 1 ] ), \n Transformation( [ 2, 5, 4, 5, 1 ] ), \n Transformation( [ 3\ , 1, 5, 1, 2 ] ), \n Transformation( [ 3, 5, 5, 1, 2 ] ), \n Transformation(\ [ 4, 1, 1, 2, 3 ] ), \n Transformation( [ 4, 2, 1, 2, 3 ] ), \n Transformat\ ion( [ 5, 2, 2, 3, 4 ] ), \n Transformation( [ 5, 3, 2, 3, 4 ] ) ]\n", count := [ 7, 3, 9, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 9, 7 ], mathmode := "Text", name := "Example", next := 349250, root := ~, start := 348542, stop := 349249 ), rec( content := "\n ", count := [ 7, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 349267, root := ~, start := 348268, stop := 349266 ) ], count := [ 7, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 349281, root := ~, start := 348092, stop := 349280 ), rec( content := "\n\n ", count := [ 7, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphHomomorphism" ), content := 0, count := [ 7, 3, 10, 2 ], mathmode := "Text", name := "Oper", next := 349422, root := ~, start := 349302, stop := 349421 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphHomomorphism" ), content := 0, count := [ 7, 3, 10, 4 ], mathmode := "Text", name := "Oper", next := 349491, root := ~, start := 349425, stop := 349490 ), rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphEpimorphism" ), content := 0, count := [ 7, 3, 10, 6 ], mathmode := "Text", name := "Oper", next := 349612, root := ~, start := 349494, stop := 349611 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEpimorphism" ), content := 0, count := [ 7, 3, 10, 8 ], mathmode := "Text", name := "Oper", next := 349680, root := ~, start := 349615, stop := 349679 ), rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphMonomorphism" ), content := 0, count := [ 7, 3, 10, 10 ], mathmode := "Text", name := "Oper", next := 349802, root := ~, start := 349683, stop := 349801 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphMonomorphism" ), content := 0, count := [ 7, 3, 10, 12 ], mathmode := "Text", name := "Oper", next := 349871, root := ~, start := 349805, stop := 349870 ), rec( attributes := rec( Arg := "digraph, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphEndomorphism" ), content := 0, count := [ 7, 3, 10, 14 ], mathmode := "Text", name := "Oper", next := 349992, root := ~, start := 349874, stop := 349991 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphEndomo\ rphism" ), content := 0, count := [ 7, 3, 10, 16 ], mathmode := "Text", name := "Oper", next := 350053, root := ~, start := 349995, stop := 350052 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "K", next := 350076, root := ~, start := 350065, stop := 350075 ), rec( content := " or ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "K", next := 350092, root := ~, start := 350080, stop := 350091 ), rec( content := ".", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Returns", next := 350103, root := ~, start := 350056, stop := 350102 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphHomomorphism", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "C", next := 350152, root := ~, start := 350124, stop := 350151 ), rec( content := " returns\ ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "K", next := 350172, root := ~, start := 350161, stop := 350171 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "A", next := 350222, root := ~, start := 350214, stop := 350221 ), rec( content := " is a ho\ momorphism from the digraph\n ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "A", next := 350272, root := ~, start := 350262, stop := 350271 ), rec( content := " to the \ digraph ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "A", next := 350298, root := ~, start := 350288, stop := 350297 ), rec( content := ".\n " , count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "P", next := 350308, root := ~, start := 350304, stop := 350307 ), rec( content := "\n\n \ ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphEpimorphism", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "C", next := 350341, root := ~, start := 350314, stop := 350340 ), rec( content := " returns\ ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "K", next := 350361, root := ~, start := 350350, stop := 350360 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "A", next := 350411, root := ~, start := 350403, stop := 350410 ), rec( content := " is a su\ rjective homomorphism from the digraph\n ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "A", next := 350472, root := ~, start := 350462, stop := 350471 ), rec( content := " to the \ digraph ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "A", next := 350498, root := ~, start := 350488, stop := 350497 ), rec( content := ".\n " , count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "P", next := 350508, root := ~, start := 350504, stop := 350507 ), rec( content := "\n\n \ ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphMonomorphism", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "C", next := 350542, root := ~, start := 350514, stop := 350541 ), rec( content := " returns\ ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "K", next := 350562, root := ~, start := 350551, stop := 350561 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "A", next := 350612, root := ~, start := 350604, stop := 350611 ), rec( content := " is an i\ njective homomorphism from the digraph\n ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "A", next := 350673, root := ~, start := 350663, stop := 350672 ), rec( content := " to the \ digraph ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "A", next := 350699, root := ~, start := 350689, stop := 350698 ), rec( content := ".\n " , count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "P", next := 350709, root := ~, start := 350705, stop := 350708 ), rec( content := "\n\n \ ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphEndomorphism", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "C", next := 350743, root := ~, start := 350715, stop := 350742 ), rec( content := " returns\ ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "K", next := 350763, root := ~, start := 350752, stop := 350762 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "A", next := 350813, root := ~, start := 350805, stop := 350812 ), rec( content := " is an e\ ndomorphism of the digraph\n ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "A", next := 350866, root := ~, start := 350852, stop := 350865 ), rec( content := ".\n " , count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "P", next := 350876, root := ~, start := 350872, stop := 350875 ), rec( content := "\n\n \ A permutation or transformation ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "A", next := 350922, root := ~, start := 350914, stop := 350921 ), rec( content := " is a ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ omomorphism", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "E", next := 350947, root := ~, start := 350928, stop := 350946 ), rec( content := " from a\\ n digraph ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "A", next := 350977, root := ~, start := 350967, stop := 350976 ), rec( content := " to a di\ graph ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "A", next := 351001, root := ~, start := 350991, stop := 351000 ), rec( content := " if the \ following hold:\n ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 23 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[u ^ ", count := [ 7, 3, 10, 23 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "x", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "A"\ , next := 351073, root := ~, start := 351065, stop := 351072 ), rec( content :\ = ", v ^ ", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ), rec( attributes := rec( ), content := [ rec( content := "x", coun\ t := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], co\ unt := [ 7, 3, 10, 23 ], mathmode := "Text", name := "A", next := 351087, root\ := ~, start := 351079, stop := 351086 ), rec( content := "]", count := [ 7, 3\ , 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 3, 10, 23 ], mathmode := "Text", name := "C", next := 351092, root := ~, star\ t := 351057, stop := 351091 ), rec( content := " is an edge of\n ", coun\ t := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "ran", count := [ 7, 3, 10, 23\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, \ 23 ], mathmode := "Text", name := "A", next := 351125, root := ~, start := 351\ 115, stop := 351124 ), rec( content := " whenever ", count := [ 7, 3, 10, 23 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[u, v]", count := [ 7, 3, 10, 23 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 23 ], mathmode := "T\ ext", name := "C", next := 351148, root := ~, start := 351135, stop := 351147 ), rec( cont\ ent := " is an\n edge of ", count := [ 7, 3, 10, 23 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "src", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 7, 3, 10, 23 ], mathmode := "Text", name := \ "A", next := 351181, root := ~, start := 351171, stop := 351180 ), rec( conten\ t := "; and ", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 3, 10, 23 ], name := "Item", next := 351194, root := ~, start := 351\ 042, stop := 351193 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x", \ count := [ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 7, 3, 10, 25 ], mathmode := "Text", name := "A", next := 351224, root :\ = ~, start := 351216, stop := 351223 ), rec( content := " fixes every ", count\ := [ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 3, 10, 25 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25\ ], mathmode := "Text", name := "C", next := 351245, root := ~, start := 35123\ 7, stop := 351244 ), rec( content := " which is not a vertex of ", count := [ \ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "src", count := [ 7, 3, 10, 25 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25\ ], mathmode := "Text", name := "A", next := 351281, root := ~, start := 35127\ 1, stop := 351280 ), rec( content := ".\n ", count := [ 7, 3, 10, 25 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25 ], name := "Ite\ m", next := 351296, root := ~, start := 351201, stop := 351295 ) ], count := [ 7, 3, 10, 22 ], mathmode := "Text", name := "List", next := 351308, root := ~, start := 351029, stop := 351307 ), rec( content := "\n Se\ e also ", count := [ 7, 3, 10, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Generat\ orsOfEndomorphismMonoid" ), content := 0, count := [ 7, 3, 10, 27 ], mathmode := "Text", name := "Ref", next := 351366, root := ~, start := 351322, stop := 351365 ), rec( content := ".", count := [ 7, 3, 10, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "P", next := 351371, root := ~, start := 351367, stop := 351370 ), rec( content := "\n\n\n \ If ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "A", next := 351392, root := ~, start := 351381, stop := 351391 ), rec( content := " and ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "A", next := 351408, root := ~, start := 351397, stop := 351407 ), rec( content := ", or ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "A", next := 351423, root := ~, start := 351413, stop := 351422 ), rec( content := ", are gi\ ven, then they must\n represent vertex colourings; see \n ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "Ref", next := 351569, root := ~, start := 351493, stop := 351568 ), rec( content := " \n f\ or details of the permissible values for these argument. The\n homomorphism\ must then also have the property:\n \n ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 30 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col[i] = col[i ^ x]", count := [ 7, 3, 10, 30 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ], math\ mode := "Text", name := "C", next := 351749, root := ~, start := 351723, stop \ := 351748 ), rec( content := " for all vertices ", count := [ 7, 3, 10, 30 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 3, 10, 30 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ], mathmode := "Text", name\ := "C", next := 351775, root := ~, start := 351767, stop := 351774 ), rec( cont\ ent := " of ", count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 10, 30 ], mathmode := "Text", name := "A", next := 351793, root :\ = ~, start := 351779, stop := 351792 ), rec( content := ",\n in the cas\ e of ", count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := [ rec( content := "IsDigraphEnd\ omorphism", count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "C", next\ := 351846, root := ~, start := 351818, stop := 351845 ), rec( content := ".", coun\ t := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ] , name := "Ite\ m", next := 351854, root := ~, start := 351708, stop := 351853 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 32 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col1[i] = col2[i ^ x]", count := [ 7, 3, 10, 32 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 32 ], math\ mode := "Text", name := "C", next := 351904, root := ~, start := 351876, stop \ := 351903 ), rec( content := " for all vertices ", count := [ 7, 3, 10, 32 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 3, 10, 32 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 32 ], mathmode := "Text", name\ := "C", next := 351930, root := ~, start := 351922, stop := 351929 ), rec( cont\ ent := " of ", count := [ 7, 3, 10, 32 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "src", count\ := [ 7, 3, 10, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 10, 32 ], mathmode := "Text", name := "A", next := 351944, root :\ = ~, start := 351934, stop := 351943 ), rec( content := ",\n in the cas\ es of the other operations.", count := [ 7, 3, 10, 32 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 32 ], name := "Item", next := 3519\ 98, root := ~, start := 351861, stop := 351997 ) ], count := [ 7, 3, 10, 29 ], mathmode := "Text", name := "List", next := 352010, root := ~, start := 351695, stop := 352009 ), rec( content := "\n\n \ See also ", count := [ 7, 3, 10, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ sRespectsColouring" ), content := 0, count := [ 7, 3, 10, 34 ], mathmode := "Text", name := "Ref", next := 352064, root := ~, start := 352025, stop := 352063 ), rec( content := ".\n \\ n ", count := [ 7, 3, 10, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> ran := Digraph([[1], [1, 2]]);\n\ngap> IsDigraphHomomorphism(src, ran, Transformation([1, 2, \ 2]));\ntrue\ngap> IsDigraphHomomorphism(src, ran, Transformation([2, 1, 2]));\ \nfalse\ngap> IsDigraphHomomorphism(src, ran, Transformation([3, 3, 3]));\nfal\ se\ngap> IsDigraphHomomorphism(src, src, Transformation([3, 3, 3]));\ntrue\nga\ p> IsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]),\n> \ [1, 2, 2], [1, 2]);\ntrue\ngap> IsDigraphHomomorphism(src, ran, Tr\ ansformation([1, 2, 2]),\n> [2, 1, 1], [1, 2]);\nfals\ e\ngap> IsDigraphEndomorphism(src, Transformation([3, 3, 3]));\ntrue\ngap> IsD\ igraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 1]);\ntrue\ngap> IsD\ igraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 2]);\nfalse\ngap> Is\ DigraphEpimorphism(src, ran, Transformation([3, 3, 3]));\nfalse\ngap> IsDigrap\ hMonomorphism(src, ran, Transformation([1, 2, 2]));\nfalse\ngap> IsDigraphEpim\ orphism(src, ran, Transformation([1, 2, 2]));\ntrue\ngap> IsDigraphMonomorphis\ m(ran, src, ());\ntrue", count := [ 7, 3, 10, 35 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 35 ], mathmode := "Text", name := "Example", next := 353266, root := ~, start := 352075, stop := 353265 ), rec( content := "\n ", count := [ 7, 3, 10, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Description", next := 353283, root := ~, start := 350106, stop := 353282 ) ], count := [ 7, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 353297, root := ~, start := 349287, stop := 353296 ), rec( content := "\n\n ", count := [ 7, 3, 10, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphEmbedding" ), content := 0, count := [ 7, 3, 11, 2 ], mathmode := "Text", name := "Oper", next := 353434, root := ~, start := 353318, stop := 353433 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEmbedding" ), content := 0, count := [ 7, 3, 11, 4 ], mathmode := "Text", name := "Oper", next := 353500, root := ~, start := 353437, stop := 353499 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "K", next := 353523, root := ~, start := 353512, stop := 353522 ), rec( content := " or ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "K", next := 353539, root := ~, start := 353527, stop := 353538 ), rec( content := ".", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 353550, root := ~, start := 353503, stop := 353549 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphEmbedding", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "C", next := 353596, root := ~, start := 353571, stop := 353595 ), rec( content := " returns\ ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "K", next := 353616, root := ~, start := 353605, stop := 353615 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 353666, root := ~, start := 353658, stop := 353665 ), rec( content := " is a em\ bedding of the digraph\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 353711, root := ~, start := 353701, stop := 353710 ), rec( content := " into th\ e digraph ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 353739, root := ~, start := 353729, stop := 353738 ), rec( content := ", while \ respecting the colourings\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 353788, root := ~, start := 353777, stop := 353787 ), rec( content := " and ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 353804, root := ~, start := 353793, stop := 353803 ), rec( content := " if give\ n.\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "P", next := 353823, root := ~, start := 353819, stop := 353822 ), rec( content := "\n\n \ A permutation or transformation ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 353869, root := ~, start := 353861, stop := 353868 ), rec( content := " is a ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ mbedding", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "E", next := 353891, root := ~, start := 353875, stop := 353890 ), rec( content := " of a di\ graph\n ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 353919, root := ~, start := 353909, stop := 353918 ), rec( content := " into a \ digraph ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 353945, root := ~, start := 353935, stop := 353944 ), rec( content := " if ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 353957, root := ~, start := 353949, stop := 353956 ), rec( content := " is a mo\ nomorphism from\n ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 353995, root := ~, start := 353985, stop := 353994 ), rec( content := " to ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 354009, root := ~, start := 353999, stop := 354008 ), rec( content := " and the\ inverse of ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 354037, root := ~, start := 354029, stop := 354036 ), rec( content := " is a mo\ nomorphism from\n the subdigraph of ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 354093, root := ~, start := 354083, stop := 354092 ), rec( content := " induced\ by the image of ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 354126, root := ~, start := 354118, stop := 354125 ), rec( content := " to\n \ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 354144, root := ~, start := 354134, stop := 354143 ), rec( content := ".\n\n \ See also ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsDigra\ phHomomorphism" ), content := 0, count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "Ref", next := 354195, root := ~, start := 354160, stop := 354194 ), rec( content := ".", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 11, 7 ], mathmode := "Text", name := "P", next := 354200, root := ~, start := 354196, stop := 354199 ), rec( content := "\n\n \ ", count := [ 7, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2]]);\n\ngap> ran := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, ());\ntrue\ngap> IsDig\ raphEmbedding(src, ran, ());\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, \ 1], [2, 1, 1]);\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, 1], [1, 2, 1]\ );\nfalse\ngap> ran := Digraph([[1, 2], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, IdentityTransf\ ormation);\ntrue\ngap> IsDigraphEmbedding(src, ran, IdentityTransformation);\n\ false", count := [ 7, 3, 11, 8 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 8 ], mathmode := "Text", name := "Example", next := 354849, root := ~, start := 354206, stop := 354848 ), rec( content := "\n ", count := [ 7, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 354866, root := ~, start := 353553, stop := 354865 ) ], count := [ 7, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 354880, root := ~, start := 353303, stop := 354879 ), rec( content := "\n\n ", count := [ 7, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "DigraphsRespectsColouring" ), content := 0, count := [ 7, 3, 12, 2 ], mathmode := "Text", name := "Oper", next := 354975, root := ~, start := 354905, stop := 354974 ), rec( attributes := rec( ), content := [ rec( content := " ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "K", next := 355001, root := ~, start := 354990, stop := 355000 ), rec( content := " or ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "K", next := 355017, root := ~, start := 355005, stop := 355016 ), rec( content := ". ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 355029, root := ~, start := 354980, stop := 355028 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphsRespectsColouring", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 355100, root := ~, start := 355068, stop := 355099 ), rec( content := " verifie\ s whether or not \n the permutation or transformation ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 355174, root := ~, start := 355166, stop := 355173 ), rec( content := " respect\ s the vertex colourings\n ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 355223, root := ~, start := 355212, stop := 355222 ), rec( content := " and ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 355239, root := ~, start := 355228, stop := 355238 ), rec( content := " of the \ digraphs ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 355266, root := ~, start := 355256, stop := 355265 ), rec( content := " and ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ ange", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 355283, root := ~, start := 355271, stop := 355282 ), rec( content := ".\n \ That is, ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphsRespectsColouring", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 355332, root := ~, start := 355300, stop := 355331 ), rec( content := " returns\ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "K", next := 355352, root := ~, start := 355341, stop := 355351 ), rec( content := " if and \ only if for\n all vertices ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 355399, root := ~, start := 355391, stop := 355398 ), rec( content := " of ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 355413, root := ~, start := 355403, stop := 355412 ), rec( content := ", ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1[i] = col2[i ^ x]", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 355443, root := ~, start := 355415, stop := 355442 ), rec( content := ".\n \ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 12, 4 ], mathmode := "Text", name := "P", next := 355455, root := ~, start := 355451, stop := 355454 ), rec( content := "\n\n \ ", count := [ 7, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2]]);\n\ngap> ran := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> DigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [1\ , 2, 1]);\ntrue\ngap> DigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [2, \ 1, 1]);\nfalse\n", count := [ 7, 3, 12, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 5 ], mathmode := "Text", name := "Example", next := 355814, root := ~, start := 355463, stop := 355813 ), rec( content := "\n ", count := [ 7, 3, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 355833, root := ~, start := 355034, stop := 355832 ) ], count := [ 7, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 355849, root := ~, start := 354888, stop := 355848 ), rec( content := "\n\n ", count := [ 7, 3, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, colors\ ][, limit]", Name := "GeneratorsOfEndomorphismMonoid" ), content := 0, count := [ 7, 3, 13, 2 ], mathmode := "Text", name := "Func", next := 355948, root := ~, start := 355870, stop := 355947 ), rec( attributes := rec( Arg := "digraph", Name := "GeneratorsOfEnd\ omorphismMonoidAttr" ), content := 0, count := [ 7, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 356014, root := ~, start := 355951, stop := 356013 ), rec( attributes := rec( ), content := [ rec( content := " A list o\ f transformations.", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Returns", next := 356063, root := ~, start := 356017, stop := 356062 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ endomorphism of ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 356117, root := ~, start := 356103, stop := 356116 ), rec( content := " is a ho\ momorphism\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Ref", next := 356173, root := ~, start := 356140, stop := 356172 ), rec( content := " from ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 356193, root := ~, start := 356179, stop := 356192 ), rec( content := " back to\ itself.\n\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfEndomorphismMonoid", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "C", next := 356252, root := ~, start := 356215, stop := 356251 ), rec( content := ", called\ with a single argument,\n returns a generating set for the monoid of all e\ ndomorphisms of\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 356371, root := ~, start := 356357, stop := 356370 ), rec( content := ". If ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 356390, root := ~, start := 356376, stop := 356389 ), rec( content := " belong\ s to ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Ref", next := 356441, root := ~, start := 356403, stop := 356440 ), rec( content := ", then t\ he value of\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfEndomorphismMonoid", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "C", next := 356502, root := ~, start := 356465, stop := 356501 ), rec( content := " will no\ t be recomputed on future\n calls.", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "P", next := 356550, root := ~, start := 356546, stop := 356549 ), rec( content := "\n\n \ If the ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olors", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "A", next := 356576, root := ~, start := 356563, stop := 356575 ), rec( content := " argumen\ t is specified, then\n ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfEndomorphismMonoid", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "C", next := 356646, root := ~, start := 356609, stop := 356645 ), rec( content := " will re\ turn a generating\n set for the monoid of endomorphisms which respect the g\ iven colouring.\n The colouring ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olors", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "A", next := 356778, root := ~, start := 356765, stop := 356777 ), rec( content := " can be \ in one of two forms: ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 13, 7 ], mathmode := "Text", name := "P", next := 356811, root := ~, start := 356807, stop := 356810 ), rec( content := "\n\n \ ", count := [ 7, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n A list of positive integers \ of size the number of vertices of\n ", count := [ 7, 3, 13, 9 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "digraph", count := [ 7, 3, 13, 9 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Tex\ t", name := "A", next := 356929, root := ~, start := 356915, stop := 356928 ), rec( cont\ ent := ", where ", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "co\ lors", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "A", next := 356\ 950, root := ~, start := 356937, stop := 356949 ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[i]", count := [ 7, 3, 13, 9 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Text", name\ := "C", next := 356960, root := ~, start := 356950, stop := 356959 ), rec( cont\ ent := " is the colour of vertex\n ", count := [ 7, 3, 13, 9 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "i", count := [ 7, 3, 13, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Text", name\ := "C", next := 357001, root := ~, start := 356993, stop := 357000 ), rec( cont\ ent := ".\n ", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], name := "Item", next := 357016, root := ~, start := 356\ 830, stop := 357015 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n A list of lists, such that ", count := [ 7, 3, 13, 11 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "colors", count := [ 7, 3, 13, 11 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 11 ], mathmode := "Text\ ", name := "A", next := 357078, root := ~, start := 357065, stop := 357077 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 3, 13, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, \ 11 ], mathmode := "Text", name := "C", next := 357088, root := ~, start := 357\ 078, stop := 357087 ), rec( content := " is a list of all\n vertices wi\ th colour ", count := [ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 7, 3, 13, 11 ], mathmode := "Text", name := "C", next := 357143, root :=\ ~, start := 357135, stop := 357142 ), rec( content := ".\n ", count := [ \ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 11 ] , name := "Ite\ m", next := 357158, root := ~, start := 357023, stop := 357157 ) ], count := [ 7, 3, 13, 8 ], mathmode := "Text", name := "List", next := 357170, root := ~, start := 356817, stop := 357169 ), rec( content := "\n\n \ If the ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "A", next := 357195, root := ~, start := 357183, stop := 357194 ), rec( content := " argumen\ t is specified, then it will return only the\n first ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "A", next := 357270, root := ~, start := 357258, stop := 357269 ), rec( content := " homomor\ phisms, where ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "A", next := 357304, root := ~, start := 357292, stop := 357303 ), rec( content := " must be\ a positive\n integer or ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "C", next := 357354, root := ~, start := 357339, stop := 357353 ), rec( content := ". ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 13, 14 ], mathmode := "Text", name := "P", next := 357360, root := ~, start := 357356, stop := 357359 ), rec( content := "\n\n \ ", count := [ 7, 3, 13, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph(List([1 .. 3], x -> [1 .. 3]));;\ngap> GeneratorsOfEndomor\ phismMonoid(gr);\n[ Transformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ),\ \n IdentityTransformation, Transformation( [ 1, 2, 1 ] ), \n Transformation\ ( [ 1, 2, 2 ] ), Transformation( [ 1, 1, 2 ] ), \n Transformation( [ 1, 1, 1 \ ] ) ]\ngap> GeneratorsOfEndomorphismMonoid(gr, 3);\n[ Transformation( [ 1, 3, \ 2 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransformation ]\ngap> gr := Co\ mpleteDigraph(3);;\ngap> GeneratorsOfEndomorphismMonoid(gr);\n[ Transformation\ ( [ 2, 3, 1 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransformation ]\ngap\ > GeneratorsOfEndomorphismMonoid(gr, [1, 2, 2]);\n[ Transformation( [ 1, 3, 2 \ ] ), IdentityTransformation ]\ngap> GeneratorsOfEndomorphismMonoid(gr, [[1], [\ 2, 3]]);\n[ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ]\n", count := [ 7, 3, 13, 15 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 15 ], mathmode := "Text", name := "Example", next := 358229, root := ~, start := 357366, stop := 358228 ), rec( content := "\n ", count := [ 7, 3, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Description", next := 358246, root := ~, start := 356066, stop := 358245 ) ], count := [ 7, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 358260, root := ~, start := 355855, stop := 358259 ), rec( content := "\n\n ", count := [ 7, 3, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, n", Label := "for a digraph \ and a number of colours", Name := "DigraphColouring" ), content := 0, count := [ 7, 3, 14, 2 ], mathmode := "Text", name := "Oper", next := 358379, root := ~, start := 358281, stop := 358378 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "K", next := 358424, root := ~, start := 358413, stop := 358423 ), rec( content := ".", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 358435, root := ~, start := 358382, stop := 358434 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "E", next := 358481, root := ~, start := 358458, stop := 358480 ), rec( content := " of a di\ graph is a labelling of its vertices in\n such a way that adjacent vertices\ have different labels. A ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper\n ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 7, 3, 14, 3 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := \ "Text", name := "C", next := 358616, root := ~, start := 358608, stop := 35861\ 5 ), rec( content := "-colouring", count := [ 7, 3, 14, 3 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "E", next := 358630, root := ~, start := 358592, stop := 358629 ), rec( content := " is a pr\ oper colouring that uses exactly ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 358679, root := ~, start := 358671, stop := 358678 ), rec( content := "\n co\ lours. Equivalently, a proper (", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 358726, root := ~, start := 358718, stop := 358725 ), rec( content := "-)colour\ ing of a digraph can be\n defined to be a ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Epimorphism" ), content := 0, count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 358810, root := ~, start := 358778, stop := 358809 ), rec( content := " from a \ digraph onto the\n complete digraph (with ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 358870, root := ~, start := 358862, stop := 358869 ), rec( content := " vertice\ s); see ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Complet\ eDigraph" ), content := 0, count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 358921, root := ~, start := 358886, stop := 358920 ), rec( content := ". Note \ that a digraph with loops (", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 358991, root := ~, start := 358956, stop := 358990 ), rec( content := ") does n\ ot have a proper ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 359024, root := ~, start := 359016, stop := 359023 ), rec( content := "-colouri\ ng for\n any value ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 359061, root := ~, start := 359053, stop := 359060 ), rec( content := ".\n " , count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "P", next := 359071, root := ~, start := 359067, stop := 359070 ), rec( content := "\n\n \ If ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 359094, root := ~, start := 359080, stop := 359093 ), rec( content := " is a di\ graph and ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 359120, root := ~, start := 359112, stop := 359119 ), rec( content := " is a no\ n-negative integer, then\n ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphColouring(", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "di\ graph", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 3591\ 91, root := ~, start := 359177, stop := 359190 ), rec( content := ", ", count := [ 7, 3, 14, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "n", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 3592\ 01, root := ~, start := 359193, stop := 359200 ), rec( content := ")", count := [ 7, 3, 14, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "C", next := 359206, root := ~, start := 359157, stop := 359205 ), rec( content := " returns\ an epimorphism\n from ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 359253, root := ~, start := 359239, stop := 359252 ), rec( content := " onto th\ e complete digraph with ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 359293, root := ~, start := 359285, stop := 359292 ), rec( content := " vertice\ s if one\n exists, else it returns ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "K", next := 359349, root := ~, start := 359338, stop := 359348 ), rec( content := ". ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "P", next := 359355, root := ~, start := 359351, stop := 359354 ), rec( content := "\n\n \ See also ", count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ GreedyColouring", Label := "for a digraph" ), content := 0, count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "Ref", next := 359428, root := ~, start := 359370, stop := 359427 ), rec( content := " and \n \ ", count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "P", next := 359442, root := ~, start := 359438, stop := 359441 ), rec( content := "\n\n \ Note that a digraph with at least two vertices has a 2-colouring if and only\n\ if it is bipartite, see ", count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "Ref", next := 359585, root := ~, start := 359553, stop := 359584 ), rec( content := ".\n\n \ ", count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphColouring(CompleteDigraph(5), 4);\nfail\ngap> DigraphColouring(Ch\ ainDigraph(10), 1);\nfail\ngap> D := ChainDigraph(10);;\ngap> t := DigraphColo\ uring(D, 2);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\ngap> IsDigra\ phColouring(D, t); \ntrue\ngap> DigraphGreedyColouring(D);\nTransformation( [ \ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\n", count := [ 7, 3, 14, 7 ], name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 14, 7 ], mathmode := "Text", name := "Example", next := 359958, root := ~, start := 359592, stop := 359957 ), rec( content := "\n ", count := [ 7, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 359975, root := ~, start := 358438, stop := 359974 ) ], count := [ 7, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 359989, root := ~, start := 358266, stop := 359988 ), rec( content := "\n\n ", count := [ 7, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, order", Label := "for a digraph \ and vertex order", Name := "DigraphGreedyColouring" ), content := 0, count := [ 7, 3, 15, 2 ], mathmode := "Text", name := "Oper", next := 360125, root := ~, start := 360010, stop := 360124 ), rec( attributes := rec( Arg := "digraph, func", Label := "for a digraph \ and vertex order function", Name := "DigraphGreedyColouring" ), content := 0, count := [ 7, 3, 15, 4 ], mathmode := "Text", name := "Oper", next := 360251, root := ~, start := 360128, stop := 360250 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "DigraphGreedyCo\ louring" ), content := 0, count := [ 7, 3, 15, 6 ], mathmode := "Text", name := "Attr", next := 360345, root := ~, start := 360254, stop := 360344 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "K", next := 360390, root := ~, start := 360379, stop := 360389 ), rec( content := ".", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Returns", next := 360401, root := ~, start := 360348, stop := 360400 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "E", next := 360447, root := ~, start := 360424, stop := 360446 ), rec( content := " of a di\ graph is a labelling of its vertices in\n such a way that adjacent vertices\ have different labels. Note that a digraph \n with loops (", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 360622, root := ~, start := 360593, stop := 360621 ), rec( content := ") does n\ ot have any proper \n colouring.\n ", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "P", next := 360673, root := ~, start := 360669, stop := 360672 ), rec( content := "\n \n\ If ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 360700, root := ~, start := 360686, stop := 360699 ), rec( content := " is a di\ graph and ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rder", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 360730, root := ~, start := 360718, stop := 360729 ), rec( content := " is a de\ nse list consisting \n of all of the vertices of ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 360803, root := ~, start := 360789, stop := 360802 ), rec( content := " (in any\ order), then \n ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGreedyColouring", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "C", next := 360860, root := ~, start := 360831, stop := 360859 ), rec( content := "\n us\ es a greedy algorithm with the specified order to obtain some proper\n colo\ uring of ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 360967, root := ~, start := 360953, stop := 360966 ), rec( content := ", which \ may not use the minimal number of\n colours. ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "P", next := 361026, root := ~, start := 361022, stop := 361025 ), rec( content := "\n \n\ If ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 361053, root := ~, start := 361039, stop := 361052 ), rec( content := " is a di\ graph and ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 361082, root := ~, start := 361071, stop := 361081 ), rec( content := " is a fu\ nction whose argument \n is a digraph, and that returns a dense list ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rder", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 361173, root := ~, start := 361161, stop := 361172 ), rec( content := ", then \\ n ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGreedyColouring(", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3612\ 25, root := ~, start := 361211, stop := 361224 ), rec( content := ", ", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "func", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3612\ 38, root := ~, start := 361227, stop := 361237 ), rec( content := ")", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 361243, root := ~, start := 361185, stop := 361242 ), rec( content := " returns\ \n ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGreedyColouring(", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3612\ 96, root := ~, start := 361282, stop := 361295 ), rec( content := ", ", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "func", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3613\ 09, root := ~, start := 361298, stop := 361308 ), rec( content := "(", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3613\ 24, root := ~, start := 361310, stop := 361323 ), rec( content := "))", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 361330, root := ~, start := 361256, stop := 361329 ), rec( content := ".\n " , count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "P", next := 361340, root := ~, start := 361336, stop := 361339 ), rec( content := "\n\n \ If the optional second argument (either a list or a function), is not \n sp\ ecified, then ", count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ WelshPowellOrder" ), content := 0, count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "Ref", next := 361474, root := ~, start := 361437, stop := 361473 ), rec( content := " is used\ by default. \n ", count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "P", next := 361504, root := ~, start := 361500, stop := 361503 ), rec( content := "\n \n\ See also\n ", count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a number of colours", Oper := "DigraphColouring" ), content := 0, count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "Ref", next := 361613, root := ~, start := 361527, stop := 361612 ), rec( content := ".\n " , count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 12 ], mathmode := "Text", name := "P", next := 361623, root := ~, start := 361619, stop := 361622 ), rec( content := "\n\n \ ", count := [ 7, 3, 15, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphGreedyColouring(ChainDigraph(10));\nTransformation( [ 2, 1, 2, 1,\ 2, 1, 2, 1, 2, 1 ] )\ngap> DigraphGreedyColouring(ChainDigraph(10), [1 .. 10]\ );\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\n", count := [ 7, 3, 15, 13 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 15, 13 ], mathmode := "Text", name := "Example", next := 361868, root := ~, start := 361629, stop := 361867 ), rec( content := "\n ", count := [ 7, 3, 15, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Description", next := 361885, root := ~, start := 360404, stop := 361884 ) ], count := [ 7, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 361899, root := ~, start := 359995, stop := 361898 ), rec( content := "\n\n ", count := [ 7, 3, 15, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphWelshPow\ ellOrder" ), content := 0, count := [ 7, 3, 16, 2 ], mathmode := "Text", name := "Attr", next := 361972, root := ~, start := 361920, stop := 361971 ), rec( attributes := rec( ), content := [ rec( content := " A list o\ f the vertices.", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "Returns", next := 362018, root := ~, start := 361975, stop := 362017 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphWelshPowellOrder", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "C", next := 362069, root := ~, start := 362039, stop := 362068 ), rec( content := " returns\ a list of all of the vertices of \n the digraph ", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "A", next := 362142, root := ~, start := 362128, stop := 362141 ), rec( content := " ordered\ according to the sum of the number of \n out- and in-neighbours, from high\ est to lowest. \n ", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 16, 4 ], mathmode := "Text", name := "P", next := 362251, root := ~, start := 362247, stop := 362250 ), rec( content := "\n\n \ ", count := [ 7, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphWelshPowellOrder(Digraph([[4], [9], [9], [], \n> \ [4, 6, 9], [1], [], [], \n> \ [4, 5], [4, 5]]));\n[ 5, 9, 4, 1, 6, 10, 2, 3, 7, 8 ]\n", count := [ 7, 3, 16, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 16, 5 ], mathmode := "Text", name := "Example", next := 362501, root := ~, start := 362257, stop := 362500 ), rec( content := "\n ", count := [ 7, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "Description", next := 362518, root := ~, start := 362021, stop := 362517 ) ], count := [ 7, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 362532, root := ~, start := 361905, stop := 362531 ), rec( content := "\n\n ", count := [ 7, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "ChromaticNumber\ " ), content := 0, count := [ 7, 3, 17, 2 ], mathmode := "Text", name := "Attr", next := 362597, root := ~, start := 362553, stop := 362596 ), rec( attributes := rec( ), content := [ rec( content := " A non-ne\ gative integer.", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 362643, root := ~, start := 362600, stop := 362642 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "E", next := 362689, root := ~, start := 362666, stop := 362688 ), rec( content := " of a di\ graph is a labelling of its\n vertices in such a way that adjacent vertices\ have different labels.\n Equivalently, a proper digraph colouring can be d\ efined to be a ", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Epimorphism" ), content := 0, count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Ref", next := 362904, root := ~, start := 362866, stop := 362903 ), rec( content := " from a \ digraph onto a complete digraph. ", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "P", next := 362949, root := ~, start := 362945, stop := 362948 ), rec( content := "\n\n \ If ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "A", next := 362972, root := ~, start := 362958, stop := 362971 ), rec( content := " is a di\ graph without loops (see ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "Ref", next := 363040, root := ~, start := 363005, stop := 363039 ), rec( content := ", then " , count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ hromaticNumber", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 363069, root := ~, start := 363047, stop := 363068 ), rec( content := " returns\ the least\n non-negative integer ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 363121, root := ~, start := 363113, stop := 363120 ), rec( content := " such th\ at there is a proper colouring of\n ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "A", next := 363181, root := ~, start := 363167, stop := 363180 ), rec( content := " with ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 363195, root := ~, start := 363187, stop := 363194 ), rec( content := " colours\ . In other words, for a digraph with at\n least one vertex, ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ hromaticNumber", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 363288, root := ~, start := 363266, stop := 363287 ), rec( content := " returns\ the least number ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 363322, root := ~, start := 363314, stop := 363321 ), rec( content := "\n su\ ch that ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphColouring(", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "di\ graph", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "A", next := 3633\ 71, root := ~, start := 363357, stop := 363370 ), rec( content := ", n)", count := [ 7, 3, 17, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 363379, root := ~, start := 363337, stop := 363378 ), rec( content := " does no\ t return\n ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "K", next := 363411, root := ~, start := 363400, stop := 363410 ), rec( content := ". See ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a number of colours", Oper := "DigraphColouring" ), content := 0, count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "Ref", next := 363499, root := ~, start := 363417, stop := 363498 ), rec( content := ".\n\n \ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ChromaticNumber(NullDigraph(10));\n1\ngap> ChromaticNumber(CompleteDigra\ ph(10));\n10\ngap> ChromaticNumber(CompleteBipartiteDigraph(5, 5));\n2\ngap> C\ hromaticNumber(Digraph([[], [3], [5], [2, 3], [4]]));\n3\ngap> ChromaticNumber\ (NullDigraph(0));\n0\ngap> D := PetersenGraph(IsMutableDigraph);\n\ngap> ChromaticNumber(D);\n3\n", count := [ 7, 3, 17, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 17, 5 ], mathmode := "Text", name := "Example", next := 363898, root := ~, start := 363506, stop := 363897 ), rec( content := "\n ", count := [ 7, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 363915, root := ~, start := 362646, stop := 363914 ) ], count := [ 7, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 363929, root := ~, start := 362538, stop := 363928 ), rec( content := "\n\n ", count := [ 7, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "DigraphCore" ), content := 0, count := [ 7, 3, 18, 2 ], mathmode := "Text", name := "Attr", next := 363992, root := ~, start := 363954, stop := 363991 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers.", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 364044, root := ~, start := 363997, stop := 364043 ), rec( attributes := rec( ), content := [ rec( content := "\n I\ f ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 364080, root := ~, start := 364072, stop := 364079 ), rec( content := " is a di\ graph, then ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCore", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "C", next := 364118, root := ~, start := 364100, stop := 364117 ), rec( content := " returns\ a list of vertices\n corresponding to the ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ore", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "C", next := 364184, root := ~, start := 364173, stop := 364183 ), rec( content := " of ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 364196, root := ~, start := 364188, stop := 364195 ), rec( content := ". In par\ ticular, the subdigraph\n of ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 364245, root := ~, start := 364237, stop := 364244 ), rec( content := " induced\ by this list is isomorphic to the core of ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 364304, root := ~, start := 364296, stop := 364303 ), rec( content := ".", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "P", next := 364309, root := ~, start := 364305, stop := 364308 ), rec( content := "\n\n \ The ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ore", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "E", next := 364330, root := ~, start := 364319, stop := 364329 ), rec( content := " of a di\ graph ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "C", next := 364352, root := ~, start := 364344, stop := 364351 ), rec( content := " is the \ minimal subdigraph ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "A", next := 364387, root := ~, start := 364379, stop := 364386 ), rec( content := " of\n \ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "C", next := 364403, root := ~, start := 364395, stop := 364402 ), rec( content := " which i\ s a homomorphic image of ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "C", next := 364444, root := ~, start := 364436, stop := 364443 ), rec( content := ". The co\ re of a digraph \n is unique up to isomorphism.\n ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(CycleDigraph(8));\n\ngap> DigraphCore(D);\n[ 1, 2 ]\ngap> D := Pet\ ersenGraph();\n\ngap> DigraphCor\ e(D);\n[ 1 .. 10 ]\ngap> D := Digraph(IsMutableDigraph, [[3], [3], [4], [5], [\ 2]]);\n\ngap> DigraphCore(D);\n[ 2, \ 3, 4, 5 ]\n", count := [ 7, 3, 18, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 5 ], mathmode := "Text", name := "Example", next := 364928, root := ~, start := 364508, stop := 364927 ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 364947, root := ~, start := 364049, stop := 364946 ) ], count := [ 7, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 364963, root := ~, start := 363937, stop := 364962 ), rec( content := "\n\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 364977, root := ~, start := 332783, stop := 364976 ) , rec( content := "\n\n", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 364989, root := ~, start := 283955, stop := 364988 ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " homomorphisms ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Finding cliques and independent sets" ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Finding cliques and indep\ endent sets", count := [ 8, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 365128, root := ~, start := 365073, stop := 365127 ) , rec( content := "\n\n In ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ", a ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "clique", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 365164, root := ~, start := 365151, stop := 365163 ), rec( content := " of a digraph is a set of mutual\ ly adjacent\n vertices of the digraph, and an ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "independent set", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 365266, root := ~, start := 365244, stop := 365265 ), rec( content := " is a set of mutually\n non-a\ djacent vertices of the digraph. A ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal clique", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 365353, root := ~, start := 365332, stop := 365352 ), rec( content := " is a clique\n which is not p\ roperly contained in another clique, and a ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal\n independen\ t set", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 365463, root := ~, start := 365427, stop := 365462 ), rec( content := " is an independent set which is \ not properly contained\n in another independent set. Using this definition\ in ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ", cliques\n and independent s\ ets are both permitted, but not required, to contain vertices\n at which th\ ere is a loop. Another name for a clique is a ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "complete\n subgraph" , count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 365771, root := ~, start := 365741, stop := 365770 ), rec( content := ".\n ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "P", next := 365781, root := ~, start := 365777, stop := 365780 ), rec( content := "\n\n ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " provides extensive functionalit\ y for computing cliques and\n independent sets of a digraph, whether maxima\ l or not. The fundamental\n algorithm used in most of the methods in ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to calculate cliques\n and i\ ndependent sets is a version of the Bron-Kerbosch algorithm.\n Calculating \ the cliques and independent sets of a digraph is a well-known\n and hard pr\ oblem, and searching for cliques or independent sets in a\n digraph can be \ very lengthy, even for a digraph with a small number of\n vertices. ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " uses several strategies to incr\ ease the performance\n of these calculations.\n ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "P", next := 366419, root := ~, start := 366415, stop := 366418 ), rec( content := "\n\n From the definition of c\ liques and independent sets, it follows that the\n presence of loops and mu\ ltiple edges in a digraph is irrelevant to the\n existence of cliques and i\ ndependent sets in the digraph. See ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "DigraphHasLoops" ), content := 0, count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "Ref", next := 366675, root := ~, start := 366640, stop := 366674 ), rec( content := " and ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMultiDigraph" ), content := 0, count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "Ref", next := 366708, root := ~, start := 366680, stop := 366707 ), rec( content := " for more\n information about\ these properties. Therefore given a digraph\n ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "A", next := 366803, root := ~, start := 366789, stop := 366802 ), rec( content := ", the cliques and independent se\ ts of ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "A", next := 366855, root := ~, start := 366841, stop := 366854 ), rec( content := " are\n equal to the cliques a\ nd independent sets of the digraph:\n\n ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphRe\ moveLoops(DigraphRemoveAllMultipleEdges(", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "C", next := 367011, root := ~, start := 366955, stop := 367010 ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "A", next := 367025, root := ~, start := 367011, stop := 367024 ), rec( attributes := rec( ), content := [ rec( content := "))", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "C", next := 367034, root := ~, start := 367025, stop := 367033 ), rec( content := ".\n ", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], name := "Item", next := 367049, root := ~, start := 366940, stop := 367048 ) ], count := [ 8, 0, 0, 6 ], mathmode := "Text", name := "List", next := 367061, root := ~, start := 366927, stop := 367060 ), rec( content := "\n\n See ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphRemoveLoops" ), content := 0, count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "Ref", next := 367103, root := ~, start := 367071, stop := 367102 ), rec( content := " and ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphRemoveAllMultipleEdges" ), content := 0, count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "Ref", next := 367157, root := ~, start := 367108, stop := 367156 ), rec( content := " for more information about thes\ e\n attributes. Furthermore, the cliques of this digraph are equal to the\ \n cliques of the digraph formed by removing any edge ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[u,v]", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "C", next := 367333, root := ~, start := 367321, stop := 367332 ), rec( content := " for which\n the correspondin\ g reverse edge ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[v,u]", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "C", next := 367391, root := ~, start := 367379, stop := 367390 ), rec( content := " is not present. Therefore,\n \ overall, the cliques of ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "A", next := 367462, root := ~, start := 367448, stop := 367461 ), rec( content := " are equal to the cliques of the\ \n symmetric digraph:\n\n ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "MaximalSy\ mmetricSubdigraphWithoutLoops(", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "C", next := 367597, root := ~, start := 367551, stop := 367596 ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "A", next := 367611, root := ~, start := 367597, stop := 367610 ), rec( attributes := rec( ), content := [ rec( content := ")", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "C", next := 367619, root := ~, start := 367611, stop := 367618 ), rec( content := ".\n ", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], name := "Item", next := 367634, root := ~, start := 367536, stop := 367633 ) ], count := [ 8, 0, 0, 10 ], mathmode := "Text", name := "List", next := 367646, root := ~, start := 367523, stop := 367645 ), rec( content := "\n\n See ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "MaximalSymmetricSubdigraphWitho\ utLoops" ), content := 0, count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "Ref", next := 367708, root := ~, start := 367656, stop := 367707 ), rec( content := " for more\n information about\ this. The ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "AutomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "Ref", next := 367810, root := ~, start := 367750, stop := 367809 ), rec( content := " of this symmetric digraph conta\ ins the\n automorphism group of ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "A", next := 367890, root := ~, start := 367876, stop := 367889 ), rec( content := " as a subgroup. By performing t\ he\n search for maximal cliques with the help of this larger automorphism g\ roup\n to reduce the search space, the computation time may be reduced. Th\ e\n functions and attributes which return representatives of cliques of\n \ ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "A", next := 368168, root := ~, start := 368154, stop := 368167 ), rec( content := " will return orbit representativ\ es of cliques under the\n action of the automorphism group of the ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal symmetric subdigr\ aph\n without loops", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "E", next := 368323, root := ~, start := 368268, stop := 368322 ), rec( content := " on sets of vertices.", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "P", next := 368348, root := ~, start := 368344, stop := 368347 ), rec( content := "\n\n The independent sets of \ a digraph are equal to the independent sets of the\n ", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphSymmetricClosure" ), content := 0, count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "Ref", next := 368470, root := ~, start := 368433, stop := 368469 ), rec( content := ". Therefore, overall, the indep\ endent\n sets of ", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "A", next := 368535, root := ~, start := 368521, stop := 368534 ), rec( content := " are equal to the independent se\ ts of the symmetric\n digraph:\n\n ", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphSy\ mmetricClosure(DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "C", next := 368722, root := ~, start := 368633, stop := 368721 ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "A", next := 368736, root := ~, start := 368722, stop := 368735 ), rec( attributes := rec( ), content := [ rec( content := ")))", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "C", next := 368746, root := ~, start := 368736, stop := 368745 ), rec( content := ".\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], name := "Item", next := 368761, root := ~, start := 368618, stop := 368760 ) ], count := [ 8, 0, 0, 15 ], mathmode := "Text", name := "List", next := 368773, root := ~, start := 368605, stop := 368772 ), rec( content := "\n\n Again, the automorphism \ group of this symmetric digraph contains the\n automorphism group of ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "A", next := 368888, root := ~, start := 368874, stop := 368887 ), rec( content := ". Since a search for independent\ sets\n is equal to a search for cliques in the ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphDual" ), content := 0, count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "Ref", next := 368995, root := ~, start := 368970, stop := 368994 ), rec( content := ", the\n methods used in ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " usually transform a search for \ independent sets\n into a search for cliques, as described above. The func\ tions and\n attributes which return representatives of independent sets of\ \n ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "A", next := 369235, root := ~, start := 369221, stop := 369234 ), rec( content := " will return orbit representativ\ es of independent sets under\n the action of the automorphism group of the \ ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "symmetric closure", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "E", next := 369368, root := ~, start := 369344, stop := 369367 ), rec( content := " of the\n digraph formed by r\ emoving loops and multiple edges.", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "P", next := 369436, root := ~, start := 369432, stop := 369435 ), rec( content := "\n\n Please note that in ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ", cliques and independent sets a\ re not\n required to be maximal. Some authors use the word clique to mean\n\ ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "E", next := 369598, root := ~, start := 369584, stop := 369597 ), rec( content := " clique, and some authors use th\ e phrase independent set to\n mean ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "E", next := 369681, root := ~, start := 369667, stop := 369680 ), rec( content := " independent set, but please not\ e that ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " does\n not use this definiti\ on.\n ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 20 ], mathmode := "Text", name := "P", next := 369773, root := ~, start := 369769, stop := 369772 ), rec( content := "\n\n ", count := [ 8, 0, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Finding cliques", count := [ 8, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 369820, root := ~, start := 369786, stop := 369819 ), rec( content := "\n ", count := [ 8, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, l", Name := "IsClique" ), content := 0, count := [ 8, 1, 1, 2 ], mathmode := "Text", name := "Oper", next := 369878, root := ~, start := 369838, stop := 369877 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalClique\ " ), content := 0, count := [ 8, 1, 1, 4 ], mathmode := "Text", name := "Oper", next := 369926, root := ~, start := 369879, stop := 369925 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 369947, root := ~, start := 369936, stop := 369946 ), rec( content := " or ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 369963, root := ~, start := 369951, stop := 369962 ), rec( content := ".", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Returns", next := 369974, root := ~, start := 369927, stop := 369973 ), rec( attributes := rec( ), content := [ rec( content := "\n If ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370008, root := ~, start := 369994, stop := 370007 ), rec( content := " is a di\ graph and ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370034, root := ~, start := 370026, stop := 370033 ), rec( content := " is a du\ plicate-free list of\n vertices of ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370091, root := ~, start := 370077, stop := 370090 ), rec( content := ", then\n\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sClique(", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 370116, root := ~, start := 370100, stop := 370115 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370130, root := ~, start := 370116, stop := 370129 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 370138, root := ~, start := 370130, stop := 370137 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370146, root := ~, start := 370138, stop := 370145 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 370154, root := ~, start := 370146, stop := 370153 ), rec( content := " returns\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 370174, root := ~, start := 370163, stop := 370173 ), rec( content := "\n if\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370190, root := ~, start := 370182, stop := 370189 ), rec( content := " is a ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ lique", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "E", next := 370209, root := ~, start := 370196, stop := 370208 ), rec( content := " of ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370227, root := ~, start := 370213, stop := 370226 ), rec( content := " and ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 370244, root := ~, start := 370232, stop := 370243 ), rec( content := " if it i\ s\n not. Similarly,\n ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalClique(", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 370302, root := ~, start := 370279, stop := 370301 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370316, root := ~, start := 370302, stop := 370315 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 370324, root := ~, start := 370316, stop := 370323 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370332, root := ~, start := 370324, stop := 370331 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 370340, root := ~, start := 370332, stop := 370339 ), rec( content := " returns\ \n ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 370364, root := ~, start := 370353, stop := 370363 ), rec( content := " if ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370376, root := ~, start := 370368, stop := 370375 ), rec( content := " is a ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal clique", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "E", next := 370403, root := ~, start := 370382, stop := 370402 ), rec( content := " of ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 370421, root := ~, start := 370407, stop := 370420 ), rec( content := " and\n \ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 370442, root := ~, start := 370430, stop := 370441 ), rec( content := " if it i\ s not. ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "P", next := 370462, root := ~, start := 370458, stop := 370461 ), rec( content := "\n\n \ A ", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ lique", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "E", next := 370483, root := ~, start := 370470, stop := 370482 ), rec( content := " of a di\ graph is a set of mutually adjacent vertices of the\n digraph. A ", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal clique", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "E", next := 370579, root := ~, start := 370558, stop := 370578 ), rec( content := " is a cl\ ique that is not properly\n contained in another clique. A clique is permit\ ted, but not required, to\n contain vertices at which there is a loop.\n \ ", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);;\ngap> IsClique(D, [1, 3, 2]);\ntrue\ngap> IsMa\ ximalClique(D, [1, 3, 2]);\nfalse\ngap> IsMaximalClique(D, DigraphVertices(D))\ ;\ntrue\ngap> D := Digraph([[1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);\n\ngap> IsClique(D, [2, 3, 4]);\nfa\ lse\ngap> IsMaximalClique(D, [1, 2, 4]);\ntrue\ngap> D := CompleteDigraph(IsMu\ tableDigraph, 4);;\ngap> IsClique(D, [1, 3, 2]);\ntrue", count := [ 8, 1, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 1, 7 ], mathmode := "Text", name := "Example", next := 371201, root := ~, start := 370741, stop := 371200 ), rec( content := "\n ", count := [ 8, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 371218, root := ~, start := 369975, stop := 371217 ) ], count := [ 8, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 371232, root := ~, start := 369825, stop := 371231 ), rec( content := "\n\n ", count := [ 8, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, hook, u\ ser_param, limit, include,\n exclude, max, size, reps", Name := "CliquesFinder" ), content := 0, count := [ 8, 1, 2, 2 ], mathmode := "Text", name := "Func", next := 371359, root := ~, start := 371253, stop := 371358 ), rec( attributes := rec( ), content := [ rec( content := "The argum\ ent ", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ser_param", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "A", next := 371401, root := ~, start := 371384, stop := 371400 ), rec( content := ".", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Returns", next := 371412, root := ~, start := 371362, stop := 371411 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function finds cliques of the digraph ", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "A", next := 371490, root := ~, start := 371476, stop := 371489 ), rec( content := " subject\ to the\n conditions imposed by the other arguments as described below. Not\ e\n that a clique is represented by the immutable list of the vertices that\ \n it contains.\n ", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "P", next := 371678, root := ~, start := 371674, stop := 371677 ), rec( content := "\n\n \ Let ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "C", next := 371696, root := ~, start := 371688, stop := 371695 ), rec( content := " denote \ the automorphism group of the maximal symmetric\n subdigraph of ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "A", next := 371784, root := ~, start := 371770, stop := 371783 ), rec( content := " without\ loops (see ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 1, 2, 4 ] , mathmode := "Text", name := "Ref", next := 371863, root := ~, start := 371804, stop := 371862 ), rec( content := " and ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraphWithoutLoops" ), content := 0, count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "Ref", next := 371926, root := ~, start := 371868, stop := 371925 ), rec( content := ").\n\n \ ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "hook", count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "A", next := 37\ 1964, root := ~, start := 371953, stop := 371963 ) ], count := [ 8, 1, 2, 5 ], name := "Mar\ k", next := 371971, root := ~, start := 371947, stop := 371970 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a f\ unction or ", count := [ 8, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 8, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 6 ], mathmode := "Text", name := "K", next := 372042, root := \ ~, start := 372031, stop := 372041 ), rec( content := ".", count := [ 8, 1, 2,\ 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := 0, count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "P", next\ := 372047, root := ~, start := 372043, stop := 372046 ), rec( content := "\n\ \n If ", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "hook", coun\ t := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 7 ], mathmode := "Text", name := "A", next := 372071, root := \ ~, start := 372060, stop := 372070 ), rec( content := " is a function, then it\ should have two arguments\n ", count := [ 8, 1, 2, 7 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "user_param", count := [ 8, 1, 2, 7 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name\ := "A", next := 372146, root := ~, start := 372129, stop := 372145 ), rec( cont\ ent := " (see below) and a clique ", count := [ 8, 1, 2, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "c", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "C", next\ := 372180, root := ~, start := 372172, stop := 372179 ), rec( content := ". T\ he function\n ", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes\ := rec( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 7 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mat\ hmode := "Text", name := "A", next := 372217, root := ~, start := 372206, stop\ := 372216 ), rec( content := "(", count := [ 8, 1, 2, 7 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec(\ content := "user_param", count := [ 8, 1, 2, 7 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name :=\ "A", next := 372235, root := ~, start := 372218, stop := 372234 ), rec( conte\ nt := ", c)", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "C", next :\ = 372243, root := ~, start := 372203, stop := 372242 ), rec( content := " is c\ alled every time a new\n clique ", count := [ 8, 1, 2, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "c", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "\ C", next := 372294, root := ~, start := 372286, stop := 372293 ), rec( content\ := " is found by ", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ liquesFinder", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "C", next \ := 372327, root := ~, start := 372307, stop := 372326 ), rec( content := ".", coun\ t := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 8, 1, 2, 8 ], mathmode := "Text", name\ := "P", next := 372332, root := ~, start := 372328, stop := 372331 ), rec( cont\ ent := "\n\n If ", count := [ 8, 1, 2, 8 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "hook", count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "A", next := 3\ 72356, root := ~, start := 372345, stop := 372355 ), rec( content := " is ", coun\ t := [ 8, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 2, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 8 ] , math\ mode := "Text", name := "K", next := 372371, root := ~, start := 372360, stop \ := 372370 ), rec( content := ", then a default function is used that\n \ simply adds every new clique found by ", count := [ 8, 1, 2, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "CliquesFinder", count := [ 8, 1, 2, 8 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 8 ], mathmode := "Text", name\ := "C", next := 372476, root := ~, start := 372456, stop := 372475 ), rec( cont\ ent := " to\n ", count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "user_param", count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "A", next \ := 372505, root := ~, start := 372488, stop := 372504 ), rec( content := ", wh\ ich must be a list in this case.\n ", count := [ 8, 1, 2, 8 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 6 ], name := "Ite\ m", next := 372555, root := ~, start := 371978, stop := 372554 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "user_param", count := [ 8, 1, 2, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 8, 1, 2, 9 ], mathmode := "Text", name := "A", next\ := 372586, root := ~, start := 372569, stop := 372585 ) ], count := [ 8, 1, 2, 9 ], name := "Mar\ k", next := 372593, root := ~, start := 372563, stop := 372592 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If ", count := [ 8, 1, 2, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode\ := "Text", name := "A", next := 372629, root := ~, start := 372618, stop := 3\ 72628 ), rec( content := " is a function, then ", count := [ 8, 1, 2, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := \ "Text", name := "A", next := 372667, root := ~, start := 372650, stop := 37266\ 6 ), rec( content := " can be any ", count := [ 8, 1, 2, 10 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "GAP", count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "\ Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := "\n\ object. The object ", count := [ 8, 1, 2, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "A" , next\ := 372729, root := ~, start := 372712, stop := 372728 ), rec( content := " is\ used as the first argument for\n the function ", count := [ 8, 1, 2, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode\ := "Text", name := "A", next := 372796, root := ~, start := 372785, stop := 3\ 72795 ), rec( content := ". For example, ", count := [ 8, 1, 2, 10 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "Te\ xt", name := "A", next := 372828, root := ~, start := 372811, stop := 372827 ) , rec( cont\ ent := " might be a\n list, and ", count := [ 8, 1, 2, 10 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( attributes := rec( ), content := [ rec( content := "hook", count := \ [ 8, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 8, 1, 2, 10 ], mathmode := "Text", name := "A", next := 372872, root := ~, \ start := 372861, stop := 372871 ), rec( content := "(", count := [ 8, 1, 2, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mat\ hmode := "Text", name := "A", next := 372890, root := ~, start := 372873, stop\ := 372889 ), rec( content := ", c)", count := [ 8, 1, 2, 10 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "C", next := 372898, root := ~, start := 372858, stop := 372897\ ), rec( content := " might add the size\n of the clique ", count := [ \ 8, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "c", count := [ 8, 1, 2, 10 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], math\ mode := "Text", name := "C", next := 372948, root := ~, start := 372940, stop \ := 372947 ), rec( content := " to the list ", count := [ 8, 1, 2, 10 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "A", next := 372978, root := ~, start := 372961, stop := 372977\ ), rec( content := ". ", count := [ 8, 1, 2, 10 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 8, 1, 2, 11 ], mathmode := "Text", name := "P", next := 372984, root := ~, star\ t := 372980, stop := 372983 ), rec( content := "\n\n If the value of ", coun\ t := [ 8, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 1\ 1 ], mathmode := "Text", name := "A", next := 373021, root := ~, start := 3730\ 10, stop := 373020 ), rec( content := " is ", count := [ 8, 1, 2, 11 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "fail", count := [ 8, 1, 2, 11 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 11 ], mathmode := "Text", name\ := "K", next := 373036, root := ~, start := 373025, stop := 373035 ), rec( cont\ ent := ", then the value of\n ", count := [ 8, 1, 2, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "user_param", count := [ 8, 1, 2, 11 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 11 ], mathmode := "Text", name\ := "A", next := 373081, root := ~, start := 373064, stop := 373080 ), rec( cont\ ent := " must be a list.\n ", count := [ 8, 1, 2, 11 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], name := "Item", next := 3731\ 11, root := ~, start := 372600, stop := 373110 ), rec( attributes := rec( ), content := [ rec( attr\ ibutes := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 2, 1\ 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, \ 12 ], mathmode := "Text", name := "A", next := 373137, root := ~, start := 373\ 125, stop := 373136 ) ], count := [ 8, 1, 2, 12 ], name := "Mark", next := 3731\ 44, root := ~, start := 373119, stop := 373143 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n This argument should be a positive integer or ", count := [ \ 8, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "infinity", count := [ 8, 1, 2, 1\ 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, \ 13 ], mathmode := "Text", name := "K", next := 373227, root := ~, start := 373\ 212, stop := 373226 ), rec( content := ".\n ", count := [ 8, 1, 2, 13 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "CliquesFinder", count := [ 8, 1, 2, 13 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 13 ], mathmode \ := "Text", name := "C", next := 373257, root := ~, start := 373237, stop := 37\ 3256 ), rec( content := " will return after it has found\n ", count := [ \ 8, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 2, 13 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 13 ] , math\ mode := "Text", name := "A", next := 373309, root := ~, start := 373297, stop \ := 373308 ), rec( content := " cliques or the search is complete.\n ", coun\ t := [ 8, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 13 ], name := "Ite\ m", next := 373358, root := ~, start := 373151, stop := 373357 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "include", count := [ 8, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 14 ], mathmode := "Text", name := "A", next\ := 373386, root := ~, start := 373372, stop := 373385 ), rec( content := " an\ d ", count := [ 8, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "exclude", count \ := [ 8, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 8, 1, 2, 14 ], mathmode := "Text", name := "A", next := 373405, root := \ ~, start := 373391, stop := 373404 ) ], count := [ 8, 1, 2, 14 ], name := "Mar\ k", next := 373412, root := ~, start := 373366, stop := 373411 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n These arguments should each\ be a (possibly empty) duplicate-free list\n of vertices of ", count :=\ [ 8, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph", count := [ 8, 1, 2,\ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2\ , 15 ], mathmode := "Text", name := "A", next := 373541, root := ~, start := 3\ 73527, stop := 373540 ), rec( content := " (i.e. positive integers less than t\ he\n number of vertices of ", count := [ 8, 1, 2, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "digraph", count := [ 8, 1, 2, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 15 ], mathmode := "Text", name \ := "A", next := 373624, root := ~, start := 373610, stop := 373623 ), rec( cont\ ent := "). ", count := [ 8, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 2, 16 ], math\ mode := "Text", name := "P", next := 373631, root := ~, start := 373627, stop \ := 373630 ), rec( content := "\n\n ", count := [ 8, 1, 2, 16 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "CliquesFinder", count := [ 8, 1, 2, 16 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 16 ], mathmode :=\ "Text", name := "C", next := 373661, root := ~, start := 373641, stop := 3736\ 60 ), rec( content := " will only look for cliques containing all of the\n \ vertices in ", count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "in\ clude", count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "A", next := 37\ 3745, root := ~, start := 373731, stop := 373744 ), rec( content := " and cont\ aining none of the vertices in\n ", count := [ 8, 1, 2, 16 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "exclude", count := [ 8, 1, 2, 16 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 16 ], mathmode := "Text", name\ := "A", next := 373807, root := ~, start := 373793, stop := 373806 ), rec( cont\ ent := ". ", count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 2, 17 ], math\ mode := "Text", name := "P", next := 373813, root := ~, start := 373809, stop \ := 373812 ), rec( content := "\n\n Note that the search may be much mor\ e efficient if each of these lists\n is invariant under the action of " , coun\ t := [ 8, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 1, 2, 17 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 17 ] , math\ mode := "Text", name := "C", next := 373943, root := ~, start := 373935, stop \ := 373942 ), rec( content := " on sets of vertices.\n ", count := [ 8, 1,\ 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 15 ], name := "Ite\ m", next := 373978, root := ~, start := 373419, stop := 373977 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "max", count := [ 8, 1, 2, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 2, 18 ], mathmode := "Text", name := "A", next := \ 374002, root := ~, start := 373992, stop := 374001 ) ], count := [ 8, 1, 2, 18 ], name := "Mar\ k", next := 374009, root := ~, start := 373986, stop := 374008 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be ", coun\ t := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "true", count := [ 8, 1, 2, 19\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 1\ 9 ], mathmode := "Text", name := "K", next := 374066, root := ~, start := 3740\ 55, stop := 374065 ), rec( content := " or ", count := [ 8, 1, 2, 19 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "false", count := [ 8, 1, 2, 19 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], mathmode := "Text" , name\ := "K", next := 374082, root := ~, start := 374070, stop := 374081 ), rec( cont\ ent := ". If ", count := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "max"\ , count := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 8, 1, 2, 19 ], mathmode := "Text", name := "A", next := 374098, root :=\ ~, start := 374088, stop := 374097 ), rec( content := " is\n true then\ ", count := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 19 ], mathmode := "Text", name := "C", next := 374140, root := ~, star\ t := 374120, stop := 374139 ), rec( content := " will only search for ", count\ := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "maximal", count := [ 8, 1, 2,\ 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2\ , 19 ], mathmode := "Text", name := "E", next := 374176, root := ~, start := 3\ 74162, stop := 374175 ), rec( content := "\n cliques. If ", count := [ \ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "max", count := [ 8, 1, 2, 19 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], math\ mode := "Text", name := "K", next := 374207, root := ~, start := 374197, stop \ := 374206 ), rec( content := " is ", count := [ 8, 1, 2, 19 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "false", count := [ 8, 1, 2, 19 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], mathmode := "Text", name :=\ "K", next := 374223, root := ~, start := 374211, stop := 374222 ), rec( conte\ nt := " then non-maximal cliques may be\n found.\n ", count := [ 8\ , 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], name := "Ite\ m", next := 374284, root := ~, start := 374016, stop := 374283 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "size", count := [ 8, 1, 2, 20 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 8, 1, 2, 20 ], mathmode := "Text", name := "A", next :=\ 374309, root := ~, start := 374298, stop := 374308 ) ], count := [ 8, 1, 2, 20 ], name := "Mar\ k", next := 374316, root := ~, start := 374292, stop := 374315 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be ", coun\ t := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 2, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 2\ 1 ], mathmode := "Text", name := "K", next := 374373, root := ~, start := 3743\ 62, stop := 374372 ), rec( content := " or a positive integer.\n If ", coun\ t := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "size", count := [ 8, 1, 2, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 2\ 1 ], mathmode := "Text", name := "A", next := 374419, root := ~, start := 3744\ 08, stop := 374418 ), rec( content := " is a positive integer then ", count :=\ [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 21 ], mathmode := "Text", name := "C", next := 374467, root := ~, star\ t := 374447, stop := 374466 ), rec( content := " will\n only search for\ cliques that contain precisely ", count := [ 8, 1, 2, 21 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "size", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "A\ ", next := 374539, root := ~, start := 374528, stop := 374538 ), rec( content \ := " vertices.\n If ", count := [ 8, 1, 2, 21 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "size", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "A", next\ := 374572, root := ~, start := 374561, stop := 374571 ), rec( content := " is\ ", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 2, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 2\ 1 ], mathmode := "Text", name := "K", next := 374587, root := ~, start := 3745\ 76, stop := 374586 ), rec( content := " then cliques of any size may be found.\ \n ", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 2, 21 ], name := "Item", next := 374640, root := ~, start := 374\ 323, stop := 374639 ), rec( attributes := rec( ), content := [ rec( attribute\ s := rec( ), content := [ rec( content := "reps", count := [ 8, 1, 2, 22 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 22 ], math\ mode := "Text", name := "A", next := 374665, root := ~, start := 374654, stop \ := 374664 ) ], count := [ 8, 1, 2, 22 ], name := "Mark", next := 374672, root := ~, start := 374\ 648, stop := 374671 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n This argument should be ", count := [ 8, 1, 2, 23 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "true", count := [ 8, 1, 2, 23 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 23 ], mathmode := "Text", name :=\ "K", next := 374729, root := ~, start := 374718, stop := 374728 ), rec( conte\ nt := " or ", count := [ 8, 1, 2, 23 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "false", cou\ nt := [ 8, 1, 2, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 23 ], mathmode := "Text", name := "K", next := 374745, root :=\ ~, start := 374733, stop := 374744 ), rec( content := ".", count := [ 8, 1, 2\ , 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := 0, count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "P" , next\ := 374750, root := ~, start := 374746, stop := 374749 ), rec( content := "\n\ \n If ", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "reps", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "A", next := 374774, root :=\ ~, start := 374763, stop := 374773 ), rec( content := " is ", count := [ 8, 1\ , 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "true", count := [ 8, 1, 2, 24 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ], math\ mode := "Text", name := "K", next := 374789, root := ~, start := 374778, stop \ := 374788 ), rec( content := " then the arguments ", count := [ 8, 1, 2, 24 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "include", count := [ 8, 1, 2, 24 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ], mathmode := "Te\ xt", name := "A", next := 374823, root := ~, start := 374809, stop := 374822 ) , rec( cont\ ent := " and\n ", count := [ 8, 1, 2, 24 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "exclude", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "A", next\ := 374850, root := ~, start := 374836, stop := 374849 ), rec( content := " ar\ e each required to be invariant under the action of\n ", count := [ 8, \ 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "G", count := [ 8, 1, 2, 24 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ], math\ mode := "Text", name := "C", next := 374921, root := ~, start := 374913, stop \ := 374920 ), rec( content := " on sets of vertices. In this case, ", count :=\ [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 24 ], mathmode := "Text", name := "C", next := 374978, root := ~, star\ t := 374958, stop := 374977 ), rec( content := " will\n find representa\ tives of the orbits of the desired cliques under the\n action of ", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 1, 2, 24 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ] , math\ mode := "Text", name := "C", next := 375086, root := ~, start := 375078, stop \ := 375085 ), rec( content := ", ", count := [ 8, 1, 2, 24 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "although representatives may be returned that\n are in \ the same orbit", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "E", next\ := 375172, root := ~, start := 375088, stop := 375171 ), rec( content := ".\n\ If ", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "reps", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "A", next := 375196, root :=\ ~, start := 375185, stop := 375195 ), rec( content := " is false then ", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 24 ], mathmode := "Text", name := "C", next := 375231, root := ~, star\ t := 375211, stop := 375230 ), rec( content := " will not take this into\n \ consideration.", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1,\ 2, 25 ], mathmode := "Text", name := "P", next := 375282, root := ~, start :=\ 375278, stop := 375281 ), rec( content := "\n\n For a digraph such tha\ t ", count := [ 8, 1, 2, 25 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "G", count := [ 8\ , 1, 2, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 8, 1, 2, 25 ], mathmode := "Text", name := "C", next := 375324, root := ~, star\ t := 375316, stop := 375323 ), rec( content := " is non-trivial, the search fo\ r\n clique representatives can be much more efficient than the search f\ or\n all cliques.\n ", count := [ 8, 1, 2, 25 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 23 ], name := "Item", next := 3754\ 68, root := ~, start := 374679, stop := 375467 ) ], count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "List", next := 375480, root := ~, start := 371934, stop := 375479 ), rec( content := "\n\n \ This function uses a version of the Bron-Kerbosch algorithm.\n ", count := [ 8, 1, 2, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(5);\n\n\ gap> user_param := [];;\ngap> f := function(a, b) # Calculate size of clique\ \n> AddSet(user_param, Size(b));\n> end;;\ngap> CliquesFinder(D, f, user_par\ am, infinity, [], [], false, fail,\n> true);\n[ 1, 2, 3, 4, 5\ ]\ngap> CliquesFinder(D, fail, [], 5, [2, 4], [3], false, fail, false);\n[ [ \ 2, 4 ], [ 1, 2, 4 ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail\ , [], 2, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> Cl\ iquesFinder(D, fail, [], infinity, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5\ ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], [], false, 3, false);\ \n[ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1, 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], i\ nfinity, [1, 3], [], true, 3, false);\n[ ]\ngap> D := CompleteDigraph(IsMutab\ leDigraph, 5);\n\ngap> user_param :\ = [];;\ngap> f := function(a, b) # Calculate size of clique\n> AddSet(user_\ param, Size(b));\n> end;;\ngap> CliquesFinder(D, f, user_param, infinity, [], \ [], false, fail,\n> true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFi\ nder(D, fail, [], 5, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 \ ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [], 2, [2, 4], [\ 3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> CliquesFinder(D, fai\ l, [], infinity, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5 ] ]\ngap> Cliques\ Finder(D, fail, [], infinity, [1, 3], [], false, 3, false);\n[ [ 1, 2, 3 ], [ \ 1, 3, 4 ], [ 1, 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], []\ , true, 3, false);\n[ ]", count := [ 8, 1, 2, 28 ], name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 28 ], mathmode := "Text", name := "Example", next := 377205, root := ~, start := 375551, stop := 377204 ), rec( content := "\n ", count := [ 8, 1, 2, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 377222, root := ~, start := 371415, stop := 377221 ) ], count := [ 8, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 377236, root := ~, start := 371238, stop := 377235 ), rec( content := "\n\n ", count := [ 8, 1, 2, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[, size]]]", Name := "DigraphClique" ), content := 0, count := [ 8, 1, 3, 2 ], mathmode := "Text", name := "Func", next := 377329, root := ~, start := 377257, stop := 377328 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalClique" ), content := 0, count := [ 8, 1, 3, 4 ], mathmode := "Text", name := "Func", next := 377415, root := ~, start := 377332, stop := 377414 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of positive integers, or ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "K", next := 377481, root := ~, start := 377470, stop := 377480 ), rec( content := ".", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Returns", next := 377492, root := ~, start := 377418, stop := 377491 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "A", next := 377530, root := ~, start := 377516, stop := 377529 ), rec( content := " is a di\ graph, then these functions returns a clique\n of ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "A", next := 377604, root := ~, start := 377590, stop := 377603 ), rec( content := " if one \ exists that satisfies the arguments, else it\n returns ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "K", next := 377680, root := ~, start := 377669, stop := 377679 ), rec( content := ". A cli\ que is defined by the set of vertices that it\n contains; see ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsCliqu\ e" ), content := 0, count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Ref", next := 377774, root := ~, start := 377752, stop := 377773 ), rec( content := " and ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alClique" ), content := 0, count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Ref", next := 377808, root := ~, start := 377779, stop := 377807 ), rec( content := ".", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "P", next := 377813, root := ~, start := 377809, stop := 377812 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 377856, root := ~, start := 377842, stop := 377855 ), rec( content := " and ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 377875, root := ~, start := 377861, stop := 377874 ), rec( content := " must ea\ ch be a\n (possibly empty) duplicate-free list of vertices of ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 377961, root := ~, start := 377947, stop := 377960 ), rec( content := ", and\n \ the optional argument ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 378004, root := ~, start := 377993, stop := 378003 ), rec( content := " must be\ a positive integer. By default,\n ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 378064, root := ~, start := 378050, stop := 378063 ), rec( content := " and ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 378083, root := ~, start := 378069, stop := 378082 ), rec( content := " are emp\ ty. These functions will search for\n a clique of ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 378157, root := ~, start := 378143, stop := 378156 ), rec( content := " that in\ cludes the vertices of ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 378202, root := ~, start := 378188, stop := 378201 ), rec( content := "\n bu\ t does not include any vertices in ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 378258, root := ~, start := 378244, stop := 378257 ), rec( content := "; if the\ argument\n ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 378291, root := ~, start := 378280, stop := 378290 ), rec( content := " is supp\ lied, then additionally the clique will be required to\n contain precisely \ ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 378387, root := ~, start := 378376, stop := 378386 ), rec( content := " vertice\ s.", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "P", next := 378401, root := ~, start := 378397, stop := 378400 ), rec( content := "\n\n \ If ", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "A", next := 378424, root := ~, start := 378410, stop := 378423 ), rec( content := " is not \ a clique, then these functions return ", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "K", next := 378481, root := ~, start := 378470, stop := 378480 ), rec( content := ".\n O\ therwise, the functions behave in the following way, depending on the\n num\ ber of arguments:\n\n ", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "One or two arguments", count := [ 8, 1\ , 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 8 ], name := "Mar\ k", next := 378634, root := ~, start := 378601, stop := 378633 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If one or two arguments are\ supplied, then ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Dig\ raphClique", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "C", next :=\ 378719, root := ~, start := 378699, stop := 378718 ), rec( content := " and\n\ ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphMaxi\ malClique", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "C", next := \ 378759, root := ~, start := 378732, stop := 378758 ), rec( content := " greedi\ ly enlarge the clique ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "include", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "A", next :=\ 378802, root := ~, start := 378788, stop := 378801 ), rec( content := "\n \ until it can not continue. The result is guaranteed to be a maximal\n \ clique. This may or may not return an answer more quickly than using\n \ ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( Func := "DigraphMaximalCliques" ), content := 0, coun\ t := [ 8, 1, 3, 9 ], mathmode := "Text", name := "Ref", next := 379000, root :\ = ~, start := 378965, stop := 378999 ), rec( content := ". with a limit of 1.\ \n ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 3, 9 ], name := "Item", next := 379035, root := ~, start := 378\ 641, stop := 379034 ), rec( attributes := rec( ), content := [ rec( content :\ = "Three arguments", count := [ 8, 1, 3, 10 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 3, 10 ], name := "Mark", next := 379070 , root := ~, start := 379\ 042, stop := 379069 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n If three arguments are supplied, then ", count := [ 8, 1, 3, 11 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphClique", count := [ 8, 1, 3, 11 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 11 ], mathmode \ := "Text", name := "C", next := 379150, root := ~, start := 379130, stop := 37\ 9149 ), rec( content := " greedily\n enlarges the clique ", count := [ \ 8, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "include", count := [ 8, 1, 3, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 1\ 1 ], mathmode := "Text", name := "A", next := 379202, root := ~, start := 3791\ 88, stop := 379201 ), rec( content := " until it can not continue, although\n \ this clique may not be maximal.", count := [ 8, 1, 3, 11 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ 0, count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "P", next := 379282, root\ := ~, start := 379278, stop := 379281 ), rec( content := "\n\n Given t\ hree arguments, ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Di\ graphMaximalClique", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "C\ ", next := 379342, root := ~, start := 379315, stop := 379341 ), rec( content \ := " returns the maximal\n clique returned by ", count := [ 8, 1, 3, 12\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "DigraphMaximalCliques(", count := [ 8, 1, 3\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, \ 3, 12 ], mathmode := "Text", name := "C", next := 379419, root := ~, start := \ 379390, stop := 379418 ), rec( attributes := rec( ), content := [ rec( conten\ t := "digraph", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "A", next\ := 379433, root := ~, start := 379419, stop := 379432 ), rec( attributes := re\ c( ), content := [ rec( content := ",\n ", count := [ 8, 1, 3, 12 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 12 ], math\ mode := "Text", name := "C", next := 379450, root := ~, start := 379433, stop \ := 379449 ), rec( attributes := rec( ), content := [ rec( content := "include\ ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "A", next := 379464, root\ := ~, start := 379450, stop := 379463 ), rec( attributes := rec( ), content \ := [ rec( content := ", ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name\ := "C", next := 379473, root := ~, start := 379464, stop := 379472 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 1, 3,\ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3\ , 12 ], mathmode := "Text", name := "A", next := 379487, root := ~, start := 3\ 79473, stop := 379486 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1)", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "C", next :\ = 379498, root := ~, start := 379487, stop := 379497 ), rec( content := " if o\ ne exists,\n else ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "fail", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "K", next\ := 379538, root := ~, start := 379527, stop := 379537 ), rec( content := ".\n\ ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 8, 1, 3, 11 ], name := "Item", next := 379553, root := ~, start := 379\ 077, stop := 379552 ), rec( attributes := rec( ), content := [ rec( content :\ = "Four arguments", count := [ 8, 1, 3, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 3, 13 ], name := "Mark", next := 379587, root := ~, start := 379\ 560, stop := 379586 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n If four arguments are supplied, then ", count := [ 8, 1, 3, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphClique", count := [ 8, 1, 3, 14 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode \ := "Text", name := "C", next := 379666, root := ~, start := 379646, stop := 37\ 9665 ), rec( content := " returns the\n clique returned by ", count := [ \ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "DigraphCliques(", count := [ 8, \ 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8,\ 1, 3, 14 ], mathmode := "Text", name := "C", next := 379728, root := ~, start\ := 379706, stop := 379727 ), rec( attributes := rec( ), content := [ rec( co\ ntent := "digraph", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "A" , next\ := 379742, root := ~, start := 379728, stop := 379741 ), rec( attributes := re\ c( ), content := [ rec( content := ",\n ", count := [ 8, 1, 3, 14 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ] , math\ mode := "Text", name := "C", next := 379761, root := ~, start := 379742, stop \ := 379760 ), rec( attributes := rec( ), content := [ rec( content := "include\ ", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "A", next := 379775, root\ := ~, start := 379761, stop := 379774 ), rec( attributes := rec( ), content \ := [ rec( content := ", ", count := [ 8, 1, 3, 14 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name\ := "C", next := 379784, root := ~, start := 379775, stop := 379783 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 1, 3,\ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3\ , 14 ], mathmode := "Text", name := "A", next := 379798, root := ~, start := 3\ 79784, stop := 379797 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1,\n ", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "\ C", next := 379818, root := ~, start := 379798, stop := 379817 ), rec( attribu\ tes := rec( ), content := [ rec( content := "size", count := [ 8, 1, 3, 14 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ] , math\ mode := "Text", name := "A", next := 379829, root := ~, start := 379818, stop \ := 379828 ), rec( attributes := rec( ), content := [ rec( content := ")", cou\ nt := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 3, 14 ], mathmode := "Text", name := "C", next := 379837, root :=\ ~, start := 379829, stop := 379836 ), rec( content := " if one exists, else " , coun\ t := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 3, 14\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 1\ 4 ], mathmode := "Text", name := "K", next := 379869, root := ~, start := 3798\ 58, stop := 379868 ), rec( content := ". This clique may\n not be maxim\ al.", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 3, 15 ], mathm\ ode := "Text", name := "P", next := 379914, root := ~, start := 379910, stop :\ = 379913 ), rec( content := " Given four arguments, ", count := [ 8, 1, 3, 15 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalClique", count := [ 8, 1, 3, 15 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], math\ mode := "Text", name := "C", next := 379964, root := ~, start := 379937, stop \ := 379963 ), rec( content := "\n returns the maximal clique returned by\ \n ", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphMaxi\ malCliques(", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "C", next\ := 380049, root := ~, start := 380020, stop := 380048 ), rec( attributes := re\ c( ), content := [ rec( content := "digraph", count := [ 8, 1, 3, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], math\ mode := "Text", name := "A", next := 380063, root := ~, start := 380049, stop \ := 380062 ), rec( attributes := rec( ), content := [ rec( content := ", ", co\ unt := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 3, 15 ], mathmode := "Text", name := "C", next := 380072, root :=\ ~, start := 380063, stop := 380071 ), rec( attributes := rec( ), content := [ \ rec( content := "include", count := [ 8, 1, 3, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name\ := "A", next := 380086, root := ~, start := 380072, stop := 380085 ), rec( attr\ ibutes := rec( ), content := [ rec( content := ",\n ", count := [ 8, 1\ , 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, \ 1, 3, 15 ], mathmode := "Text", name := "C", next := 380103, root := ~, start \ := 380086, stop := 380102 ), rec( attributes := rec( ), content := [ rec( con\ tent := "exclude", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "A", next\ := 380117, root := ~, start := 380103, stop := 380116 ), rec( attributes := re\ c( ), content := [ rec( content := ", 1, ", count := [ 8, 1, 3, 15 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmo\ de := "Text", name := "C", next := 380129, root := ~, start := 380117, stop :=\ 380128 ), rec( attributes := rec( ), content := [ rec( content := "size", co\ unt := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 3, 15 ], mathmode := "Text", name := "A", next := 380140, root :=\ ~, start := 380129, stop := 380139 ), rec( attributes := rec( ), content := [ \ rec( content := ")", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "C\ ", next := 380148, root := ~, start := 380140, stop := 380147 ), rec( content \ := " if one exists, else\n ", count := [ 8, 1, 3, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "fail", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := \ "K", next := 380188, root := ~, start := 380177, stop := 380187 ), rec( conten\ t := ".\n ", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 3, 14 ], name := "Item", next := 380203, root := ~, start := 379\ 594, stop := 380202 ) ], count := [ 8, 1, 3, 8 ], mathmode := "Text", name := "List", next := 380215, root := ~, start := 378588, stop := 380214 ), rec( content := "\n\n \ ", count := [ 8, 1, 3, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> Digraph\ Clique(D);\n[ 5 ]\ngap> DigraphMaximalClique(D);\n[ 5 ]\ngap> DigraphClique(D,\ [1, 2]);\n[ 1, 2, 3 ]\ngap> DigraphMaximalClique(D, [1, 3]);\n[ 1, 3, 2 ]\nga\ p> DigraphClique(D, [1], [2]);\n[ 1, 4 ]\ngap> DigraphMaximalClique(D, [1], [3\ , 4]);\nfail\ngap> DigraphClique(D, [1, 5]);\nfail\ngap> DigraphClique(D, [], \ [], 2);\n[ 1, 2 ]\ngap> D := Digraph(IsMutableDigraph,\n> [[2,\ 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> DigraphClique(D);\n[ 5 ]", count := [ 8, 1, 3, 18 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 3, 18 ], mathmode := "Text", name := "Example", next := 380920, root := ~, start := 380221, stop := 380919 ), rec( content := "\n ", count := [ 8, 1, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Description", next := 380937, root := ~, start := 377495, stop := 380936 ) ], count := [ 8, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 380951, root := ~, start := 377242, stop := 380950 ), rec( content := "\n\n ", count := [ 8, 1, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[, limit[, size]]]]", Name := "DigraphMaximalCliques" ), content := 0, count := [ 8, 1, 4, 2 ], mathmode := "Text", name := "Func", next := 381061, root := ~, start := 380972, stop := 381060 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n limit[, size]]]]", Name := "DigraphMaximalCliquesReps" ), content := 0, count := [ 8, 1, 4, 4 ], mathmode := "Text", name := "Func", next := 381161, root := ~, start := 381064, stop := 381160 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n limit[, size]]]]", Name := "DigraphCliques" ), content := 0, count := [ 8, 1, 4, 6 ], mathmode := "Text", name := "Func", next := 381250, root := ~, start := 381164, stop := 381249 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphCliquesReps" ), content := 0, count := [ 8, 1, 4, 8 ], mathmode := "Text", name := "Func", next := 381343, root := ~, start := 381253, stop := 381342 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesAttr" ), content := 0, count := [ 8, 1, 4, 10 ], mathmode := "Text", name := "Attr", next := 381400, root := ~, start := 381346, stop := 381399 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesRepsAttr" ), content := 0, count := [ 8, 1, 4, 12 ], mathmode := "Text", name := "Attr", next := 381461, root := ~, start := 381403, stop := 381460 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of lists of positive integers.", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Returns", next := 381531, root := ~, start := 381464, stop := 381530 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "A", next := 381569, root := ~, start := 381555, stop := 381568 ), rec( content := " is digr\ aph, then these functions and attributes use ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Cliques\ Finder" ), content := 0, count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Ref", next := 381655, root := ~, start := 381622, stop := 381654 ), rec( content := " to retu\ rn cliques of ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "A", next := 381691, root := ~, start := 381677, stop := 381690 ), rec( content := ". A\n \ clique is defined by the set of vertices that it contains; see ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsCliqu\ e" ), content := 0, count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Ref", next := 381790, root := ~, start := 381762, stop := 381789 ), rec( content := " and ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alClique" ), content := 0, count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Ref", next := 381824, root := ~, start := 381795, stop := 381823 ), rec( content := ".", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "P", next := 381829, root := ~, start := 381825, stop := 381828 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 381872, root := ~, start := 381858, stop := 381871 ), rec( content := " and ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 381891, root := ~, start := 381877, stop := 381890 ), rec( content := " must ea\ ch be a\n (possibly empty) list of vertices of ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 381962, root := ~, start := 381948, stop := 381961 ), rec( content := ", the op\ tional argument\n ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 382002, root := ~, start := 381990, stop := 382001 ), rec( content := " must be\ either a positive integer or ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "C", next := 382055, root := ~, start := 382040, stop := 382054 ), rec( content := ", and th\ e\n optional argument ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 382098, root := ~, start := 382087, stop := 382097 ), rec( content := " must be\ a positive integer. If not\n specified, then ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 382169, root := ~, start := 382155, stop := 382168 ), rec( content := " and ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 382188, root := ~, start := 382174, stop := 382187 ), rec( content := " are cho\ sen to be empty\n lists, and ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 382239, root := ~, start := 382227, stop := 382238 ), rec( content := " is set \ to ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "C", next := 382265, root := ~, start := 382250, stop := 382264 ), rec( content := ". ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "P", next := 382271, root := ~, start := 382267, stop := 382270 ), rec( content := "\n\n \ The functions will return as many suitable cliques as possible, up to the\n \ number ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 382374, root := ~, start := 382362, stop := 382373 ), rec( content := ". These\ functions will find cliques that contain all\n of the vertices of ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 382465, root := ~, start := 382451, stop := 382464 ), rec( content := " but do \ not contain any of the\n vertices of ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 382526, root := ~, start := 382512, stop := 382525 ), rec( content := ". The a\ rgument ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 382553, root := ~, start := 382542, stop := 382552 ), rec( content := " restric\ ts the search\n to those cliques that contain precisely ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 382630, root := ~, start := 382619, stop := 382629 ), rec( content := " vertice\ s.\n If the function or attribute has ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximal", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "C", next := 382692, root := ~, start := 382678, stop := 382691 ), rec( content := " in its \ name, then only\n maximal cliques will be returned; otherwise non-maximal c\ liques may be\n returned. ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "P", next := 382809, root := ~, start := 382805, stop := 382808 ), rec( content := "\n\n \ Let ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "C", next := 382827, root := ~, start := 382819, stop := 382826 ), rec( content := " denote \ the automorphism group of maximal symmetric subdigraph\n of ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "A", next := 382911, root := ~, start := 382897, stop := 382910 ), rec( content := " without\ loops (see ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "Ref", next := 382990, root := ~, start := 382931, stop := 382989 ), rec( content := " and ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraphWithoutLoops" ), content := 0, count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "Ref", next := 383053, root := ~, start := 382995, stop := 383052 ), rec( content := ").\n\n \ ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Distinct cliques", count := [ 8, 1, 4,\ 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 17 ], name := "Mar\ k", next := 383103, root := ~, start := 383074, stop := 383102 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 8, 1, 4, 18 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalCliques", count := [ 8, 1, 4, 18 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], math\ mode := "Text", name := "C", next := 383153, root := ~, start := 383125, stop \ := 383152 ), rec( content := " and ", count := [ 8, 1, 4, 18 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "DigraphCliques", count := [ 8, 1, 4, 18 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], mathmode := "Text\ ", name := "C", next := 383179, root := ~, start := 383158, stop := 383178 ), rec( cont\ ent := " each return a\n duplicate-free list of at most ", count := [ 8\ , 1, 4, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 4, 18 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ] , math\ mode := "Text", name := "A", next := 383245, root := ~, start := 383233, stop \ := 383244 ), rec( content := " cliques of ", count := [ 8, 1, 4, 18 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "digraph", count := [ 8, 1, 4, 18 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], mathmode := "Text\ ", name := "A", next := 383271, root := ~, start := 383257, stop := 383270 ), rec( cont\ ent := "\n that satisfy the arguments.", count := [ 8, 1, 4, 18 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := 0, count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "P", next := 3\ 83311, root := ~, start := 383307, stop := 383310 ), rec( content := "\n\n \ The computation may be significantly faster if ", count := [ 8, 1, 4, 19 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "include", count := [ 8, 1, 4, 19 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 19 ], mathmode := "Te\ xt", name := "A", next := 383382, root := ~, start := 383368, stop := 383381 ) , rec( cont\ ent := " and\n ", count := [ 8, 1, 4, 19 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "exclude", count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "A", next\ := 383409, root := ~, start := 383395, stop := 383408 ), rec( content := " ar\ e invariant under the action of ", count := [ 8, 1, 4, 19 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "G", count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "C", next\ := 383452, root := ~, start := 383444, stop := 383451 ), rec( content := "\n \ on sets of vertices.\n ", count := [ 8, 1, 4, 19 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], name := "Item\ ", next := 383495, root := ~, start := 383110, stop := 383494 ), rec( attributes :\ = rec( ), content := [ rec( content := "Orbit representatives of cliques", coun\ t := [ 8, 1, 4, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 20 ], name := "Mar\ k", next := 383548, root := ~, start := 383503, stop := 383547 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n To use ", count := [ 8, 1, \ 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "DigraphMaximalCliquesReps", count := [\ 8, 1, 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 8, 1, 4, 21 ], mathmode := "Text", name := "C", next := 383609, root := ~, star\ t := 383577, stop := 383608 ), rec( content := " or ", count := [ 8, 1, 4, 21 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphCliquesReps", count := [ 8, 1, 4, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], math\ mode := "Text", name := "C", next := 383638, root := ~, start := 383613, stop \ := 383637 ), rec( content := ",\n the arguments ", count := [ 8, 1, 4, \ 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "include", count := [ 8, 1, 4, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], math\ mode := "Text", name := "A", next := 383676, root := ~, start := 383662, stop \ := 383675 ), rec( content := " and ", count := [ 8, 1, 4, 21 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "exclude", count := [ 8, 1, 4, 21 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], mathmode := "Text", name\ := "A", next := 383695, root := ~, start := 383681, stop := 383694 ), rec( cont\ ent := " must each be invariant\n under the action of ", count := [ 8, \ 1, 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "G", count := [ 8, 1, 4, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], math\ mode := "Text", name := "C", next := 383755, root := ~, start := 383747, stop \ := 383754 ), rec( content := " on sets of vertices.", count := [ 8, 1, 4, 21 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := 0, count := [ 8, 1, 4, 22 ], mathmode := "Text", name := "P", next := 3\ 83780, root := ~, start := 383776, stop := 383779 ), rec( content := "\n\n \ If this is the case, then ", count := [ 8, 1, 4, 22 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "DigraphMaximalCliquesReps", count := [ 8, 1, 4, 22 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ], mathmode := "Tex\ t", name := "C", next := 383848, root := ~, start := 383816, stop := 383847 ), rec( cont\ ent := " and\n ", count := [ 8, 1, 4, 22 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "DigraphCliquesReps", count := [ 8, 1, 4, 22 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ], mathmode := "Text", name :\ = "C", next := 383886, root := ~, start := 383861, stop := 383885 ), rec( cont\ ent := " each return a duplicate-free list of at most\n ", count := [ 8\ , 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 4, 22 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ] , math\ mode := "Text", name := "A", next := 383952, root := ~, start := 383940, stop \ := 383951 ), rec( content := " orbits representatives (under the action of ", coun\ t := [ 8, 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 1, 4, 22 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ] , math\ mode := "Text", name := "C", next := 384005, root := ~, start := 383997, stop \ := 384004 ), rec( content := " on\n sets vertices) of cliques of ", coun\ t := [ 8, 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph", count := [ 8, 1, 4,\ 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4\ , 22 ], mathmode := "Text", name := "A", next := 384060, root := ~, start := 3\ 84046, stop := 384059 ), rec( content := " that satisfy the\n arguments\ .", count := [ 8, 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 8, 1, 4, 23 ], mathmode := "Text", name\ := "P", next := 384100, root := ~, start := 384096, stop := 384099 ), rec( cont\ ent := "\n\n The representatives are not guaranteed to be in distinct o\ rbits.\n However, if fewer than ", count := [ 8, 1, 4, 23 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "lim", count := [ 8, 1, 4, 23 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 23 ], mathmode := "Text", name\ := "A", next := 384216, root := ~, start := 384206, stop := 384215 ), rec( cont\ ent := " results are returned, then there\n will be at least one repres\ entative from each orbit of maximal cliques.\n ", count := [ 8, 1, 4, 23 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], name := "Ite\ m", next := 384343, root := ~, start := 383555, stop := 384342 ) ], count := [ 8, 1, 4, 17 ], mathmode := "Text", name := "List", next := 384355, root := ~, start := 383061, stop := 384354 ), rec( content := "\n\n \ ", count := [ 8, 1, 4, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]])\ ;\n\ngap> IsSymmetricDigraph(D);\ \ntrue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4)\ ])\ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]\ng\ ap> DigraphMaximalCliquesReps(D);\n[ [ 1, 2, 3 ], [ 3, 4 ] ]\ngap> Orbit(G, [1\ , 2, 3], OnSets);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]\ngap> Orbit(G, [3, 4], OnSets)\ ;\n[ [ 3, 4 ] ]\ngap> DigraphMaximalCliquesReps(D, [3, 4], [], 1);\n[ [ 3, 4 ]\ ]\ngap> DigraphMaximalCliques(D, [1, 2], [5, 6], 1, 2);\n[ ]\ngap> DigraphCl\ iques(D, [1], [5, 6], infinity, 2);\n[ [ 1, 2 ], [ 1, 3 ] ]\ngap> D := Digraph\ (IsMutableDigraph, [\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]])\ ;\n\ngap> IsSymmetricDigraph(D);\nt\ rue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ \ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]", count := [ 8, 1, 4, 26 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 4, 26 ], mathmode := "Text", name := "Example", next := 385370, root := ~, start := 384361, stop := 385369 ), rec( content := "\n ", count := [ 8, 1, 4, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Description", next := 385387, root := ~, start := 381534, stop := 385386 ) ], count := [ 8, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 385401, root := ~, start := 380957, stop := 385400 ), rec( content := "\n\n ", count := [ 8, 1, 4, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "CliqueNumber" ) , content := 0, count := [ 8, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 385463, root := ~, start := 385422, stop := 385462 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer.", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 385508, root := ~, start := 385466, stop := 385507 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "A", next := 385546, root := ~, start := 385532, stop := 385545 ), rec( content := " is a di\ graph, then ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ liqueNumber(", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 8, 1, 5, 3 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := \ "Text", name := "A", next := 385596, root := ~, start := 385582, stop := 38559\ 5 ), rec( content := ")", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 385601, root := ~, start := 385566, stop := 385600 ), rec( content := "\n re\ turns the largest integer ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 385642, root := ~, start := 385634, stop := 385641 ), rec( content := " such th\ at ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "A", next := 385667, root := ~, start := 385653, stop := 385666 ), rec( content := " contain\ s a\n clique with ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 385703, root := ~, start := 385695, stop := 385702 ), rec( content := " vertice\ s as an induced subdigraph.\n ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "P", next := 385747, root := ~, start := 385743, stop := 385746 ), rec( content := "\n\n \ A ", count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ lique", count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "E", next := 385768, root := ~, start := 385755, stop := 385767 ), rec( content := " of a di\ graph is a set of mutually adjacent vertices of the\n digraph. Loops and mu\ ltiple edges are ignored for the purpose of\n determining the clique number\ of a digraph. \n ", count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);;\ngap> CliqueNumber(D);\n4\ngap> D := Digraph([\ [1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);\n\ngap> CliqueNumber(D);\n3\ngap> D := CompleteDigraph(IsMutabl\ eDigraph, 4);;\ngap> CliqueNumber(D);\n4", count := [ 8, 1, 5, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 5, 5 ], mathmode := "Text", name := "Example", next := 386246, root := ~, start := 385950, stop := 386245 ), rec( content := "\n", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 386261, root := ~, start := 385511, stop := 386260 ) ], count := [ 8, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 386275, root := ~, start := 385407, stop := 386274 ), rec( content := "\n\n ", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 386289, root := ~, start := 369777, stop := 386288 ) , rec( content := "\n\n ", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Finding independe\ nt sets", count := [ 8, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 386345, root := ~, start := 386302, stop := 386344 ), rec( content := "\n ", count := [ 8, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, l", Name := "IsIndependentSe\ t" ), content := 0, count := [ 8, 2, 1, 2 ], mathmode := "Text", name := "Oper", next := 386413, root := ~, start := 386365, stop := 386412 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalIndepe\ ndentSet" ), content := 0, count := [ 8, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 386471, root := ~, start := 386416, stop := 386470 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 386494, root := ~, start := 386483, stop := 386493 ), rec( content := " or ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 386510, root := ~, start := 386498, stop := 386509 ), rec( content := ".", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 386521, root := ~, start := 386474, stop := 386520 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386559, root := ~, start := 386545, stop := 386558 ), rec( content := " is a di\ graph and ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386585, root := ~, start := 386577, stop := 386584 ), rec( content := " is a du\ plicate-free list of\n vertices of ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386644, root := ~, start := 386630, stop := 386643 ), rec( content := ", then\n\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sIndependentSet(", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 386679, root := ~, start := 386655, stop := 386678 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386693, root := ~, start := 386679, stop := 386692 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 386701, root := ~, start := 386693, stop := 386700 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386709, root := ~, start := 386701, stop := 386708 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 386717, root := ~, start := 386709, stop := 386716 ), rec( content := " returns\ \n ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 386741, root := ~, start := 386730, stop := 386740 ), rec( content := " if ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386753, root := ~, start := 386745, stop := 386752 ), rec( content := " is an " , count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ndependent set", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "E", next := 386782, root := ~, start := 386760, stop := 386781 ), rec( content := " of ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386800, root := ~, start := 386786, stop := 386799 ), rec( content := " and\n \ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 386821, root := ~, start := 386809, stop := 386820 ), rec( content := " if it i\ s not. Similarly,\n ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalIndependentSet(", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 386883, root := ~, start := 386852, stop := 386882 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386897, root := ~, start := 386883, stop := 386896 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 386905, root := ~, start := 386897, stop := 386904 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386913, root := ~, start := 386905, stop := 386912 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 386921, root := ~, start := 386913, stop := 386920 ), rec( content := "\n re\ turns ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 386945, root := ~, start := 386934, stop := 386944 ), rec( content := " if ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 386957, root := ~, start := 386949, stop := 386956 ), rec( content := " is a ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal independent set", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "E", next := 386993, root := ~, start := 386963, stop := 386992 ), rec( content := " of\n \ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 387015, root := ~, start := 387001, stop := 387014 ), rec( content := " and ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 387032, root := ~, start := 387020, stop := 387031 ), rec( content := " if it i\ s not. ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "P", next := 387052, root := ~, start := 387048, stop := 387051 ), rec( content := "\n\n \ An ", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ndependent set", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "E", next := 387083, root := ~, start := 387061, stop := 387082 ), rec( content := " of a di\ graph is a set of mutually non-adjacent\n vertices of the digraph. A ", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal independent set", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "E", next := 387192, root := ~, start := 387162, stop := 387191 ), rec( content := " is an i\ ndependent\n set that is not properly contained in another independent set.\ An\n independent set is permitted, but not required, to contain vertices a\ t\n which there is a loop.\n ", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(4);;\ngap> IsIndependentSet(D, [1]);\ntrue\ngap> IsMax\ imalIndependentSet(D, [1]);\nfalse\ngap> IsIndependentSet(D, [1, 4, 3]);\nfals\ e\ngap> IsIndependentSet(D, [2, 4]);\ntrue\ngap> IsMaximalIndependentSet(D, [2\ , 4]);\ntrue\ngap> D := CycleDigraph(IsMutableDigraph, 4);;\ngap> IsIndependen\ tSet(D, [1]);\ntrue", count := [ 8, 2, 1, 7 ], name := "PCDATA", root := ~ ) ] , count := [ 8, 2, 1, 7 ], mathmode := "Text", name := "Example", next := 387736, root := ~, start := 387387, stop := 387735 ), rec( content := "\n ", count := [ 8, 2, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 387753, root := ~, start := 386524, stop := 387752 ) ], count := [ 8, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 387767, root := ~, start := 386350, stop := 387766 ), rec( content := "\n\n ", count := [ 8, 2, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[, size]]]", Name := "DigraphIndependentSet" ), content := 0, count := [ 8, 2, 2, 2 ], mathmode := "Text", name := "Func", next := 387872, root := ~, start := 387788, stop := 387871 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalIndependentSet" ), content := 0, count := [ 8, 2, 2, 4 ], mathmode := "Text", name := "Func", next := 387970, root := ~, start := 387875, stop := 387969 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of positive integers, or ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "K", next := 388036, root := ~, start := 388025, stop := 388035 ), rec( content := ".", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Returns", next := 388047, root := ~, start := 387973, stop := 388046 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "A", next := 388085, root := ~, start := 388071, stop := 388084 ), rec( content := " is a di\ graph, then these functions returns an independent\n set of ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "A", next := 388169, root := ~, start := 388155, stop := 388168 ), rec( content := " if one \ exists that satisfies the arguments, else it\n returns ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "K", next := 388245, root := ~, start := 388234, stop := 388244 ), rec( content := ". An in\ dependent set is defined by the set of vertices\n that it contains; see ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsIndep\ endentSet" ), content := 0, count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Ref", next := 388357, root := ~, start := 388327, stop := 388356 ), rec( content := " and ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alIndependentSet" ), content := 0, count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Ref", next := 388405, root := ~, start := 388362, stop := 388404 ), rec( content := ".", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "P", next := 388410, root := ~, start := 388406, stop := 388409 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388453, root := ~, start := 388439, stop := 388452 ), rec( content := " and ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388472, root := ~, start := 388458, stop := 388471 ), rec( content := " must ea\ ch be a\n (possibly empty) duplicate-free list of vertices of ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388558, root := ~, start := 388544, stop := 388557 ), rec( content := ", and th\ e\n optional argument ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388601, root := ~, start := 388590, stop := 388600 ), rec( content := " must be\ a positive integer. By default,\n ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388661, root := ~, start := 388647, stop := 388660 ), rec( content := " and ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388680, root := ~, start := 388666, stop := 388679 ), rec( content := " are emp\ ty. These functions will search\n for an independent set of ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388764, root := ~, start := 388750, stop := 388763 ), rec( content := " that in\ cludes the vertices of\n ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388813, root := ~, start := 388799, stop := 388812 ), rec( content := " but doe\ s not include any vertices in ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388865, root := ~, start := 388851, stop := 388864 ), rec( content := ";\n i\ f the argument ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 388898, root := ~, start := 388887, stop := 388897 ), rec( content := " is supp\ lied, then additionally the independent\n set will be required to contain p\ recisely ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 389003, root := ~, start := 388992, stop := 389002 ), rec( content := " vertice\ s.", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "P", next := 389017, root := ~, start := 389013, stop := 389016 ), rec( content := "\n\n \ If ", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "A", next := 389040, root := ~, start := 389026, stop := 389039 ), rec( content := " is not \ an independent set, then these functions return\n ", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "K", next := 389111, root := ~, start := 389100, stop := 389110 ), rec( content := ". Other\ wise, the functions behave in the following way,\n depending on the number \ of arguments:\n\n ", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "One or two arguments", count := [ 8, 2\ , 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 8 ], name := "Mar\ k", next := 389261, root := ~, start := 389228, stop := 389260 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If one or two arguments are\ supplied, then ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Dig\ raphIndependentSet", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "C", next\ := 389354, root := ~, start := 389326, stop := 389353 ), rec( content := "\n \ and ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphMaxi\ malIndependentSet", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "C", next\ := 389402, root := ~, start := 389367, stop := 389401 ), rec( content := " gr\ eedily enlarge the\n independent set ", count := [ 8, 2, 2, 9 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "include", count := [ 8, 2, 2, 9 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], mathmode := "Text" , name\ := "A", next := 389462, root := ~, start := 389448, stop := 389461 ), rec( cont\ ent := " until it can not continue. The result\n is guaranteed to be a\ maximal independent set. This may or may not\n return an answer more q\ uickly than using ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Func := "DigraphMaximalIndependentS\ ets" ), content := 0, count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "Re\ f", next := 389679, root := ~, start := 389626, stop := 389678 ), rec( content\ := ". with a limit of 1.\n ", count := [ 8, 2, 2, 9 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], name := "Item", next := 3897\ 14, root := ~, start := 389268, stop := 389713 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "Three arguments", count := [ 8, 2, 2, 10 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 10 ], name := "Mark", next := 3897\ 49, root := ~, start := 389721, stop := 389748 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n If three arguments are supplied, then ", count := [ 8, 2, 2,\ 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "DigraphIndependentSet", count := [ 8, 2, \ 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2,\ 2, 11 ], mathmode := "Text", name := "C", next := 389837, root := ~, start :=\ 389809, stop := 389836 ), rec( content := "\n greedily enlarges the in\ dependent set ", count := [ 8, 2, 2, 11 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "incl\ ude", count := [ 8, 2, 2, 11 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 8, 2, 2, 11 ], mathmode := "Text", name := "A", next := 3898\ 98, root := ~, start := 389884, stop := 389897 ), rec( content := " until it c\ an not\n continue, although this independent set may not be maximal.", coun\ t := [ 8, 2, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 8, 2, 2, 12 ], mathmode := "Text", name\ := "P", next := 389987, root := ~, start := 389983, stop := 389986 ), rec( cont\ ent := "\n\n Given three arguments, ", count := [ 8, 2, 2, 12 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "DigraphMaximalIndependentSet", count := [ 8, 2, 2, 12\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 1\ 2 ], mathmode := "Text", name := "C", next := 390055, root := ~, start := 3900\ 20, stop := 390054 ), rec( content := " returns the\n maximal independe\ nt set returned by\n ", count := [ 8, 2, 2, 12 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "DigraphMaximalIndependentSets(", count := [ 8, 2, 2, 12 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 12 ], mathmode :=\ "Text", name := "C", next := 390157, root := ~, start := 390120, stop := 3901\ 56 ), rec( attributes := rec( ), content := [ rec( content := "digraph", coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "A", next := 390171, root :=\ ~, start := 390157, stop := 390170 ), rec( attributes := rec( ), content := [ \ rec( content := ",\n ", count := [ 8, 2, 2, 12 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 12 ], mathmode := "Text", name\ := "C", next := 390188, root := ~, start := 390171, stop := 390187 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "include", count := [ 8, 2, 2,\ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 12 ], mathmode := "Text", name := "A", next := 390202, root := ~, start := 3\ 90188, stop := 390201 ), rec( attributes := rec( ), content := [ rec( content\ := ", ", count := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 2, 2, 12 ], mathmode := "Text", name := "C", next := \ 390211, root := ~, start := 390202, stop := 390210 ), rec( attributes := rec( \ ), content := [ rec( content := "exclude", count := [ 8, 2, 2, 12 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 12 ], mathmod\ e := "Text", name := "A", next := 390225, root := ~, start := 390211, stop := \ 390224 ), rec( attributes := rec( ), content := [ rec( content := ", 1)", cou\ nt := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "C", next := 390236, root :=\ ~, start := 390225, stop := 390235 ), rec( content := " if one exists,\n \ else ", count := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "K", next := 390276, root :=\ ~, start := 390265, stop := 390275 ), rec( content := ".\n ", count := [ \ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 11 ], name := "Ite\ m", next := 390291, root := ~, start := 389756, stop := 390290 ), rec( attributes :\ = rec( ), content := [ rec( content := "Four arguments", count := [ 8, 2, 2, \ 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 13 ], name := "Mar\ k", next := 390325, root := ~, start := 390298, stop := 390324 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If four arguments are suppl\ ied, then ", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphInde\ pendentSet", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "C", next \ := 390412, root := ~, start := 390384, stop := 390411 ), rec( content := "\n \ returns the independent set returned by\n ", count := [ 8, 2, 2, \ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "DigraphIndependentSets(", count := [ 8, 2\ , 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, \ 2, 2, 14 ], mathmode := "Text", name := "C", next := 390499, root := ~, start \ := 390469, stop := 390498 ), rec( attributes := rec( ), content := [ rec( con\ tent := "digraph", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "A", next\ := 390513, root := ~, start := 390499, stop := 390512 ), rec( attributes := re\ c( ), content := [ rec( content := ", ", count := [ 8, 2, 2, 14 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode \ := "Text", name := "C", next := 390522, root := ~, start := 390513, stop := 39\ 0521 ), rec( attributes := rec( ), content := [ rec( content := "include", co\ unt := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 14 ], mathmode := "Text", name := "A", next := 390536, root :=\ ~, start := 390522, stop := 390535 ), rec( attributes := rec( ), content := [ \ rec( content := ",\n ", count := [ 8, 2, 2, 14 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name\ := "C", next := 390553, root := ~, start := 390536, stop := 390552 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 2, 2,\ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 14 ], mathmode := "Text", name := "A", next := 390567, root := ~, start := 3\ 90553, stop := 390566 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1, ", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "C", next \ := 390579, root := ~, start := 390567, stop := 390578 ), rec( attributes := re\ c( ), content := [ rec( content := "size", count := [ 8, 2, 2, 14 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmod\ e := "Text", name := "A", next := 390590, root := ~, start := 390579, stop := \ 390589 ), rec( attributes := rec( ), content := [ rec( content := ")", count \ := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 8, 2, 2, 14 ], mathmode := "Text", name := "C", next := 390598, root := \ ~, start := 390590, stop := 390597 ), rec( content := " if one exists, else\n \ ", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 14 ], mathmode := "Text", name := "K", next := 390638, root :=\ ~, start := 390627, stop := 390637 ), rec( content := ". This independent set\ may not be maximal.", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, \ 2, 2, 15 ], mathmode := "Text", name := "P", next := 390684, root := ~, start \ := 390680, stop := 390683 ), rec( content := " Given four\n arguments, \ ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphMaximalIndependentSet"\ , count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 8, 2, 2, 15 ], mathmode := "Text", name := "C", next := 390750, root :=\ ~, start := 390715, stop := 390749 ), rec( content := " returns the maximal\n\ independent set returned by\n ", count := [ 8, 2, 2, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalIndependentSets(", count := [ 8, 2, 2,\ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 15 ], mathmode := "Text", name := "C", next := 390852, root := ~, start := 3\ 90815, stop := 390851 ), rec( attributes := rec( ), content := [ rec( content\ := "digraph", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "A", next\ := 390866, root := ~, start := 390852, stop := 390865 ), rec( attributes := re\ c( ), content := [ rec( content := ",\n ", count := [ 8, 2, 2, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ] , math\ mode := "Text", name := "C", next := 390885, root := ~, start := 390866, stop \ := 390884 ), rec( attributes := rec( ), content := [ rec( content := "include\ ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "A", next := 390899, root\ := ~, start := 390885, stop := 390898 ), rec( attributes := rec( ), content \ := [ rec( content := ", ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name\ := "C", next := 390908, root := ~, start := 390899, stop := 390907 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 2, 2,\ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 15 ], mathmode := "Text", name := "A", next := 390922, root := ~, start := 3\ 90908, stop := 390921 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1,\n ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "\ C", next := 390942, root := ~, start := 390922, stop := 390941 ), rec( attribu\ tes := rec( ), content := [ rec( content := "size", count := [ 8, 2, 2, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ] , math\ mode := "Text", name := "A", next := 390953, root := ~, start := 390942, stop \ := 390952 ), rec( attributes := rec( ), content := [ rec( content := ")", cou\ nt := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 15 ], mathmode := "Text", name := "C", next := 390961, root :=\ ~, start := 390953, stop := 390960 ), rec( content := " if one exists, else " , coun\ t := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 2, 2, 15\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 1\ 5 ], mathmode := "Text", name := "K", next := 390993, root := ~, start := 3909\ 82, stop := 390992 ), rec( content := ".\n ", count := [ 8, 2, 2, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], name := "Ite\ m", next := 391008, root := ~, start := 390332, stop := 391007 ) ], count := [ 8, 2, 2, 8 ], mathmode := "Text", name := "List", next := 391020, root := ~, start := 389215, stop := 391019 ), rec( content := "\n ", count := [ 8, 2, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := ChainDigraph(6);\n\ngap> D\ igraphIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphMaximalIndependentSet(D);\n\ [ 6, 4, 2 ]\ngap> DigraphIndependentSet(D, [2, 4]);\n[ 2, 4, 6 ]\ngap> Digraph\ MaximalIndependentSet(D, [1, 3]);\n[ 1, 3, 6 ]\ngap> DigraphIndependentSet(D, \ [2, 4], [6]);\n[ 2, 4 ]\ngap> DigraphMaximalIndependentSet(D, [2, 4], [6]);\nf\ ail\ngap> DigraphIndependentSet(D, [1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximal\ IndependentSet(D, [1], [], 2);\nfail\ngap> DigraphMaximalIndependentSet(D, [1]\ , [], 3);\n[ 1, 3, 5 ]\ngap> D := ChainDigraph(IsMutableDigraph, 6);\n\ngap> DigraphIndependentSet(D);\n[ 6, 4, 2 \ ]\ngap> DigraphMaximalIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphIndependent\ Set(D, [2, 4]);\n[ 2, 4, 6 ]\ngap> DigraphMaximalIndependentSet(D, [1, 3]);\n[\ 1, 3, 6 ]\ngap> DigraphIndependentSet(D, [2, 4], [6]);\n[ 2, 4 ]\ngap> Digrap\ hMaximalIndependentSet(D, [2, 4], [6]);\nfail\ngap> DigraphIndependentSet(D, [\ 1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximalIndependentSet(D, [1], [], 2);\nfail\ \ngap> DigraphMaximalIndependentSet(D, [1], [], 3);\n[ 1, 3, 5 ]", count := [ 8, 2, 2, 18 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 2, 18 ], mathmode := "Text", name := "Example", next := 392173, root := ~, start := 391025, stop := 392172 ), rec( content := "\n ", count := [ 8, 2, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Description", next := 392190, root := ~, start := 388050, stop := 392189 ) ], count := [ 8, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 392204, root := ~, start := 387773, stop := 392203 ), rec( content := "\n\n ", count := [ 8, 2, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n limit[, size]]]]", Name := "DigraphMaximalIndependentSets" ), content := 0, count := [ 8, 2, 3, 2 ], mathmode := "Text", name := "Func", next := 392326, root := ~, start := 392225, stop := 392325 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphMaximalIndependentSetsRe\ ps" ), content := 0, count := [ 8, 2, 3, 4 ], mathmode := "Text", name := "Func", next := 392434, root := ~, start := 392329, stop := 392433 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphIndependentSets" ), content := 0, count := [ 8, 2, 3, 6 ], mathmode := "Text", name := "Func", next := 392531, root := ~, start := 392437, stop := 392530 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphIndependentSetsReps" ), content := 0, count := [ 8, 2, 3, 8 ], mathmode := "Text", name := "Func", next := 392632, root := ~, start := 392534, stop := 392631 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsAttr" ), content := 0, count := [ 8, 2, 3, 10 ], mathmode := "Text", name := "Attr", next := 392697, root := ~, start := 392635, stop := 392696 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsRepsAttr" ), content := 0, count := [ 8, 2, 3, 12 ], mathmode := "Text", name := "Attr", next := 392766, root := ~, start := 392700, stop := 392765 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of lists of positive integers.", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Returns", next := 392836, root := ~, start := 392769, stop := 392835 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "A", next := 392874, root := ~, start := 392860, stop := 392873 ), rec( content := " is digr\ aph, then these functions and attributes use ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Cliques\ Finder" ), content := 0, count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Ref", next := 392960, root := ~, start := 392927, stop := 392959 ), rec( content := " to retu\ rn independent sets of ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "A", next := 393005, root := ~, start := 392991, stop := 393004 ), rec( content := ". An\n \ independent set is defined by the set of vertices that it contains; see\n \ ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alIndependentSet" ), content := 0, count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Ref", next := 393127, root := ~, start := 393090, stop := 393126 ), rec( content := " and ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsIndep\ endentSet" ), content := 0, count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Ref", next := 393168, root := ~, start := 393132, stop := 393167 ), rec( content := ".", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "P", next := 393173, root := ~, start := 393169, stop := 393172 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 393216, root := ~, start := 393202, stop := 393215 ), rec( content := " and ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 393235, root := ~, start := 393221, stop := 393234 ), rec( content := " must ea\ ch be a\n (possibly empty) list of vertices of ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 393306, root := ~, start := 393292, stop := 393305 ), rec( content := ", the op\ tional argument\n ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 393346, root := ~, start := 393334, stop := 393345 ), rec( content := " must be\ either a positive integer or ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "C", next := 393399, root := ~, start := 393384, stop := 393398 ), rec( content := ", and th\ e\n optional argument ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 393442, root := ~, start := 393431, stop := 393441 ), rec( content := " must be\ a positive integer. If not\n specified, then ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 393513, root := ~, start := 393499, stop := 393512 ), rec( content := " and ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 393532, root := ~, start := 393518, stop := 393531 ), rec( content := " are cho\ sen to be empty\n lists, and ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 393583, root := ~, start := 393571, stop := 393582 ), rec( content := " is set \ to ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "C", next := 393609, root := ~, start := 393594, stop := 393608 ), rec( content := ". ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "P", next := 393615, root := ~, start := 393611, stop := 393614 ), rec( content := "\n\n \ The functions will return as many suitable independent sets as possible, up\n \ to the number ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 393727, root := ~, start := 393715, stop := 393726 ), rec( content := ". These\ functions will find independent sets\n that contain all of the vertices of\ ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 393827, root := ~, start := 393813, stop := 393826 ), rec( content := " but do \ not\n contain any of the vertices of ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 393888, root := ~, start := 393874, stop := 393887 ), rec( content := " The arg\ ument ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 393913, root := ~, start := 393902, stop := 393912 ), rec( content := "\n re\ stricts the search to those cliques that contain precisely ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 393990, root := ~, start := 393979, stop := 393989 ), rec( content := "\n ve\ rtices. If the function or attribute has ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximal", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "C", next := 394053, root := ~, start := 394039, stop := 394052 ), rec( content := " in its \ name,\n then only maximal independent sets will be returned; otherwise non-\ maximal\n independent sets may be returned.\n ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "P", next := 394192, root := ~, start := 394188, stop := 394191 ), rec( content := "\n\n \ Let ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "C", next := 394210, root := ~, start := 394202, stop := 394209 ), rec( content := " denote \ the ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "Ref", next := 394275, root := ~, start := 394222, stop := 394274 ), rec( content := "\n of\ the ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ SymmetricClosure" ), content := 0, count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "Ref", next := 394324, root := ~, start := 394287, stop := 394323 ), rec( content := " of the \ digraph formed from\n ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "A", next := 394370, root := ~, start := 394356, stop := 394369 ), rec( content := " by remo\ ving loops and ignoring the multiplicity of edges.\n\n ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Distinct independent sets", count := [ \ 8, 2, 3, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 17 ], name := "Mar\ k", next := 394485, root := ~, start := 394447, stop := 394484 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 8, 2, 3, 18 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalIndependentSets", count := [ 8, 2, 3, \ 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3,\ 18 ], mathmode := "Text", name := "C", next := 394543, root := ~, start := 39\ 4507, stop := 394542 ), rec( content := " and ", count := [ 8, 2, 3, 18 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphIndependentSets", count := [ 8, 2, 3, 18 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 18 ], math\ mode := "Text", name := "C", next := 394577, root := ~, start := 394548, stop \ := 394576 ), rec( content := "\n each return a duplicate-free list of a\ t most ", count := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "limit", cou\ nt := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 3, 18 ], mathmode := "Text", name := "A", next := 394643, root :=\ ~, start := 394631, stop := 394642 ), rec( content := " independent\n \ sets of ", count := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 3, 18 ], mathmode := "Text", name := "A", next := 394686, root :=\ ~, start := 394672, stop := 394685 ), rec( content := " that satisfy the argu\ ments.", count := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 19 ], math\ mode := "Text", name := "P", next := 394718, root := ~, start := 394714, stop \ := 394717 ), rec( content := "\n\n The computation may be significantly\ faster if ", count := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "include", c\ ount := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 3, 19 ], mathmode := "Text", name := "A", next := 394789, root :=\ ~, start := 394775, stop := 394788 ), rec( content := " and\n ", count\ := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 2, 3,\ 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3\ , 19 ], mathmode := "Text", name := "A", next := 394816, root := ~, start := 3\ 94802, stop := 394815 ), rec( content := " are invariant under the action of " , coun\ t := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 2, 3, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 19 ] , math\ mode := "Text", name := "C", next := 394859, root := ~, start := 394851, stop \ := 394858 ), rec( content := " on sets of\n vertices.\n ", count :\ = [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 18 ], name := "Ite\ m", next := 394902, root := ~, start := 394492, stop := 394901 ), rec( attributes :\ = rec( ), content := [ rec( content := "Representatives of distinct orbits of\ independent sets", count := [ 8, 2, 3, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 3, 20 ], name := "Mark", next := 394977, root := ~, start := 394\ 910, stop := 394976 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n To use ", count := [ 8, 2, 3, 21 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "DigraphMaximalIndependentSetsReps", count := [ 8, 2, 3, 21 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "\ Text", name := "C", next := 395046, root := ~, start := 395006, stop := 395045\ ), rec( content := " or\n ", count := [ 8, 2, 3, 21 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "DigraphIndependentSetsReps", count := [ 8, 2, 3, 21 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmo\ de := "Text", name := "C", next := 395091, root := ~, start := 395058, stop :=\ 395090 ), rec( content := ", the arguments ", count := [ 8, 2, 3, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "include", count := [ 8, 2, 3, 21 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "Te\ xt", name := "A", next := 395121, root := ~, start := 395107, stop := 395120 ) , rec( cont\ ent := " and\n ", count := [ 8, 2, 3, 21 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "exclude", count := [ 8, 2, 3, 21 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "Text", name := "A", next\ := 395148, root := ~, start := 395134, stop := 395147 ), rec( content := " mu\ st each be invariant under the action of ", count := [ 8, 2, 3, 21 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "G", count := [ 8, 2, 3, 21 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "Text", name\ := "C", next := 395200, root := ~, start := 395192, stop := 395199 ), rec( cont\ ent := " on\n sets of vertices.", count := [ 8, 2, 3, 21 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ 0, count := [ 8, 2, 3, 22 ], mathmode := "Text", name := "P", next := 395233, root\ := ~, start := 395229, stop := 395232 ), rec( content := "\n\n If this\ is the case, then ", count := [ 8, 2, 3, 22 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "DigraphMaximalIndependentSetsReps", count := [ 8, 2, 3, 22 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], mathmode := "T\ ext", name := "C", next := 395309, root := ~, start := 395269, stop := 395308 ), rec( cont\ ent := " and\n ", count := [ 8, 2, 3, 22 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "DigraphIndependentSetsReps", count := [ 8, 2, 3, 22 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], mathmode := "Text" , name\ := "C", next := 395355, root := ~, start := 395322, stop := 395354 ), rec( cont\ ent := " each return a list of\n at most ", count := [ 8, 2, 3, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "limit", count := [ 8, 2, 3, 22 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], mathmode := "Text\ ", name := "A", next := 395406, root := ~, start := 395394, stop := 395405 ), rec( cont\ ent := " orbits representatives (under the action of\n ", count := [ 8,\ 2, 3, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "G", count := [ 8, 2, 3, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], math\ mode := "Text", name := "C", next := 395467, root := ~, start := 395459, stop \ := 395466 ), rec( content := " on sets of vertices) of independent sets of ", coun\ t := [ 8, 2, 3, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph", count := [ 8, 2, 3,\ 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3\ , 22 ], mathmode := "Text", name := "A", next := 395526, root := ~, start := 3\ 95512, stop := 395525 ), rec( content := "\n that satisfy the arguments\ . ", count := [ 8, 2, 3, 22 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 23 ], mathmo\ de := "Text", name := "P", next := 395567, root := ~, start := 395563, stop :=\ 395566 ), rec( content := "\n\n The representatives are not guaranteed\ to be in distinct orbits.\n However, if ", count := [ 8, 2, 3, 23 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "lim", count := [ 8, 2, 3, 23 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 23 ], mathmode := "Text", name\ := "A", next := 395672, root := ~, start := 395662, stop := 395671 ), rec( cont\ ent := " is not specified, or fewer than ", count := [ 8, 2, 3, 23 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "lim", count := [ 8, 2, 3, 23 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 23 ], mathmode := "Text", name\ := "A", next := 395715, root := ~, start := 395705, stop := 395714 ), rec( cont\ ent := "\n results are returned, then there will be at least one repres\ entative\n from each orbit of maximal independent sets.\n ", count\ := [ 8, 2, 3, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], name := "Ite\ m", next := 395859, root := ~, start := 394984, stop := 395858 ) ], count := [ 8, 2, 3, 17 ], mathmode := "Text", name := "List", next := 395871, root := ~, start := 394434, stop := 395870 ), rec( content := "\n\n \ ", count := [ 8, 2, 3, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(5);\n\ngap> D\ igraphMaximalIndependentSetsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSet\ sReps(D);\n[ [ 1 ], [ 1, 3 ] ]\ngap> Set(DigraphMaximalIndependentSets(D));\n[\ [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndepe\ ndentSets(D, [1]);\n[ [ 1, 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [],\ [4, 5]);\n[ [ 1 ], [ 2 ], [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [\ ], [4, 5], 1, 2);\n[ [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\ \n\ngap> DigraphMaximalIndependentSe\ tsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSetsReps(D);\n[ [ 1 ], [ 1, 3\ ] ]\ngap> Set(DigraphMaximalIndependentSets(D));\n[ [ 1, 3 ], [ 1, 4 ], [ 2, \ 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndependentSets(D, [1]);\n[ [ 1,\ 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5]);\n[ [ 1 ], [ 2 ],\ [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5], 1, 2);\n[ [ 1, \ 3 ] ]", count := [ 8, 2, 3, 26 ], name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 26 ], mathmode := "Text", name := "Example", next := 396899, root := ~, start := 395877, stop := 396898 ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Description", next := 396916, root := ~, start := 392839, stop := 396915 ) ], count := [ 8, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 396930, root := ~, start := 392210, stop := 396929 ), rec( content := "\n\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 396944, root := ~, start := 386293, stop := 396943 ) , rec( content := "\n", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 396955, root := ~, start := 365016, stop := 396954 ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " cliques ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Visualising and IO" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Visualising and IO", count := [ 9, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 397049, root := ~, start := 397012, stop := 397048 ) , rec( content := "\n\n ", count := [ 9, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Visualising a dig\ raph", count := [ 9, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 397102, root := ~, start := 397062, stop := 397101 ), rec( content := "\n ", count := [ 9, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str, [opts]", Name := "Splash" ), content := 0, count := [ 9, 1, 1, 2 ], mathmode := "Text", name := "Func", next := 397165, root := ~, start := 397126, stop := 397164 ), rec( attributes := rec( ), content := [ rec( content := "Nothing." , count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 397197, root := ~, start := 397170, stop := 397196 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function attempts to convert the string ", count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "A", next := 397277, root := ~, start := 397267, stop := 397276 ), rec( content := " into a \ pdf\n document and open this document, i.e. to splash it all over your mo\ nitor.", count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "P", next := 397372, root := ~, start := 397368, stop := 397371 ), rec( content := "\n\n \ The string ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "A", next := 397401, root := ~, start := 397391, stop := 397400 ), rec( content := " must co\ rrespond to a valid ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 397439, root := ~, start := 397429, stop := 397438 ), rec( content := " or\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ aTeX", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 397461, root := ~, start := 397449, stop := 397460 ), rec( content := " text fi\ le and you must have have ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 397510, root := ~, start := 397495, stop := 397509 ), rec( content := " and\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ dflatex", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 397536, root := ~, start := 397521, stop := 397535 ), rec( content := " install\ ed on your computer. For details about these file\n formats, see ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.latex-project.org", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "URL", next := 397653, root := ~, start := 397614, stop := 397652 ), rec( content := " and\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "URL", next := 397698, root := ~, start := 397664, stop := 397697 ), rec( content := ".", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "P", next := 397703, root := ~, start := 397699, stop := 397702 ), rec( content := "\n\n \ This function is provided to allow convenient, immediate viewing of the\n \ pictures produced by the function ", count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DotDigr\ aph" ), content := 0, count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "Ref", next := 397847, root := ~, start := 397823, stop := 397846 ), rec( content := ".", count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "P", next := 397852, root := ~, start := 397848, stop := 397851 ), rec( content := "\n\n \ The optional second argument ", count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ pts", count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "A", next := 397900, root := ~, start := 397889, stop := 397899 ), rec( content := " should \ be a record with\n components corresponding to various options, given bel\ ow.\n\n ", count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "path", count := [ 9, 1, 1, 7 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 7 ], name := "Mar\ k", next := 398028, root := ~, start := 398011, stop := 398027 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the path to the directory where\n you want ", count := [ \ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "Splash", count := [ 9, 1, 1, 8 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 8 ], math\ mode := "Text", name := "C", next := 398155, root := ~, start := 398142, stop \ := 398154 ), rec( content := " to do its work. The default value of this\n \ option is ", count := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"~\ /\"", count := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 9, 1, 1, 8 ], mathmode := "Text", name := "C", next := 398229 , root\ := ~, start := 398218, stop := 398228 ), rec( content := ".\n ", count\ := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 8 ], name := "Ite\ m", next := 398246, root := ~, start := 398037, stop := 398245 ), rec( attributes :\ = rec( ), content := [ rec( content := "directory", count := [ 9, 1, 1, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 9 ], name := "Mar\ k", next := 398278, root := ~, start := 398256, stop := 398277 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the name of the directory in\n ", count := [ 9, 1, 1, 10 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "path", count := [ 9, 1, 1, 10 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 10 ], mathmode := "Text" , name\ := "C", next := 398391, root := ~, start := 398380, stop := 398390 ), rec( cont\ ent := " where you want ", count := [ 9, 1, 1, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "Splash", count := [ 9, 1, 1, 10 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 9, 1, 1, 10 ], mathmode := "Text", name := "C", next\ := 398420, root := ~, start := 398407, stop := 398419 ), rec( content := " to\ do its work. This function\n will create this directory if does not \ already exist. ", count := [ 9, 1, 1, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1,\ 11 ], mathmode := "Text", name := "P", next := 398519, root := ~, start := 39\ 8515, stop := 398518 ), rec( content := "\n\n The default value of th\ is option is ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"tmp.viz\"\ ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "C", next := 398583, root\ := ~, start := 398567, stop := 398582 ), rec( content := " if the option\n \ ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "path", coun\ t := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 11 ], mathmode := "Text", name := "C", next := 398619, root :=\ ~, start := 398608, stop := 398618 ), rec( content := " is present, and the r\ esult of\n ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref", Func := "Dire\ ctoryTemporary" ), content := 0, count := [ 9, 1, 1, 11 ], mathmode := "Text", name\ := "Ref", next := 398707, root := ~, start := 398660, stop := 398706 ), rec( cont\ ent := " is used otherwise.\n ", count := [ 9, 1, 1, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 10 ], name := "Ite\ m", next := 398742, root := ~, start := 398287, stop := 398741 ), rec( attributes :\ = rec( ), content := [ rec( content := "filename", count := [ 9, 1, 1, 12 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 12 ], name := "Mar\ k", next := 398773, root := ~, start := 398752, stop := 398772 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the name of the file where\n ", count := [ 9, 1, 1, 13 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "str", count := [ 9, 1, 1, 13 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 13 ], mathmode := "Text", name\ := "A", next := 398883, root := ~, start := 398873, stop := 398882 ), rec( cont\ ent := " will be written. The default value of this option is\n ", coun\ t := [ 9, 1, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "\"vizpicture\"", count := [ 9\ , 1, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 9, 1, 1, 13 ], mathmode := "Text", name := "C", next := 398967, root := ~, star\ t := 398948, stop := 398966 ), rec( content := ".\n ", count := [ 9, 1,\ 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 13 ], name := "Ite\ m", next := 398984, root := ~, start := 398782, stop := 398983 ), rec( attributes :\ = rec( ), content := [ rec( content := "viewer", count := [ 9, 1, 1, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 14 ], name := "Mar\ k", next := 399013, root := ~, start := 398994, stop := 399012 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the name of the program which\n should open the files pro\ duced by ", count := [ 9, 1, 1, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "GraphViz", \ count := [ 9, 1, 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 15 ], mathmode := "Text", name := "C", next := 399165, root :=\ ~, start := 399150, stop := 399164 ), rec( content := " or ", count := [ 9, 1\ , 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "pdflatex", count := [ 9, 1, 1, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 15 ] , math\ mode := "Text", name := "C", next := 399184, root := ~, start := 399169, stop \ := 399183 ), rec( content := ".\n ", count := [ 9, 1, 1, 15 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 15 ], name := "Ite\ m", next := 399201, root := ~, start := 399022, stop := 399200 ), rec( attributes :\ = rec( ), content := [ rec( content := "type", count := [ 9, 1, 1, 16 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 16 ], name := "Mar\ k", next := 399228, root := ~, start := 399211, stop := 399227 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this option can be used t\ o specify that the string ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "str", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "A", next \ := 399315, root := ~, start := 399305, stop := 399314 ), rec( content := " con\ tains\n a ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( Only := "LaTeX" ), content := [ r\ ec( content := "{\\LaTeX}", count := [ 9, 1, 1, 17 ], name := "PCDATA", root :\ = ~ ) ], count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "Alt", next := \ 33, root := ~, start := 1, stop := 32 ), rec( attributes := rec( Not := "LaTeX\ " ), content := [ rec( content := "LaTeX", count := [ 9, 1, 1, 17 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], name := \ "Alt", next := 61, root := ~, start := 33, stop := 60 ), rec( content := " or \ ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "dot", count := [ 9, 1, 1, 17 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17\ ], mathmode := "Text", name := "C", next := 399358, root := ~, start := 39934\ 8, stop := 399357 ), rec( content := " document. You can specify this option i\ n\n ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "str", count\ := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 17 ], mathmode := "Text", name := "A", next := 399420, root :=\ ~, start := 399410, stop := 399419 ), rec( content := " directly by making th\ e first line ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"%latex\""\ , count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 9, 1, 1, 17 ], mathmode := "Text", name := "C", next := 399470, root :=\ ~, start := 399455, stop := 399469 ), rec( content := " or\n ", coun\ t := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "\"//dot\"", count := [ 9, 1, \ 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1,\ 1, 17 ], mathmode := "Text", name := "C", next := 399498, root := ~, start :=\ 399484, stop := 399497 ), rec( content := ". There is no default value for t\ his option, this\n option must be specified in ", count := [ 9, 1, 1,\ 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "str", count := [ 9, 1, 1, 17 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], mathmode\ := "Text", name := "A", next := 399597, root := ~, start := 399587, stop := 3\ 99596 ), rec( content := " or in ", count := [ 9, 1, 1, 17 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "opt.type", count := [ 9, 1, 1, 17 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], mathmode := "Text", name \ := "A", next := 399619, root := ~, start := 399604, stop := 399618 ), rec( cont\ ent := ".\n ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], name := "Item", next := 39963\ 6, root := ~, start := 399237, stop := 399635 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "engine", count := [ 9, 1, 1, 18 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 1, 1, 18 ], name := "Mark", next := 399665, root := ~, start := 399\ 646, stop := 399664 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n this option can be used to specify the GraphViz engine to use\n\ to render a ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "dot", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := \ 399785, root := ~, start := 399775, stop := 399784 ), rec( content := " docume\ nt. The valid choices are ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "\"dot\"", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next\ := 399830, root := ~, start := 399818, stop := 399829 ), rec( content := ",\n\ ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"neato\"",\ count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := 399856, root :=\ ~, start := 399842, stop := 399855 ), rec( content := ", ", count := [ 9, 1, \ 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "\"circo\"", count := [ 9, 1, 1, 19 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 19 ], math\ mode := "Text", name := "C", next := 399872, root := ~, start := 399858, stop \ := 399871 ), rec( content := ", ", count := [ 9, 1, 1, 19 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "\"twopi\"", count := [ 9, 1, 1, 19 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name \ := "C", next := 399888, root := ~, start := 399874, stop := 399887 ), rec( cont\ ent := ", ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"fdp\"", c\ ount := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := 399902, root :=\ ~, start := 399890, stop := 399901 ), rec( content := ",\n ", count \ := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "\"sfdp\"", count := [ 9, 1, 1\ , 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, \ 1, 19 ], mathmode := "Text", name := "C", next := 399927, root := ~, start := \ 399914, stop := 399926 ), rec( content := ", and ", count := [ 9, 1, 1, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "\"patchwork\"", count := [ 9, 1, 1, 19 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode \ := "Text", name := "C", next := 399951, root := ~, start := 399933, stop := 39\ 9950 ), rec( content := ". Please refer to the\n GraphViz documentati\ on for details on these engines.\n The default value for this option \ is ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "\"dot\"", count\ := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := 400095, root :=\ ~, start := 400083, stop := 400094 ), rec( content := ", and it\n mu\ st be specified in ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "opt.engine", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "A", next\ := 400152, root := ~, start := 400135, stop := 400151 ), rec( content := ".\n\ ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 19 ], name := "Item", next := 400169, root := ~, start := 399\ 674, stop := 400168 ), rec( attributes := rec( ), content := [ rec( content :\ = "filetype", count := [ 9, 1, 1, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 20 ], name := "Mark", next := 400200, root := ~, start := 400\ 179, stop := 400199 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n this should be a string representing the type of file which\n \ ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Splash", co\ unt := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 21 ], mathmode := "Text", name := "C", next := 400309, root :=\ ~, start := 400296, stop := 400308 ), rec( content := " should produce. For " , coun\ t := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( Only := "LaTeX" ), content := [ rec( content := "{\\LaTeX}", co\ unt := [ 9, 1, 1, 21 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 2\ 1 ], mathmode := "Text", name := "Alt", next := 33, root := ~, start := 1, stop\ := 32 ), rec( attributes := rec( Not := "LaTeX" ), content := [ rec( content \ := "LaTeX", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 9, 1, 1, 21 ], name := "Alt", next := 61, root := ~, star\ t := 33, stop := 60 ), rec( content := " files, this option is\n igno\ red and the default value ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "\"pdf\"", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "C", next\ := 400412, root := ~, start := 400400, stop := 400411 ), rec( content := " is\ used.\n ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 1, 21 ], name := "Item", next := 400437, root := ~, start := 400\ 209, stop := 400436 ) ], count := [ 9, 1, 1, 7 ], mathmode := "Text", name := "List", next := 400451, root := ~, start := 397996, stop := 400450 ), rec( content := "\n\n \ This function was written by Attila Egri-Nagy and Manuel Delgado with some\n\ minor changes by J. D. Mitchell.", count := [ 9, 1, 1, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 24 ], mathmode := "Text", name := "P", next := 400576, root := ~, start := 400572, stop := 400575 ), rec( content := "\n\n \ ", count := [ 9, 1, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> Splash(DotDigraph(RandomDigraph(4)));\n", count := [ 9, 1, 1, 25 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 25 ], mathmode := "Text", name := "Log", next := 400651, root := ~, start := 400584, stop := 400650 ), rec( content := "\n ", count := [ 9, 1, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 400670, root := ~, start := 397202, stop := 400669 ) ], count := [ 9, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 400686, root := ~, start := 397109, stop := 400685 ), rec( content := "\n\n ", count := [ 9, 1, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotDigraph" ), content := 0, count := [ 9, 1, 2, 2 ], mathmode := "Text", name := "Attr", next := 400750, root := ~, start := 400711, stop := 400749 ), rec( attributes := rec( Arg := "digraph", Name := "DotVertexLabell\ edDigraph" ), content := 0, count := [ 9, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 400808, root := ~, start := 400755, stop := 400807 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 400841, root := ~, start := 400813, stop := 400840 ), rec( attributes := rec( ), content := [ rec( content := "\n " , count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "C", next := 400883, root := ~, start := 400866, stop := 400882 ), rec( content := " produce\ s a graphical representation of the digraph\n ", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "A", next := 400955, root := ~, start := 400941, stop := 400954 ), rec( content := ". Vertic\ es are displayed as circles, numbered consistently\n with ", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "A", next := 401039, root := ~, start := 401025, stop := 401038 ), rec( content := ". Edges \ are displayed as arrowed lines between\n vertices, with the arrowhead of \ each line pointing towards the range\n of the edge.", count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "P", next := 401183, root := ~, start := 401179, stop := 401182 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexLabelledDigraph", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "C", next := 401222, root := ~, start := 401191, stop := 401221 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "C", next := 401253, root := ~, start := 401236, stop := 401252 ), rec( content := " only in\ \n that the values in ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 401320, root := ~, start := 401287, stop := 401319 ), rec( content := " are use\ d to label\n the vertices in the produced picture rather than the numbers\ ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "C", next := 401414, root := ~, start := 401406, stop := 401413 ), rec( content := " to\n \ the number of vertices of the digraph. ", count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "P", next := 401467, root := ~, start := 401463, stop := 401466 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "C", next := 401502, root := ~, start := 401492, stop := 401501 ), rec( content := " format \ (also known as ", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "C", next := 401540, root := ~, start := 401525, stop := 401539 ), rec( content := ")\n \ format. For details about this file format, and information about how to\n \ display or edit this format see ", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "URL", next := 401693, root := ~, start := 401659, stop := 401692 ), rec( content := ". ", count := [ 9, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "P", next := 401699, root := ~, start := 401695, stop := 401698 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "C", next := 401747, root := ~, start := 401730, stop := 401746 ), rec( content := " or\n \ ", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexLabelledDigraph", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "C", next := 401788, root := ~, start := 401757, stop := 401787 ), rec( content := " can be \ written to a file using\n the command ", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "Ref", next := 401880, root := ~, start := 401838, stop := 401879 ), rec( content := ".", count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 9 ], mathmode := "Text", name := "P", next := 401885, root := ~, start := 401881, stop := 401884 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> adj := List([1 .. 4], x -> [1, 1, 1, 1]);\n[ [ 1, 1, 1, 1 ], [ 1, 1, 1, \ 1 ], [ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ] ]\ngap> adj[1][3] := 0;\n0\ngap> gr := Dig\ raphByAdjacencyMatrix(adj);\n\nga\ p> FileString(\"dot/k4.dot\", DotDigraph(gr));\n154", count := [ 9, 1, 2, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 10 ], mathmode := "Text", name := "Log", next := 402192, root := ~, start := 401893, stop := 402191 ), rec( content := "\n ", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 402211, root := ~, start := 400846, stop := 402210 ) ], count := [ 9, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 402227, root := ~, start := 400694, stop := 402226 ), rec( content := "\n\n ", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotSymmetricDig\ raph" ), content := 0, count := [ 9, 1, 3, 2 ], mathmode := "Text", name := "Attr", next := 402300, root := ~, start := 402252, stop := 402299 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 402333, root := ~, start := 402305, stop := 402332 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of the symmetric\n digra\ ph ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "A", next := 402453, root := ~, start := 402439, stop := 402452 ), rec( content := ". ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "C", next := 402481, root := ~, start := 402455, stop := 402480 ), rec( content := " will re\ turn an\n error if ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "A", next := 402526, root := ~, start := 402512, stop := 402525 ), rec( content := " is not \ a symmetric digraph. See\n ", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 402597, root := ~, start := 402565, stop := 402596 ), rec( content := ".", count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "P", next := 402602, root := ~, start := 402598, stop := 402601 ), rec( content := "\n\n \ Vertices are displayed as circles, numbered consistently with\n ", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "A", next := 402692, root := ~, start := 402678, stop := 402691 ), rec( content := ". Since \ ", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "A", next := 402714, root := ~, start := 402700, stop := 402713 ), rec( content := " is symm\ etric, for every non-loop\n edge there is a complementary edge with oppos\ ite source and range.\n ", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "C", next := 402853, root := ~, start := 402827, stop := 402852 ), rec( content := " display\ s each pair of complementary edges\n as a single line between the relevan\ t vertices, with no arrowhead.", count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "P", next := 402972, root := ~, start := 402968, stop := 402971 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "C", next := 403007, root := ~, start := 402997, stop := 403006 ), rec( content := " format \ (also known as ", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "C", next := 403045, root := ~, start := 403030, stop := 403044 ), rec( content := ")\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n ", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "URL", next := 403204, root := ~, start := 403170, stop := 403203 ), rec( content := ". ", count := [ 9, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "P", next := 403210, root := ~, start := 403206, stop := 403209 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "C", next := 403267, root := ~, start := 403241, stop := 403266 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "Ref", next := 403359, root := ~, start := 403317, stop := 403358 ), rec( content := ".", count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 7 ], mathmode := "Text", name := "P", next := 403364, root := ~, start := 403360, stop := 403363 ), rec( content := "\n\n \ ", count := [ 9, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> star := Digraph([[2, 2, 3, 4], [1, 1], [1], [1, 4]]);\n\ngap> IsSymmetricDigraph(star);\ntrue\ngap> F\ ileString(\"dot/star.dot\", DotSymmetricDigraph(gr));\n83", count := [ 9, 1, 3, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 8 ], mathmode := "Text", name := "Log", next := 403601, root := ~, start := 403372, stop := 403600 ), rec( content := "\n ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 403620, root := ~, start := 402338, stop := 403619 ) ], count := [ 9, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 403636, root := ~, start := 402235, stop := 403635 ), rec( content := "\n\n ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotPartialOrder\ Digraph" ), content := 0, count := [ 9, 1, 4, 2 ], mathmode := "Text", name := "Attr", next := 403712, root := ~, start := 403661, stop := 403711 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 403745, root := ~, start := 403717, stop := 403744 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of a partial order\n dig\ raph ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "A", next := 403867, root := ~, start := 403853, stop := 403866 ), rec( content := ". ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPartialOrderDigraph", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "C", next := 403898, root := ~, start := 403869, stop := 403897 ), rec( content := " will re\ turn an error\n if ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "A", next := 403943, root := ~, start := 403929, stop := 403942 ), rec( content := " is not \ a partial order digraph. See ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Ref", next := 404023, root := ~, start := 403980, stop := 404022 ), rec( content := ".", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "P", next := 404028, root := ~, start := 404024, stop := 404027 ), rec( content := "\n\n \ Since ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "A", next := 404056, root := ~, start := 404042, stop := 404055 ), rec( content := " is a pa\ rtial order, it is both reflexive and\n transitive. The output of ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPartialOrderDigraph", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "C", next := 404163, root := ~, start := 404134, stop := 404162 ), rec( content := " is the\\ n ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DotDigr\ aph" ), content := 0, count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 404201, root := ~, start := 404177, stop := 404200 ), rec( content := " of the\\ n ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveReduction" ), content := 0, count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 404264, root := ~, start := 404215, stop := 404263 ), rec( content := " of ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "A", next := 404282, root := ~, start := 404268, stop := 404281 ), rec( content := ".", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "P", next := 404287, root := ~, start := 404283, stop := 404286 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "C", next := 404322, root := ~, start := 404312, stop := 404321 ), rec( content := " format \ (also known as ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "C", next := 404360, root := ~, start := 404345, stop := 404359 ), rec( content := ")\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "URL", next := 404519, root := ~, start := 404485, stop := 404518 ), rec( content := ". ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "P", next := 404525, root := ~, start := 404521, stop := 404524 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPartialOrderDigraph", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "C", next := 404585, root := ~, start := 404556, stop := 404584 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "Ref", next := 404677, root := ~, start := 404635, stop := 404676 ), rec( content := ".", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 7 ], mathmode := "Text", name := "P", next := 404682, root := ~, start := 404678, stop := 404681 ), rec( content := "\n\n \ ", count := [ 9, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> poset := Digraph([[1, 4], [2], [2, 3, 4], [4]);\ngap> IsPartialOrderDigr\ aph(gr);\ntrue\ngap> FileString(\"dot/poset.dot\", DotPartialOrderDigraph(gr))\ ;\n83", count := [ 9, 1, 4, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 8 ], mathmode := "Text", name := "Log", next := 404868, root := ~, start := 404690, stop := 404867 ), rec( content := "\n ", count := [ 9, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 404887, root := ~, start := 403750, stop := 404886 ) ], count := [ 9, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 404903, root := ~, start := 403644, stop := 404902 ), rec( content := "\n\n ", count := [ 9, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotPreorderDigr\ aph" ), content := 0, count := [ 9, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 404975, root := ~, start := 404928, stop := 404974 ), rec( attributes := rec( Arg := "digraph", Name := "DotQuasiorderDi\ graph" ), content := 0, count := [ 9, 1, 5, 4 ], mathmode := "Text", name := "Attr", next := 405029, root := ~, start := 404980, stop := 405028 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Returns", next := 405062, root := ~, start := 405034, stop := 405061 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of a preorder\n digraph \ ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "A", next := 405179, root := ~, start := 405165, stop := 405178 ), rec( content := ". ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPreorderDigraph", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "C", next := 405206, root := ~, start := 405181, stop := 405205 ), rec( content := " will re\ turn an error\n if ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "A", next := 405251, root := ~, start := 405237, stop := 405250 ), rec( content := " is not \ a preorder digraph. See ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPreor\ derDigraph" ), content := 0, count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Ref", next := 405322, root := ~, start := 405283, stop := 405321 ), rec( content := ".", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "P", next := 405327, root := ~, start := 405323, stop := 405326 ), rec( content := "\n\n \ A preorder digraph is reflexive and transitive but in general it is\n n\ ot anti-symmetric and may have strongly connected components\n containing\ more than one vertex. The ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Quotien\ tDigraph" ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "Ref", next := 405543, root := ~, start := 405514, stop := 405542 ), rec( content := "\n \ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 405558, root := ~, start := 405550, stop := 405557 ), rec( content := " obtaine\ d by forming the quotient of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 405609, root := ~, start := 405595, stop := 405608 ), rec( content := " by the\\ n partition of its vertices into the strongly connected components\n \ satisfies ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "Ref", next := 405739, root := ~, start := 405704, stop := 405738 ), rec( content := ". Thus e\ very vertex of\n ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 405776, root := ~, start := 405768, stop := 405775 ), rec( content := " corresp\ onds to a strongly connected component of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 405840, root := ~, start := 405826, stop := 405839 ), rec( content := ".\n \ The output of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPreorderDigraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "C", next := 405887, root := ~, start := 405862, stop := 405886 ), rec( content := " display\ s the \n ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveReduction" ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "Ref", next := 405957, root := ~, start := 405908, stop := 405956 ), rec( content := " of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 405969, root := ~, start := 405961, stop := 405968 ), rec( content := " with\n \ vertices displayed as rounded rectangles labelled by all of the vertices\ \n of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 406077, root := ~, start := 406063, stop := 406076 ), rec( content := " in the \ corresponding strongly connected component. ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "P", next := 406133, root := ~, start := 406129, stop := 406132 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "C", next := 406168, root := ~, start := 406158, stop := 406167 ), rec( content := " format \ (also known as ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "C", next := 406206, root := ~, start := 406191, stop := 406205 ), rec( content := ")\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "URL", next := 406365, root := ~, start := 406331, stop := 406364 ), rec( content := ". ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "P", next := 406371, root := ~, start := 406367, stop := 406370 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPreorderDigraph", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "C", next := 406427, root := ~, start := 406402, stop := 406426 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "Ref", next := 406519, root := ~, start := 406477, stop := 406518 ), rec( content := ".", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 9 ], mathmode := "Text", name := "P", next := 406524, root := ~, start := 406520, stop := 406523 ), rec( content := "\n\n \ ", count := [ 9, 1, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> preset := Digraph([[1, 2, 4, 5], [1, 2, 4, 5], [3, 4], [4], [1, 2, 4, 5]\ );\ngap> IsPreorderDigraph(gr);\ntrue\ngap> FileString(\"dot/preset.dot\", Dot\ ProrderDigraph(gr));\n83", count := [ 9, 1, 5, 10 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 10 ], mathmode := "Text", name := "Log", next := 406729, root := ~, start := 406532, stop := 406728 ), rec( content := "\n ", count := [ 9, 1, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Description", next := 406748, root := ~, start := 405067, stop := 406747 ) ], count := [ 9, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 406764, root := ~, start := 404911, stop := 406763 ), rec( content := "\n\n ", count := [ 9, 1, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, verts [\ , colour1, colour2]", Name := "DotHighlightedDigraph" ), content := 0, count := [ 9, 1, 6, 2 ], mathmode := "Text", name := "Oper", next := 406867, root := ~, start := 406789, stop := 406866 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 406900, root := ~, start := 406872, stop := 406899 ), rec( attributes := rec( ), content := [ rec( content := "\n " , count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otHighlightedDigraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "C", next := 406953, root := ~, start := 406925, stop := 406952 ), rec( content := " produce\ s a graphical represenation of the\n digraph ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 407024, root := ~, start := 407010, stop := 407023 ), rec( content := ", where \ the vertices in the list ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 407069, root := ~, start := 407057, stop := 407068 ), rec( content := ", and\n \ edges between them, are drawn with colour ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour1", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 407137, root := ~, start := 407123, stop := 407136 ), rec( content := " and all\ other\n vertices and edges in ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 407194, root := ~, start := 407180, stop := 407193 ), rec( content := " are dra\ wn with colour ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour2", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 407231, root := ~, start := 407217, stop := 407230 ), rec( content := ".\n \ If ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour1", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 407256, root := ~, start := 407242, stop := 407255 ), rec( content := " and ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour2", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 407275, root := ~, start := 407261, stop := 407274 ), rec( content := " are not\ given then\n ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otHighlightedDigraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "C", next := 407329, root := ~, start := 407301, stop := 407328 ), rec( content := " uses bl\ ack and grey respectively. ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "P", next := 407368, root := ~, start := 407364, stop := 407367 ), rec( content := "\n\n \ Note that ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otHighlightedDigraph", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "C", next := 407414, root := ~, start := 407386, stop := 407413 ), rec( content := " does no\ t validate the colours\n ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour1", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "A", next := 407465, root := ~, start := 407451, stop := 407464 ), rec( content := " and ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour2", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "A", next := 407484, root := ~, start := 407470, stop := 407483 ), rec( content := " - consu\ lt the GraphViz documentation to\n see what is available.\n\n See ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DotDigr\ aph" ), content := 0, count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "Ref", next := 407589, root := ~, start := 407565, stop := 407588 ), rec( content := " for mor\ e details on the output.", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 6, 5 ], mathmode := "Text", name := "P", next := 407625, root := ~, start := 407621, stop := 407624 ), rec( content := "\n\n \ ", count := [ 9, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := Digraph([[2, 3], [2], [1, 3]]);\n\ngap> FileString(\"dot/my_digraph.dot\",\n> DotHighlightedDigraph(digra\ ph, [1, 2], \"red\", \"black\"));\n264", count := [ 9, 1, 6, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 6 ], mathmode := "Text", name := "Log", next := 407840, root := ~, start := 407633, stop := 407839 ), rec( content := "\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 407859, root := ~, start := 406905, stop := 407858 ) ], count := [ 9, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 407875, root := ~, start := 406772, stop := 407874 ), rec( content := "\n\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 407889, root := ~, start := 397053, stop := 407888 ) , rec( content := "\n\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Reading and writi\ ng digraphs to a file", count := [ 9, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 407959, root := ~, start := 407902, stop := 407958 ), rec( content := "\n This section descr\ ibes different ways to store and read graphs\n from a file in the ", count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package.\n ", count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := " Graph6 " , count := [ 9, 2, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 4 ], name := "Mark", next := 408108, root := ~, start := 408087, stop := 408107 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Graph6 ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "E", next := 408138, root := ~, start := 408123, stop := 408137 ), rec( content := " is a gr\ aph data format for storing undirected graphs\n\twith no multiple edges nor lo\ ops of size up to ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ 2^{36} - 1 ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "M", next := 408259, root := ~, start := 408240, stop := 408258 ), rec( content := " in\n\tp\ rintable chracters. The format consists of two parts. The first part\n\tuses o\ ne to eight bytes to store the number of vertices. And the second\n\tpart is t\ he upper half of the adjacency matrix converted into ASCII\n\tcharacters. For \ a more detail description see ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "Graph6"\ ), content := [ rec( content := "\n\thttp://cs.anu.edu.au/~bdm/data/formats.t\ xt", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "URL", next := 408592, root := ~, start := 408523, stop := 408591 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], name := "Item", next := 408607, root := ~, start := 408115, stop := 408606 ), rec( attributes := rec( ), content := [ rec( content := " Sparse6 \ ", count := [ 9, 2, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 6 ], name := "Mark", next := 408637, root := ~, start := 408615, stop := 408636 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "E", next := 408668, root := ~, start := 408652, stop := 408667 ), rec( content := " is a gr\ aph data format for storing undirected graphs\n\twith possibly multiple edges \ or loops. The maximal number of vertices\n\tallowed is ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ 2^{36} - 1 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "M", next := 408824, root := ~, start := 408805, stop := 408823 ), rec( content := ". The fo\ rmat consists of two parts. The\n\tfirst part uses one to eight bytes to store\ the number of vertices.\n\tAnd the second part only stores information about \ the edges.\n\tTherefore, the ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "E", next := 409027, root := ~, start := 409011, stop := 409026 ), rec( content := " format \ return a more compact encoding\n\tthen ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Graph6 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "E", next := 409087, root := ~, start := 409072, stop := 409086 ), rec( content := " for spa\ rse graph, i.e. graphs where the number\n\tof edges is much less than the num\ ber of vertices squared. For a more\n\tdetail description see ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "Sparse6\ " ), content := [ rec( content := "\n http://cs.anu.edu.au/~bdm/data/fo\ rmats.txt", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "URL", next := 409309, root := ~, start := 409232, stop := 409308 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], name := "Item", next := 409324, root := ~, start := 408644, stop := 409323 ), rec( attributes := rec( ), content := [ rec( content := " Digraph6\ ", count := [ 9, 2, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 8 ], name := "Mark", next := 409355, root := ~, start := 409332, stop := 409354 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraph6", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "E", next := 409385, root := ~, start := 409370, stop := 409384 ), rec( content := " is a ne\ w format based on ", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Graph6 ", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "E", next := 409426, root := ~, start := 409411, stop := 409425 ), rec( content := ", but de\ signed\n\tfor digraphs. The entire adjacency matrix is stored, and\n\ttherefor\ e there is support for directed edges and single-vertex loops.\n\tHowever, mul\ tiple edges are not supported.\n ", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 9 ], name := "Item", next := 409628, root := ~, start := 409362, stop := 409627 ), rec( attributes := rec( ), content := [ rec( content := " DiSparse\ 6 ", count := [ 9, 2, 0, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 10 ], name := "Mark", next := 409660, root := ~, start := 409636, stop := 409659 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ iSparse6", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "E", next := 409691, root := ~, start := 409675, stop := 409690 ), rec( content := " is a ne\ w format based on ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "E", next := 409733, root := ~, start := 409717, stop := 409732 ), rec( content := ", but\n\\ tdesigned for digraphs. In this format the list of edges is\n\tpartitioned int\ o inceasing and decreasing edges, depending whether the\n\tedge has its source\ bigger than the range. Then both sets of edges are\n\twritten separetly in ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "E", next := 409981, root := ~, start := 409965, stop := 409980 ), rec( content := " format \ with a separation symbol\n\tin between.\n ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], name := "Item", next := 410040, root := ~, start := 409667, stop := 410039 ) ], count := [ 9, 2, 0, 4 ], mathmode := "Text", name := "List", next := 410052, root := ~, start := 408074, stop := 410051 ), rec( content := "\n ", count := [ 9, 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "String" ), content := 0, count := [ 9, 2, 1, 2 ], mathmode := "Text", name := "Attr", next := 410107, root := ~, start := 410072, stop := 410106 ), rec( attributes := rec( Arg := "digraph", Name := "PrintString" ), content := 0, count := [ 9, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 410150, root := ~, start := 410110, stop := 410149 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 410181, root := ~, start := 410153, stop := 410180 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns a string ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tring", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "C", next := 410232, root := ~, start := 410219, stop := 410231 ), rec( content := " such th\ at ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ valString(string)", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "C", next := 410268, root := ~, start := 410243, stop := 410267 ), rec( content := "\n is\ equal to ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "A", next := 410299, root := ~, start := 410285, stop := 410298 ), rec( content := ", and ha\ s the same mutability.\n See ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "EvalString" ), content := 0, count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Ref", next := 410378, root := ~, start := 410338, stop := 410377 ), rec( content := ".\n " , count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "P", next := 410388, root := ~, start := 410384, stop := 410387 ), rec( content := "\n\n \ The methods installed for ", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ tring", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "C", next := 410433, root := ~, start := 410420, stop := 410432 ), rec( content := " make so\ me attempts to\n ensure that ", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tring", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "C", next := 410485, root := ~, start := 410472, stop := 410484 ), rec( content := " has as \ short a length as possible, but\n there may exist shorter strings that also\ evaluate to ", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "A", next := 410597, root := ~, start := 410583, stop := 410596 ), rec( content := ".\n " , count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "P", next := 410607, root := ~, start := 410603, stop := 410606 ), rec( content := "\n\n \ It is possible that ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tring", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 410646, root := ~, start := 410633, stop := 410645 ), rec( content := " may con\ tain escaped special\n characters. To obtain a representation of ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "A", next := 410735, root := ~, start := 410721, stop := 410734 ), rec( content := " that\n \ can be entered as GAP input, please use\n ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "Print" ), content := 0, count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "Ref", next := 410824, root := ~, start := 410789, stop := 410823 ), rec( content := ".\n N\ ote that ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ rint", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 410852, root := ~, start := 410840, stop := 410851 ), rec( content := " for a d\ igraph delegates to\n ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ rintString", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 410903, root := ~, start := 410885, stop := 410902 ), rec( content := ", which \ delegates to ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ tring", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 410937, root := ~, start := 410924, stop := 410936 ), rec( content := ".\n " , count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(3);\n\ngap> P\ rint(D);\nCycleDigraph(3);\ngap> G := PetersenGraph(IsMutableDigraph);\n\ngap> String(G);\n\"DigraphFromGraph6St\ ring(IsMutableDigraph, \\\"IheA@GUAo\\\");\"\ngap> Print(last);\nDigraphFromGr\ aph6String(IsMutableDigraph, \"IheA@GUAo\");\ngap> DigraphFromGraph6String(IsM\ utableDigraph, \"IheA@GUAo\");\n\n\ ", count := [ 9, 2, 1, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 8 ], mathmode := "Text", name := "Log", next := 411412, root := ~, start := 410943, stop := 411411 ), rec( content := "\n ", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 411429, root := ~, start := 410184, stop := 411428 ) ], count := [ 9, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 411443, root := ~, start := 410057, stop := 411442 ), rec( content := "\n\n ", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromGrap\ h6String" ), content := 0, count := [ 9, 2, 2, 2 ], mathmode := "Text", name := "Oper", next := 411523, root := ~, start := 411464, stop := 411522 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDigr\ aph6String" ), content := 0, count := [ 9, 2, 2, 4 ], mathmode := "Text", name := "Oper", next := 411585, root := ~, start := 411526, stop := 411584 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromSpar\ se6String" ), content := 0, count := [ 9, 2, 2, 6 ], mathmode := "Text", name := "Oper", next := 411647, root := ~, start := 411588, stop := 411646 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDiSp\ arse6String" ), content := 0, count := [ 9, 2, 2, 8 ], mathmode := "Text", name := "Oper", next := 411709, root := ~, start := 411650, stop := 411708 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "Returns", next := 411741, root := ~, start := 411712, stop := 411740 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "A", next := 411775, root := ~, start := 411765, stop := 411774 ), rec( content := " is a st\ ring encoding a graph in Graph6, Digraph6, Sparse6 or\n DiSparse6 format, t\ hen the corresponding function returns a digraph. In the\n case of either G\ raph6 or Sparse6, formats which do not support directed\n edges, this will \ be a digraph such that for every edge, the edge going in\n the opposite dir\ ection is also present.", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "P", next := 412118, root := ~, start := 412114, stop := 412117 ), rec( content := "\n\n \ Each of these functions takes an optional first argument ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "A", next := 412192, root := ~, start := 412181, stop := 412191 ), rec( content := ",\n w\ hich should be either ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "Ref", next := 412251, root := ~, start := 412221, stop := 412250 ), rec( content := "\n or\ ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "Ref", next := 412291, root := ~, start := 412259, stop := 412290 ), rec( content := ",\n a\ nd which specifies whether the output digraph shall\n be mutable or immutab\ le.\n If no first argument is provided, then an immutable\n digraph is r\ eturned by default.\n ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphFromGraph6String(\"?\");\n\ngap> DigraphFromGraph6String(\"C]\");\n\ngap> DigraphFromGraph6String(\"H?AAEM{\");\n\ngap> DigraphFromDigraph6String(\"&?\");\n\ngap> DigraphFromDigraph6String(IsMutableDigraph,\ \"&DOOOW?\");\n\ngap> DigraphFromDi\ graph6String(\"&CQFG\");\n\ngap> D\ igraphFromDigraph6String(\"&IM[SrKLc~lhesbU[F_\");\n\ngap> DigraphFromDiSparse6String(\".CaWBGA?b\");\n\n", count := [ 9, 2, 2, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 2, 11 ], mathmode := "Text", name := "Example", next := 413220, root := ~, start := 412475, stop := 413219 ), rec( content := "\n ", count := [ 9, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "Description", next := 413239, root := ~, start := 411744, stop := 413238 ) ], count := [ 9, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 413253, root := ~, start := 411449, stop := 413252 ), rec( content := "\n\n ", count := [ 9, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "Graph6String" ) , content := 0, count := [ 9, 2, 3, 2 ], mathmode := "Text", name := "Oper", next := 413315, root := ~, start := 413274, stop := 413314 ), rec( attributes := rec( Arg := "digraph", Name := "Digraph6String"\ ), content := 0, count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 413361, root := ~, start := 413318, stop := 413360 ), rec( attributes := rec( Arg := "digraph", Name := "Sparse6String" ), content := 0, count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "Oper", next := 413406, root := ~, start := 413364, stop := 413405 ), rec( attributes := rec( Arg := "digraph", Name := "DiSparse6String\ " ), content := 0, count := [ 9, 2, 3, 8 ], mathmode := "Text", name := "Oper", next := 413453, root := ~, start := 413409, stop := 413452 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "Returns", next := 413484, root := ~, start := 413456, stop := 413483 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se four functions return a highly compressed string fully describing the\n \ digraph ", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "A", next := 413607, root := ~, start := 413593, stop := 413606 ), rec( content := ". ", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "P", next := 413613, root := ~, start := 413609, stop := 413612 ), rec( content := "\n\n \ Graph6 and Digraph6 are formats best used on small, dense graphs, if\n appl\ icable. For larger, sparse graphs use ", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ parse6", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "E", next := 413748, root := ~, start := 413734, stop := 413747 ), rec( content := " and\n \ ", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ isparse6", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "E", next := 413773, root := ~, start := 413757, stop := 413772 ), rec( content := " (this l\ atter also preserves multiple edges). ", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "P", next := 413823, root := ~, start := 413819, stop := 413822 ), rec( content := "\n\n \ See ", count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "Ref", next := 413860, root := ~, start := 413833, stop := 413859 ), rec( content := ".\n " , count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2, 3], [1], [1]]);\n\ngap> Sparse6String(gr);\n\":Bc\"\ngap> DiSparse6String(gr);\n\".Bc{f\ \"\n", count := [ 9, 2, 3, 12 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 12 ], mathmode := "Text", name := "Example", next := 414047, root := ~, start := 413866, stop := 414046 ), rec( content := "\n ", count := [ 9, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "Description", next := 414064, root := ~, start := 413487, stop := 414063 ) ], count := [ 9, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 414078, root := ~, start := 413259, stop := 414077 ), rec( content := "\n\n ", count := [ 9, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename [, code\ r][, mode]", Name := "DigraphFile" ), content := 0, count := [ 9, 2, 4, 2 ], mathmode := "Text", name := "Func", next := 414158, root := ~, start := 414099, stop := 414157 ), rec( attributes := rec( ), content := [ rec( content := "An IO fil\ e object.", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 414198, root := ~, start := 414161, stop := 414197 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "A", next := 414237, root := ~, start := 414222, stop := 414236 ), rec( content := " is a st\ ring representing the name of a file, then\n ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFile", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "C", next := 414310, root := ~, start := 414292, stop := 414309 ), rec( content := " returns\ an ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 4, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 4, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " package\ file object for that file.\n ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "P", next := 414370, root := ~, start := 414366, stop := 414369 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oder", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "A", next := 414413, root := ~, start := 414401, stop := 414412 ), rec( content := " is spec\ ified\n and is a function which either encodes a digraph as a string, or de\ codes a\n string into a digraph, then this function will be used when readi\ ng or\n writing to the returned file object. If the optional argument ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oder", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "A", next := 414659, root := ~, start := 414647, stop := 414658 ), rec( content := "\n is\ not specified, then the encoding of the digraphs in the returned file\n ob\ ject must be specified in the the file extension. The file extension must\n \ be one of: ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ g6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 414842, root := ~, start := 414832, stop := 414841 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ s6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 414854, root := ~, start := 414844, stop := 414853 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ d6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 414866, root := ~, start := 414856, stop := 414865 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ds6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 414879, root := ~, start := 414868, stop := 414878 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ txt", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 414892, root := ~, start := 414881, stop := 414891 ), rec( content := ",\n " , count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ p", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 414907, root := ~, start := 414898, stop := 414906 ), rec( content := ", or ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ pickle", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 414926, root := ~, start := 414912, stop := 414925 ), rec( content := "; more d\ etails of these file formats is given\n below. ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "P", next := 414987, root := ~, start := 414983, stop := 414986 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ode", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "A", next := 415029, root := ~, start := 415018, stop := 415028 ), rec( content := " is spec\ ified, then it must be one of:\n ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "w\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 415082, root := ~, start := 415072, stop := 415081 ), rec( content := " (for wr\ ite), ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "a\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 415106, root := ~, start := 415096, stop := 415105 ), rec( content := " (for ap\ pend), or ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "r\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 415134, root := ~, start := 415124, stop := 415133 ), rec( content := " (for re\ ad).\n If ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ode", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "A", next := 415165, root := ~, start := 415154, stop := 415164 ), rec( content := " is not \ specified, then ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "r\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 415199, root := ~, start := 415189, stop := 415198 ), rec( content := " is used\ by default. ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "P", next := 415224, root := ~, start := 415220, stop := 415223 ), rec( content := "\n\n \ If ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "A", next := 415248, root := ~, start := 415233, stop := 415247 ), rec( content := " ends in\ one of: ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 415275, root := ~, start := 415265, stop := 415274 ), rec( content := ", ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 415288, root := ~, start := 415277, stop := 415287 ), rec( content := ", or\n \ ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 415307, root := ~, start := 415297, stop := 415306 ), rec( content := ", then t\ he digraphs which are read from, or written to, the\n returned file object \ are decompressed, or compressed, appropriately.\n ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "P", next := 415448, root := ~, start := 415444, stop := 415447 ), rec( content := "\n\n \ The file object returned by ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFile", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "C", next := 415500, root := ~, start := 415482, stop := 415499 ), rec( content := " can be \ given as the first\n argument for either of the functions ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "ReadDig\ raphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 415594, root := ~, start := 415568, stop := 415593 ), rec( content := " or ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 415631, root := ~, start := 415598, stop := 415630 ), rec( content := ". The pu\ rpose of this is to reduce the overhead of\n recreating the file object ins\ ide the functions ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "ReadDig\ raphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 415760, root := ~, start := 415734, stop := 415759 ), rec( content := "\n or\ ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 415795, root := ~, start := 415768, stop := 415794 ), rec( content := " when, f\ or example, reading or writing many\n digraphs in a loop.\n ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 8 ], mathmode := "Text", name := "P", next := 415871, root := ~, start := 415867, stop := 415870 ), rec( content := "\n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n ", count := [ 9, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 4, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 9 ], name := "Mar\ k", next := 416002, root := ~, start := 415977, stop := 416001 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n ", count := [ 9, 2, \ 4, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 10 ], name := "Ite\ m", next := 416338, root := ~, start := 416009, stop := 416337 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 4, 1\ 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 11 ], name := "Mar\ k", next := 416371, root := ~, start := 416345, stop := 416370 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tUnlike graph6, sparse6 has suppor\ t for loops and multiple edges.\n\tHowever, its use is still limited to symmet\ ric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those wit\ h few edges per vertex.\n ", count := [ 9, 2, 4, 12 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 12 ], name := "Item", next := 4166\ 06, root := ~, start := 416378, stop := 416605 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 4, 13 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 13 ], name := "Mark", next := 4166\ 40, root := ~, start := 416613, stop := 416639 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n ", coun\ t := [ 9, 2, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 14 ], name := "Ite\ m", next := 416870, root := ~, start := 416647, stop := 416869 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 4\ , 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 15 ], name := "Mar\ k", next := 416906, root := ~, start := 416877, stop := 416905 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tAny type of digraph can be encode\ d in disparse6: directions, loops, and\n\tmultiple edges are all allowed. Sim\ ilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\ \n ", count := [ 9, 2, 4, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 4, 16 ], name := "Item", next := 417118, root := ~, start := 416\ 913, stop := 417117 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 4, 17 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 4, 17 ], name := "Mark", next := 4171\ 55, root := ~, start := 417125, stop := 417154 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 4, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "0 7 0 8 1\ 7 2 8 3 8 4 8 5 8 6 8", count := [ 9, 2, 4, 18 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 18 ], mathmode := "Text", name\ := "C", next := 417287, root := ~, start := 417242, stop := 417286 ), rec( cont\ ent := " i.e. pairs of vertices\n describing edges in a graph. More spe\ cifically, the vertices making up\n one edge must be separated by a sin\ gle space, and pairs of vertices must\n be separated by two spaces. ", coun\ t := [ 9, 2, 4, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 4, 19 ], mathmode := "Text", name\ := "P", next := 417511, root := ~, start := 417507, stop := 417510 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 4, 19 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 4, 19 ], mathmode := "Text", name := "Ref\ ", next := 417559, root := ~, start := 417525, stop := 417558 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 4, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 4, 20 ], mathmode := "Text", name\ := "P", next := 417636, root := ~, start := 417632, stop := 417635 ), rec( cont\ ent := "\n ", count := [ 9, 2, 4, 20 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 4, 18 ], name := "Item", next := 417650, root := ~, start := 417\ 162, stop := 417649 ), rec( attributes := rec( ), content := [ rec( content :\ = "pickled (", count := [ 9, 2, 4, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := ".p", count \ := [ 9, 2, 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 4, 21 ], mathmode := "Text", name := "F", next := 417682, root := \ ~, start := 417673, stop := 417681 ), rec( content := " or ", count := [ 9, 2,\ 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 4, 21 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 21 ], math\ mode := "Text", name := "F", next := 417700, root := ~, start := 417686, stop \ := 417699 ), rec( content := ")", count := [ 9, 2, 4, 21 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 21 ], name := "Mark", next := 4177\ 08, root := ~, start := 417658, stop := 417707 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n Digraphs are pickled using the ", count := [ 9, 2, 4, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := \ "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ re\ c( content := "\n http://gap-packages.github.io/io/\n", count := [ 9, 2, 4\ , 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, \ 4, 22 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17\ , stop := 77 ) ], count := [ 9, 2, 4, 22 ], name := "Alt", next := 84, root :=\ ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ), conten\ t := [ rec( attributes := rec( ), content := [ rec( content := "IO", count :=\ [ 9, 2, 4, 22 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 22 ], n\ ame := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count\ := [ 9, 2, 4, 22 ], mathmode := "Text", name := "Alt", next := 128, root := ~ , star\ t := 84, stop := 127 ), rec( content := " package. This is particularly good\n\ when the ", count := [ 9, 2, 4, 22 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGroup" ), content :\ = 0, count := [ 9, 2, 4, 22 ], mathmode := "Text", name := "Ref", next := 4178\ 44, root := ~, start := 417818, stop := 417843 ), rec( content := " is non-tri\ vial.\n ", count := [ 9, 2, 4, 22 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 9, 2, 4, 22 ], name := "Item", next := 417874, root := ~, start := 417\ 715, stop := 417873 ) ], count := [ 9, 2, 4, 9 ], mathmode := "Text", name := "List", next := 417886, root := ~, start := 415964, stop := 417885 ), rec( content := "\n\n \ ", count := [ 9, 2, 4, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap\ > file := DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> WriteD\ igraphs(file, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Close(file);;\n\ gap> file := DigraphFile(filename, \"r\");;\ngap> ReadDigraphs(file, 9);\n", count := [ 9, 2, 4, 25 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 4, 25 ], mathmode := "Text", name := "Example", next := 418261, root := ~, start := 417892, stop := 418260 ), rec( content := "\n ", count := [ 9, 2, 4, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 418280, root := ~, start := 414201, stop := 418279 ) ], count := [ 9, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 418294, root := ~, start := 414084, stop := 418293 ), rec( content := "\n\n ", count := [ 9, 2, 4, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename[, decod\ er][, n]", Name := "ReadDigraphs" ), content := 0, count := [ 9, 2, 5, 2 ], mathmode := "Text", name := "Func", next := 418373, root := ~, start := 418315, stop := 418372 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or a list of digraphs.", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 418428, root := ~, start := 418376, stop := 418427 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 418467, root := ~, start := 418452, stop := 418466 ), rec( content := " is a st\ ring containing the name of a file containing\n encoded digraphs or an ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 5, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 5, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " file ob\ ject created using ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ File" ), content := 0, count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Ref", next := 418610, root := ~, start := 418579, stop := 418609 ), rec( content := ", then " , count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 418636, root := ~, start := 418617, stop := 418635 ), rec( content := " returns\ the digraphs\n encoded in the file as a list. Note that if ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 418722, root := ~, start := 418707, stop := 418721 ), rec( content := " is a\n \ compressed file, which has been compressed appropriately to give a filename\ \n extension of ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 418835, root := ~, start := 418825, stop := 418834 ), rec( content := ", ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 418848, root := ~, start := 418837, stop := 418847 ), rec( content := ", or ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 418863, root := ~, start := 418853, stop := 418862 ), rec( content := ", then\n\ ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 418893, root := ~, start := 418874, stop := 418892 ), rec( content := " can rea\ d ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 418918, root := ~, start := 418903, stop := 418917 ), rec( content := " without\ it first needing to be\n decompressed.\n ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "P", next := 418976, root := ~, start := 418972, stop := 418975 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 419021, root := ~, start := 419007, stop := 419020 ), rec( content := " is spec\ ified\n and is a function which decodes a string into a digraph,\n then \ ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "C", next := 419124, root := ~, start := 419105, stop := 419123 ), rec( content := " will us\ e ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 419148, root := ~, start := 419134, stop := 419147 ), rec( content := " to deco\ de the digraphs\n contained in ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 419204, root := ~, start := 419189, stop := 419203 ), rec( content := ".", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "P", next := 419209, root := ~, start := 419205, stop := 419208 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "A", next := 419248, root := ~, start := 419240, stop := 419247 ), rec( content := " is spec\ ified, then ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "C", next := 419287, root := ~, start := 419268, stop := 419286 ), rec( content := "\n re\ turns the ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "A", next := 419312, root := ~, start := 419304, stop := 419311 ), rec( content := "th digra\ ph encoded in the file ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "A", next := 419358, root := ~, start := 419343, stop := 419357 ), rec( content := ".\n " , count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "P", next := 419368, root := ~, start := 419364, stop := 419367 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "A", next := 419413, root := ~, start := 419399, stop := 419412 ), rec( content := " is not \ specified, then\n ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 419460, root := ~, start := 419441, stop := 419459 ), rec( content := " will de\ duce which decoder to use based on the filename\n extension of ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "A", next := 419548, root := ~, start := 419533, stop := 419547 ), rec( content := " (after \ removing the compression-related\n filename extensions ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 419623, root := ~, start := 419613, stop := 419622 ), rec( content := ", ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 419636, root := ~, start := 419625, stop := 419635 ), rec( content := ", and ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 419652, root := ~, start := 419642, stop := 419651 ), rec( content := "). For \ example,\n if the filename extension is ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ g6", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 419712, root := ~, start := 419702, stop := 419711 ), rec( content := ", then " , count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 419738, root := ~, start := 419719, stop := 419737 ), rec( content := " will us\ e\n the graph6 decoder ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ FromGraph6String" ), content := 0, count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "Ref", next := 419808, root := ~, start := 419771, stop := 419807 ), rec( content := ".\n " , count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 7 ], mathmode := "Text", name := "P", next := 419818, root := ~, start := 419814, stop := 419817 ), rec( content := "\n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n ", count := [ 9, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 5, 8 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 8 ], name := "Mar\ k", next := 419949, root := ~, start := 419924, stop := 419948 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n ", count := [ 9, 2, \ 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 9 ], name := "Ite\ m", next := 420285, root := ~, start := 419956, stop := 420284 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 5, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 10 ], name := "Mar\ k", next := 420318, root := ~, start := 420292, stop := 420317 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tUnlike graph6, sparse6 has suppor\ t for loops and multiple edges.\n\tHowever, its use is still limited to symmet\ ric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those wit\ h few edges per vertex.\n ", count := [ 9, 2, 5, 11 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 11 ], name := "Item", next := 4205\ 53, root := ~, start := 420325, stop := 420552 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 5, 12 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 12 ], name := "Mark", next := 4205\ 87, root := ~, start := 420560, stop := 420586 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n ", coun\ t := [ 9, 2, 5, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 13 ], name := "Ite\ m", next := 420817, root := ~, start := 420594, stop := 420816 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 5\ , 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 14 ], name := "Mar\ k", next := 420853, root := ~, start := 420824, stop := 420852 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tAny type of digraph can be encode\ d in disparse6: directions, loops, and\n\tmultiple edges are all allowed. Sim\ ilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\ \n ", count := [ 9, 2, 5, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 5, 15 ], name := "Item", next := 421065, root := ~, start := 420\ 860, stop := 421064 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 5, 16 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 5, 16 ], name := "Mark", next := 4211\ 02, root := ~, start := 421072, stop := 421101 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 5, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "0 7 0 8 1\ 7 2 8 3 8 4 8 5 8 6 8", count := [ 9, 2, 5, 17 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 17 ], mathmode := "Text", name\ := "C", next := 421234, root := ~, start := 421189, stop := 421233 ), rec( cont\ ent := " i.e. pairs of vertices\n describing edges in a graph. More spe\ cifically, the vertices making up\n one edge must be separated by a sin\ gle space, and pairs of vertices must\n be separated by two spaces. ", coun\ t := [ 9, 2, 5, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 5, 18 ], mathmode := "Text", name\ := "P", next := 421458, root := ~, start := 421454, stop := 421457 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 5, 18 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 5, 18 ], mathmode := "Text", name := "Ref\ ", next := 421506, root := ~, start := 421472, stop := 421505 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 5, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 5, 19 ], mathmode := "Text", name\ := "P", next := 421583, root := ~, start := 421579, stop := 421582 ), rec( cont\ ent := "\n", count := [ 9, 2, 5, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( content := "\n Just so I remember:\n ReadPlainT\ extDigraph(\"data-local/soc-Epinions1.txt.gz\", \"\\t\", 1,\n '#');\ \n", count := [ 9, 2, 5, 19 ], mathmode := "Text", name := "XMLCOMMENT", root \ := ~ ), rec( content := "\n ", count := [ 9, 2, 5, 19 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 17 ], name := "Item", next := 4217\ 28, root := ~, start := 421109, stop := 421727 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "pickled (", count := [ 9, 2, 5, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ p", count := [ 9, 2, 5, 20 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 9, 2, 5, 20 ], mathmode := "Text", name := "F", next := 421760 , root\ := ~, start := 421751, stop := 421759 ), rec( content := " or ", count := [ 9\ , 2, 5, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 5, 20 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 20\ ], mathmode := "Text", name := "F", next := 421778, root := ~, start := 42176\ 4, stop := 421777 ), rec( content := ")", count := [ 9, 2, 5, 20 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 20 ], name := "Mar\ k", next := 421786, root := ~, start := 421736, stop := 421785 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n Digraphs are pickled using \ the ", count := [ 9, 2, 5, 21 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 5, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 5, 21 ], mathmode := "Text", name := "URL\ ", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 5, 21 ] , name\ := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes :=\ rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ \ rec( content := "IO", count := [ 9, 2, 5, 21 ], name := "PCDATA", root := ~ ) \ ], count := [ 9, 2, 5, 21 ], name := "Package", next := 122, root := ~, start \ := 101, stop := 121 ) ], count := [ 9, 2, 5, 21 ], mathmode := "Text", name :=\ "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " \ package. This is particularly good\n when the ", count := [ 9, 2, 5, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ Attr := "DigraphGroup" ), content := 0, count := [ 9, 2, 5, 21 ], mathmode :=\ "Text", name := "Ref", next := 421922, root := ~, start := 421896, stop := 42\ 1921 ), rec( content := " is non-trivial.\n ", count := [ 9, 2, 5, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 21 ], name := "Ite\ m", next := 421952, root := ~, start := 421793, stop := 421951 ) ], count := [ 9, 2, 5, 8 ], mathmode := "Text", name := "List", next := 421964, root := ~, start := 419911, stop := 421963 ), rec( content := "\n\n \ ", count := [ 9, 2, 5, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), \ 10);\n\ngap> ReadDigraphs(\n> Conc\ atenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 17);\n\ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \ \"/data/tree9.4.txt\"));\n[ , \n \ , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \ \n , \n ]", count := [ 9, 2, 5, 24 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 24 ], mathmode := "Text", name := "Example", next := 423010, root := ~, start := 421970, stop := 423009 ), rec( content := "\n ", count := [ 9, 2, 5, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 423029, root := ~, start := 418431, stop := 423028 ) ], count := [ 9, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 423043, root := ~, start := 418300, stop := 423042 ), rec( content := "\n\n ", count := [ 9, 2, 5, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename, digrap\ hs[, encoder] [, mode]", Name := "WriteDigraphs" ), content := 0, count := [ 9, 2, 6, 2 ], mathmode := "Text", name := "Func", next := 423137, root := ~, start := 423064, stop := 423136 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "A", next := 423176, root := ~, start := 423161, stop := 423175 ), rec( content := " is a li\ st of digraphs or a digraph and ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "A", next := 423231, root := ~, start := 423216, stop := 423230 ), rec( content := "\n is\ a string or an ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 6, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 6, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " file ob\ ject created using ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ File" ), content := 0, count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Ref", next := 423310, root := ~, start := 423285, stop := 423309 ), rec( content := ",\n t\ hen ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ riteDigraphs", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 423341, root := ~, start := 423321, stop := 423340 ), rec( content := " writes \ the digraphs to the file represented by\n ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "A", next := 423408, root := ~, start := 423393, stop := 423407 ), rec( content := ". If th\ e supplied filename ends in one of the extensions\n ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 423480, root := ~, start := 423470, stop := 423479 ), rec( content := ", ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 423493, root := ~, start := 423482, stop := 423492 ), rec( content := ", or ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 423508, root := ~, start := 423498, stop := 423507 ), rec( content := ", then t\ he file will be compressed\n appropriately. Excluding these extensions, if\ the file ends with an\n extension in the list below, the corresponding gra\ ph format will be used to\n encode it. If such an extension is not include\ d, an appropriate format\n will be chosen intelligently, and an extension a\ ppended, to minimise file\n size.\n ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "P", next := 423868, root := ~, start := 423864, stop := 423867 ), rec( content := "\n\n \ ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " TODO: e\ ncode and mode. ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n\n \ For more verbose information on the progress of the function, set the info\n \ level of ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nfoDigraphs", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "A", next := 424018, root := ~, start := 423999, stop := 424017 ), rec( content := " to 1 or\ higher, using ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etInfoLevel", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "C", next := 424060, root := ~, start := 424041, stop := 424059 ), rec( content := ".", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 6, 5 ], mathmode := "Text", name := "P", next := 424065, root := ~, start := 424061, stop := 424064 ), rec( content := "\n\n \ The currently supported file formats are:\n ", count := [ 9, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 6, 6 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 6 ], name := "Mar\ k", next := 424155, root := ~, start := 424130, stop := 424154 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n ", count := [ 9, 2, \ 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 7 ], name := "Ite\ m", next := 424491, root := ~, start := 424162, stop := 424490 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 6, 8\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 8 ], name := "Mar\ k", next := 424524, root := ~, start := 424498, stop := 424523 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tUnlike graph6, sparse6 has suppor\ t for loops and multiple edges.\n\tHowever, its use is still limited to symmet\ ric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those wit\ h few edges per vertex.\n ", count := [ 9, 2, 6, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 9 ], name := "Item", next := 4247\ 59, root := ~, start := 424531, stop := 424758 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 6, 10 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 10 ], name := "Mark", next := 4247\ 93, root := ~, start := 424766, stop := 424792 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n ", coun\ t := [ 9, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 11 ], name := "Ite\ m", next := 425023, root := ~, start := 424800, stop := 425022 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 6\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 12 ], name := "Mar\ k", next := 425059, root := ~, start := 425030, stop := 425058 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tAny type of digraph can be encode\ d in disparse6: directions, loops, and\n\tmultiple edges are all allowed. Sim\ ilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\ \n ", count := [ 9, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 6, 13 ], name := "Item", next := 425271, root := ~, start := 425\ 066, stop := 425270 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 6, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 6, 14 ], name := "Mark", next := 4253\ 09, root := ~, start := 425279, stop := 425308 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "0 7 0 8 1\ 7 2 8 3 8 4 8 5 8 6 8", count := [ 9, 2, 6, 15 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 15 ], mathmode := "Text", name\ := "C", next := 425441, root := ~, start := 425396, stop := 425440 ), rec( cont\ ent := " i.e. pairs of vertices\n describing edges in a graph. More spe\ cifically, the vertices making up\n one edge must be separated by a sin\ gle space, and pairs of vertices must\n be separated by two spaces. ", coun\ t := [ 9, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 6, 16 ], mathmode := "Text", name\ := "P", next := 425665, root := ~, start := 425661, stop := 425664 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 6, 16 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 6, 16 ], mathmode := "Text", name := "Ref\ ", next := 425713, root := ~, start := 425679, stop := 425712 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 6, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 6, 17 ], mathmode := "Text", name\ := "P", next := 425790, root := ~, start := 425786, stop := 425789 ), rec( cont\ ent := "\n", count := [ 9, 2, 6, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( content := "\n Just so I remember:\n ReadPlainT\ extDigraph(\"data-local/soc-Epinions1.txt.gz\", \"\\t\", 1, '#');\n", coun\ t := [ 9, 2, 6, 17 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( cont\ ent := "\n ", count := [ 9, 2, 6, 17 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 6, 15 ], name := "Item", next := 425927, root := ~, start := 425\ 316, stop := 425926 ), rec( attributes := rec( ), content := [ rec( content :\ = "pickled (", count := [ 9, 2, 6, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := ".p", count \ := [ 9, 2, 6, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 6, 18 ], mathmode := "Text", name := "F", next := 425958, root := \ ~, start := 425949, stop := 425957 ), rec( content := " or ", count := [ 9, 2,\ 6, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 6, 18 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 18 ], math\ mode := "Text", name := "F", next := 425976, root := ~, start := 425962, stop \ := 425975 ), rec( content := ")", count := [ 9, 2, 6, 18 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 18 ], name := "Mark", next := 4259\ 84, root := ~, start := 425934, stop := 425983 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n Digraphs are pickled using the ", count := [ 9, 2, 6, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := \ "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ re\ c( content := "\n http://gap-packages.github.io/io/\n", count := [ 9, 2, 6\ , 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, \ 6, 19 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17\ , stop := 77 ) ], count := [ 9, 2, 6, 19 ], name := "Alt", next := 84, root :=\ ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ), conten\ t := [ rec( attributes := rec( ), content := [ rec( content := "IO", count :=\ [ 9, 2, 6, 19 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 19 ], n\ ame := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count\ := [ 9, 2, 6, 19 ], mathmode := "Text", name := "Alt", next := 128, root := ~ , star\ t := 84, stop := 127 ), rec( content := " package. This is particularly good\n\ when the ", count := [ 9, 2, 6, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGroup" ), content :\ = 0, count := [ 9, 2, 6, 19 ], mathmode := "Text", name := "Ref", next := 4261\ 20, root := ~, start := 426094, stop := 426119 ), rec( content := " is non-tri\ vial.\n ", count := [ 9, 2, 6, 19 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 9, 2, 6, 19 ], name := "Item", next := 426150, root := ~, start := 425\ 991, stop := 426149 ) ], count := [ 9, 2, 6, 6 ], mathmode := "Text", name := "List", next := 426162, root := ~, start := 424117, stop := 426161 ), rec( content := "\n\n \ ", count := [ 9, 2, 6, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> grs := [];;\ngap> grs[1] := Digraph([]);\n\ngap> grs[2] := Digraph([[1, 3], [2], [1, 2]]);\n\ngap> grs[3] := Digraph([\n> [6, 7], [6, 9], [1, 3\ , 4, 5, 8, 9],\n> [1, 2, 3, 4, 5, 6, 7, 10], [1, 5, 6, 7, 10], [2, 4, 5, 9, 10\ ],\n> [3, 4, 5, 6, 7, 8, 9, 10], [1, 3, 5, 7, 8, 9], [1, 2, 5],\n> [1, 2, 4, 6\ , 7, 8]]);\n\ngap> filename := C\ oncatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap> WriteDigraphs(fil\ ename, grs, \"w\");\nIO_OK\ngap> ReadDigraphs(filename);\n[ , \n , \n \ ]", count := [ 9, 2, 6, 22 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 6, 22 ], mathmode := "Text", name := "Example", next := 426929, root := ~, start := 426168, stop := 426928 ), rec( content := "\n ", count := [ 9, 2, 6, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 426948, root := ~, start := 423140, stop := 426947 ) ], count := [ 9, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 426962, root := ~, start := 423049, stop := 426961 ), rec( content := "\n\n ", count := [ 9, 2, 6, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename [, deco\ der]", Name := "IteratorFromDigraphFile" ), content := 0, count := [ 9, 2, 7, 2 ], mathmode := "Text", name := "Func", next := 427048, root := ~, start := 426983, stop := 427047 ), rec( attributes := rec( ), content := [ rec( content := "An iterat\ or.", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Returns", next := 427082, root := ~, start := 427051, stop := 427081 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "A", next := 427121, root := ~, start := 427106, stop := 427120 ), rec( content := " is a st\ ring representing the name of a file containing\n encoded digraphs, then ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ teratorFromDigraphFile", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "C", next := 427234, root := ~, start := 427204, stop := 427233 ), rec( content := " returns\ an iterator\n for which the value of ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Oper := "NextIterator" ), content := 0, count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Ref", next := 427323, root := ~, start := 427282, stop := 427322 ), rec( content := " is the\\ n next digraph encoded in the file.\n ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "P", next := 427377, root := ~, start := 427373, stop := 427376 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "A", next := 427422, root := ~, start := 427408, stop := 427421 ), rec( content := " is spec\ ified and is a function\n which decodes a string into a digraph, then ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ teratorFromDigraphFile", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "C", next := 427532, root := ~, start := 427502, stop := 427531 ), rec( content := "\n wi\ ll use ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "A", next := 427560, root := ~, start := 427546, stop := 427559 ), rec( content := " to deco\ de the digraphs contained in ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "A", next := 427612, root := ~, start := 427597, stop := 427611 ), rec( content := ".\n " , count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 5 ], mathmode := "Text", name := "P", next := 427622, root := ~, start := 427618, stop := 427621 ), rec( content := "\n\n \ The purpose of this function is to easily allow looping over digraphs\n enc\ oded in a file when loading all of the encoded digraphs would require\n too\ much memory. ", count := [ 9, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "P", next := 427800, root := ~, start := 427796, stop := 427799 ), rec( content := "\n\n \ To see what file types are available, see ", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "Ref", next := 427875, root := ~, start := 427848, stop := 427874 ), rec( content := ".\n\n \ ", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap\ > file := DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> Writ\ eDigraphs(file, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Close(file);;\ \ngap> iter := IteratorFromDigraphFile(filename);\n\ngap> for x in i\ ter do od;\n", count := [ 9, 2, 7, 7 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 7 ], mathmode := "Text", name := "Example", next := 428224, root := ~, start := 427882, stop := 428223 ), rec( content := "\n ", count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Description", next := 428241, root := ~, start := 427085, stop := 428240 ) ], count := [ 9, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 428255, root := ~, start := 426968, stop := 428254 ), rec( content := "\n\n ", count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "delimiter1, [del\ imiter2,] offset", Name := "DigraphPlainTextLineEncoder" ), content := 0, count := [ 9, 2, 8, 2 ], mathmode := "Text", name := "Func", next := 428357, root := ~, start := 428276, stop := 428356 ), rec( attributes := rec( Arg := "delimiter1, [del\ imiter2,] offset", Name := "DigraphPlainTextLineDecoder" ), content := 0, count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "Oper", next := 428441, root := ~, start := 428360, stop := 428440 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "Returns", next := 428472, root := ~, start := 428444, stop := 428471 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se two functions return a function which encodes or decodes a digraph in\n \ a plain text format.", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "P", next := 428597, root := ~, start := 428593, stop := 428596 ), rec( content := "\n\n \ ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineEncoder", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 428637, root := ~, start := 428603, stop := 428636 ), rec( content := " returns\ a function which takes a single\n digraph as an argument. The function re\ turns a string describing the edges\n of that digraph; each edge is written\ as a pair of integers separated by the\n string ", count := [ 9, 2, 8, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter2", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 428867, root := ~, start := 428850, stop := 428866 ), rec( content := ", and th\ e edges themselves are separated by the\n string ", count := [ 9, 2, 8, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter1", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 428943, root := ~, start := 428926, stop := 428942 ), rec( content := ". ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineDecoder", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 428980, root := ~, start := 428946, stop := 428979 ), rec( content := " returns\ the\n corresponding decoder function, which takes a string argument in thi\ s format\n and returns a digraph.", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "P", next := 429104, root := ~, start := 429100, stop := 429103 ), rec( content := "\n\n \ If only one delimiter is passed as an argument to\n ", count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineDecoder", count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "A", next := 429198, root := ~, start := 429164, stop := 429197 ), rec( content := ", it wil\ l return a function which decodes\n a single edge, returning its contents a\ s a list of integers.", count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "P", next := 429308, root := ~, start := 429304, stop := 429307 ), rec( content := "\n\n \ The argument ", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "A", next := 429340, root := ~, start := 429327, stop := 429339 ), rec( content := " should \ be an integer, which will describe a\n number to be added to each vertex be\ fore it is encoded, or after it is\n decoded. This may be used, for exampl\ e, to label vertices starting at 0\n instead of 1.", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 9 ], mathmode := "Text", name := "P", next := 429558, root := ~, start := 429554, stop := 429557 ), rec( content := "\n\n \ Note that the number of vertices of a digraph is not stored, and so vertices\n\ which are not connected to any edge may be lost.\n\n ", count := [ 9, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2, 3], [1], [1]]);\n\ngap> enc := DigraphPlainTextLineEncoder(\" \", \" \", -1);;\ngap> d\ ec := DigraphPlainTextLineDecoder(\" \", \" \", 1);;\ngap> enc(gr);\n\"0 1 0\ 2 1 0 2 0\"\ngap> dec(last);\n\ \n", count := [ 9, 2, 8, 10 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 10 ], mathmode := "Text", name := "Example", next := 430025, root := ~, start := 429699, stop := 430024 ), rec( content := "\n ", count := [ 9, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "Description", next := 430042, root := ~, start := 428475, stop := 430041 ) ], count := [ 9, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 430056, root := ~, start := 428261, stop := 430055 ), rec( content := "\n\n ", count := [ 9, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "TournamentLineD\ ecoder" ), content := 0, count := [ 9, 2, 9, 2 ], mathmode := "Text", name := "Oper", next := 430123, root := ~, start := 430077, stop := 430122 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Returns", next := 430155, root := ~, start := 430126, stop := 430154 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function takes a string ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "A", next := 430215, root := ~, start := 430205, stop := 430214 ), rec( content := ", decode\ s it, and then returns the\n tournament [see ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTourn\ ament" ), content := 0, count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Ref", next := 430296, root := ~, start := 430270, stop := 430295 ), rec( content := "] which \ it defines, according to\n the following rules. ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "P", next := 430358, root := ~, start := 430354, stop := 430357 ), rec( content := "\n\n \ The characters of the string ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "A", next := 430403, root := ~, start := 430393, stop := 430402 ), rec( content := " represe\ nt the entries in the upper\n triangle of a tournament's adjacency matrix. \ The number of vertices\n ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "C", next := 430523, root := ~, start := 430515, stop := 430522 ), rec( content := " will be\ detected from the length of the string and will be as\n large as possible.\ ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "P", next := 430614, root := ~, start := 430610, stop := 430613 ), rec( content := "\n\n \ The first character represents the possible edge ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 2", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 430682, root := ~, start := 430669, stop := 430681 ), rec( content := ", the se\ cond\n represents ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 3", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 430723, root := ~, start := 430710, stop := 430722 ), rec( content := " and so \ on until ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> n", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 430753, root := ~, start := 430740, stop := 430752 ), rec( content := "; then t\ he following\n character represents ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -> 3", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 430812, root := ~, start := 430799, stop := 430811 ), rec( content := ", and so\ on up to the character which\n represents the edge ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1 -> n", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 430889, root := ~, start := 430874, stop := 430888 ), rec( content := ". ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "P", next := 430895, root := ~, start := 430891, stop := 430894 ), rec( content := "\n\n \ If a character of the string with corresponding edge ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 430967, root := ~, start := 430954, stop := 430966 ), rec( content := " is equa\ l\n to ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 430992, root := ~, start := 430984, stop := 430991 ), rec( content := ", then t\ he edge ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 431021, root := ~, start := 431008, stop := 431020 ), rec( content := " is pres\ ent in the tournament.\n Otherwise, the edge ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 431089, root := ~, start := 431076, stop := 431088 ), rec( content := " is pres\ ent instead. In this way, all the\n possible edges are encoded one-by-one.\ \n\n ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := TournamentLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 1, 2, 4 ], [ 1, 2 ] \ ]\n", count := [ 9, 2, 9, 7 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 7 ], mathmode := "Text", name := "Example", next := 431364, root := ~, start := 431180, stop := 431363 ), rec( content := "\n ", count := [ 9, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Description", next := 431381, root := ~, start := 430158, stop := 431380 ) ], count := [ 9, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 431395, root := ~, start := 430062, stop := 431394 ), rec( content := "\n\n ", count := [ 9, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "AdjacencyMatrix\ UpperTriangleLineDecoder" ), content := 0, count := [ 9, 2, 10, 2 ], mathmode := "Text", name := "Oper", next := 431480, root := ~, start := 431416, stop := 431479 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Returns", next := 431512, root := ~, start := 431483, stop := 431511 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function takes a string ", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "A", next := 431572, root := ~, start := 431562, stop := 431571 ), rec( content := ", decode\ s it, and then returns the\n topologically sorted digraph [see ", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Ref", next := 431681, root := ~, start := 431645, stop := 431680 ), rec( content := "]\n w\ hich it defines, according to the following rules. ", count := [ 9, 2, 10, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "P", next := 431743, root := ~, start := 431739, stop := 431742 ), rec( content := "\n\n \ The characters of the string ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "A", next := 431788, root := ~, start := 431778, stop := 431787 ), rec( content := " represe\ nt the entries in the upper\n triangle of a digraph's adjacency matrix. The\ number of vertices ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "C", next := 431901, root := ~, start := 431893, stop := 431900 ), rec( content := "\n wi\ ll be detected from the length of the string and will be as large as\n poss\ ible. ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "P", next := 431996, root := ~, start := 431992, stop := 431995 ), rec( content := "\n\n \ The first character represents the possible edge ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 2", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 432064, root := ~, start := 432051, stop := 432063 ), rec( content := ", the se\ cond\n represents ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 3", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 432105, root := ~, start := 432092, stop := 432104 ), rec( content := " and so \ on until ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> n", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 432135, root := ~, start := 432122, stop := 432134 ), rec( content := "; then t\ he following\n character represents ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -> 3", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 432194, root := ~, start := 432181, stop := 432193 ), rec( content := ", and so\ on up to the character which\n represents the edge ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1 -> n", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 432271, root := ~, start := 432256, stop := 432270 ), rec( content := ". If a \ character of the string with\n corresponding edge ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 432344, root := ~, start := 432331, stop := 432343 ), rec( content := " is equa\ l to ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 432365, root := ~, start := 432357, stop := 432364 ), rec( content := ", then t\ his edge is\n present in the digraph. Otherwise, it is not present. In thi\ s way, all the\n possible edges are encoded one-by-one. ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "P", next := 432512, root := ~, start := 432508, stop := 432511 ), rec( content := "\n\n \ In particular, note that there exists no edge ", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "C", next := 432577, root := ~, start := 432564, stop := 432576 ), rec( content := " if ", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ \\leq\n i", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "M", next := 432602, root := ~, start := 432581, stop := 432601 ), rec( content := ". In or\ der words, the digraph will be topologically sorted.\n\n ", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := AdjacencyMatrixUpperTriangleLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 4 \ ], [ ] ]\ngap> gr := AdjacencyMatrixUpperTriangleLineDecoder(\"111111x111\");\ \n\ngap> OutNeighbours(gr);\n[ [ 2\ , 3, 4, 5 ], [ 3, 4 ], [ 4, 5 ], [ 5 ], [ ] ]\n", count := [ 9, 2, 10, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 10, 7 ], mathmode := "Text", name := "Example", next := 433047, root := ~, start := 432668, stop := 433046 ), rec( content := "\n ", count := [ 9, 2, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 433064, root := ~, start := 431515, stop := 433063 ) ], count := [ 9, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 433078, root := ~, start := 431401, stop := 433077 ), rec( content := "\n\n ", count := [ 9, 2, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "TCodeDecoder" ) , content := 0, count := [ 9, 2, 11, 2 ], mathmode := "Text", name := "Oper", next := 433136, root := ~, start := 433099, stop := 433135 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "Returns", next := 433168, root := ~, start := 433139, stop := 433167 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "A", next := 433202, root := ~, start := 433192, stop := 433201 ), rec( content := " is a st\ ring consisting of at least two non-negative integers\n separated by spaces\ , then this function will attempt to return the digraph\n which it defines \ as a TCode string. ", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "P", next := 433387, root := ~, start := 433383, stop := 433386 ), rec( content := "\n\n \ The first integer of the string defines the number of vertices ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 433464, root := ~, start := 433456, stop := 433463 ), rec( content := " in\n \ the digraph, and the second defines the number of edges ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 433536, root := ~, start := 433528, stop := 433535 ), rec( content := ". The\n\ following ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ e", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 433566, root := ~, start := 433557, stop := 433565 ), rec( content := " integer\ s should be vertex numbers in the range ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 0 ..\n v-1]", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 433637, root := ~, start := 433614, stop := 433636 ), rec( content := ". These \ integers are read in pairs and define the digraph's\n edges. This function\ will return an error if ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "A", next := 433757, root := ~, start := 433747, stop := 433756 ), rec( content := " has few\ er than\n ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ e+2", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 433788, root := ~, start := 433777, stop := 433787 ), rec( content := " entries\ . ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "P", next := 433802, root := ~, start := 433798, stop := 433801 ), rec( content := "\n\n \ Note that the vertex numbers will be incremented by 1 in the digraph\n retu\ rned. Hence the string fragment ", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ 6", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "C", next := 433928, root := ~, start := 433918, stop := 433927 ), rec( content := " will de\ scribe the edge\n ", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1,7]", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "C", next := 433968, root := ~, start := 433956, stop := 433967 ), rec( content := ".\n " , count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := TCodeDecoder(\"3 2 0 2 2 1\");\n\ngap> OutNeighbours(gr);\n[ [ 3 ], [ ], [ 2 ] ]\ngap> gr := TCodeD\ ecoder(\"12 3 0 10 5 2 8 8\");\n\ \ngap> OutNeighbours(gr);\n[ [ 11 ], [ ], [ ], [ ], [ ], [ 3 ], [ ], [ ]\ , [ 9 ], [ ], \n [ ], [ ] ]\n", count := [ 9, 2, 11, 6 ], name := "PCDATA" , root := ~ ) ], count := [ 9, 2, 11, 6 ], mathmode := "Text", name := "Example", next := 434336, root := ~, start := 433974, stop := 434335 ), rec( content := "\n ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "Description", next := 434353, root := ~, start := 433171, stop := 434352 ) ], count := [ 9, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 434367, root := ~, start := 433084, stop := 434366 ), rec( content := "\n\n ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "PlainTextString\ " ), content := 0, count := [ 9, 2, 12, 2 ], mathmode := "Text", name := "Oper", next := 434432, root := ~, start := 434388, stop := 434431 ), rec( attributes := rec( Arg := "s", Name := "DigraphFromPlai\ nTextString" ), content := 0, count := [ 9, 2, 12, 4 ], mathmode := "Text", name := "Oper", next := 434484, root := ~, start := 434435, stop := 434483 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "Returns", next := 434515, root := ~, start := 434487, stop := 434514 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lainTextString", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "A", next := 434558, root := ~, start := 434536, stop := 434557 ), rec( content := " takes a\ single digraph, and returns a string\n describing the edges of that digrap\ h. ", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFromPlainTextString", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "A", next := 434680, root := ~, start := 434647, stop := 434679 ), rec( content := "\n ta\ kes such a string and returns the digraph which it describes. Each edge\n \ is written as a pair of integers separated by a single space. The edges\n \ themselves are separated by a double space. Vertex numbers are reduced by\n \ 1 when they are encoded, so that vertices in the string are labelled\n st\ arting at 0.", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "P", next := 435011, root := ~, start := 435007, stop := 435010 ), rec( content := "\n\n \ Note that the number of vertices of a digraph is not stored, and so vertices\n\ which are not connected to any edge may be lost.", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "P", next := 435150, root := ~, start := 435146, stop := 435149 ), rec( content := "\n\n \ \n The operation ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFromPlainTextString", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "C", next := 435208, root := ~, start := 435175, stop := 435207 ), rec( content := "\n ta\ kes an optional first argument ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "Ref", next := 435276, root := ~, start := 435246, stop := 435275 ), rec( content := "\n or\ ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "Ref", next := 435316, root := ~, start := 435284, stop := 435315 ), rec( content := ", which \ specifies whether the output digraph shall\n be mutable or immutable. If no\ first argument is provided, then an immutable\n digraph is returned by def\ ault.\n\n ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2, 3], [1], [1]]);\n\ngap> PlainTextString(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> DigraphFromP\ lainTextString(last);\n\n", count := [ 9, 2, 12, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 8 ], mathmode := "Text", name := "Example", next := 435737, root := ~, start := 435489, stop := 435736 ), rec( content := "\n ", count := [ 9, 2, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "Description", next := 435754, root := ~, start := 434518, stop := 435753 ) ], count := [ 9, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 435768, root := ~, start := 434373, stop := 435767 ), rec( content := "\n\n ", count := [ 9, 2, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename, digrap\ h, delimiter, offset", Name := "WritePlainTextDigraph" ), content := 0, count := [ 9, 2, 13, 2 ], mathmode := "Text", name := "Func", next := 435868, root := ~, start := 435789, stop := 435867 ), rec( attributes := rec( Arg := "filename, delimi\ ter, offset, ignore", Name := "ReadPlainTextDigraph" ), content := 0, count := [ 9, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 435948, root := ~, start := 435871, stop := 435947 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se functions write and read a single digraph in a human-readable plain\n te\ xt format as follows: each line contains a single edge, and each edge is\n \ written as a pair of integers separated by the string ", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "A", next := 436196, root := ~, start := 436180, stop := 436195 ), rec( content := ".", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "P", next := 436201, root := ~, start := 436197, stop := 436200 ), rec( content := "\n\n \ ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 436222, root := ~, start := 436207, stop := 436221 ), rec( content := " should \ be the name of a file which will be written to or\n read from, and ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 436312, root := ~, start := 436299, stop := 436311 ), rec( content := " should \ be an integer which is added to each\n vertex number as it is written or re\ ad. For example, if\n ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ ritePlainTextDigraph", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 436450, root := ~, start := 436422, stop := 436449 ), rec( content := " is call\ ed with ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 436479, root := ~, start := 436466, stop := 436478 ), rec( content := " ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "-\ 1", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 436489, root := ~, start := 436480, stop := 436488 ), rec( content := ", then\n\ the vertices will be numbered in the file starting from 0 instead of 1 -\n\ ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadPlainTextDigraph", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 436604, root := ~, start := 436577, stop := 436603 ), rec( content := " would t\ hen need to be called with ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 436652, root := ~, start := 436639, stop := 436651 ), rec( content := "\n ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 436665, root := ~, start := 436657, stop := 436664 ), rec( content := " to conv\ ert back to the original graph.", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "P", next := 436708, root := ~, start := 436704, stop := 436707 ), rec( content := "\n\n \ ", count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ gnore", count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "A", next := 436727, root := ~, start := 436714, stop := 436726 ), rec( content := " should \ be a list of characters which will be ignored when\n reading the graph.\n \ ", count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[1, 2, 3], [1, 1], [2]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out\ /plain.txt\");;\ngap> WritePlainTextDigraph(filename, gr, \",\", -1);\ngap> Re\ adPlainTextDigraph(filename, \",\", 1, ['/', '%']);\n\n", count := [ 9, 2, 13, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 8 ], mathmode := "Text", name := "Example", next := 437170, root := ~, start := 436813, stop := 437169 ), rec( content := "\n ", count := [ 9, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 437187, root := ~, start := 435951, stop := 437186 ) ], count := [ 9, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 437201, root := ~, start := 435774, stop := 437200 ), rec( content := "\n\n ", count := [ 9, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename, digrap\ h", Name := "WriteDIMACSDigraph" ), content := 0, count := [ 9, 2, 14, 2 ], mathmode := "Text", name := "Oper", next := 437279, root := ~, start := 437222, stop := 437278 ), rec( attributes := rec( Arg := "filename", Name := "ReadDIMACSDigra\ ph" ), content := 0, count := [ 9, 2, 14, 4 ], mathmode := "Text", name := "Oper", next := 437329, root := ~, start := 437282, stop := 437328 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations write or read the single symmetric digraph ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "A", next := 437424, root := ~, start := 437410, stop := 437423 ), rec( content := "\n to\ or from a file in DIMACS format, as appropriate. The operation\n ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ riteDIMACSDigraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "C", next := 437524, root := ~, start := 437499, stop := 437523 ), rec( content := " records\ the vertices and edges of ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "A", next := 437573, root := ~, start := 437559, stop := 437572 ), rec( content := ".\n T\ he vertex labels of ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "A", next := 437614, root := ~, start := 437600, stop := 437613 ), rec( content := " will be\ recorded only if they are\n integers. See ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "Ref", next := 437699, root := ~, start := 437667, stop := 437698 ), rec( content := " and ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "Ref", next := 437743, root := ~, start := 437704, stop := 437742 ), rec( content := ".", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "P", next := 437748, root := ~, start := 437744, stop := 437747 ), rec( content := "\n\n \ The first argument ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "A", next := 437788, root := ~, start := 437773, stop := 437787 ), rec( content := " should \ be the name of the file which will\n be written to or read from. A file ca\ n contain one symmetric digraph in\n DIMACS format. If ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "A", next := 437945, root := ~, start := 437930, stop := 437944 ), rec( content := " ends in\ one of ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "C", next := 437971, root := ~, start := 437961, stop := 437970 ), rec( content := ", ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "C", next := 437984, root := ~, start := 437973, stop := 437983 ), rec( content := ",\n o\ r ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "C", next := 438003, root := ~, start := 437993, stop := 438002 ), rec( content := ", then t\ he file is compressed, or decompressed, appropriately.\n ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 14, 7 ], mathmode := "Text", name := "P", next := 438074, root := ~, start := 438070, stop := 438073 ), rec( content := "\n\n \ The DIMACS format is described as follows. Each line in the DIMACS file has\n\ one of four types:\n ", count := [ 9, 2, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n A line beginning with ", coun\ t := [ 9, 2, 14, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "c", count := [ 9, 2, 14, 9 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 9 ] , math\ mode := "Text", name := "C", next := 438242, root := ~, start := 438234, stop \ := 438241 ), rec( content := " and followed by any number of characters\n \ is a comment line, and is ignored.\n ", count := [ 9, 2, 14, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 9 ], name := "Ite\ m", next := 438340, root := ~, start := 438197, stop := 438339 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n A line beginning with ", coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "p", count := [ 9, 2, 14, 11 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 11\ ], mathmode := "Text", name := "C", next := 438392, root := ~, start := 43838\ 4, stop := 438391 ), rec( content := " defines the numbers of vertices and edg\ es\n the digraph. This line has the format ", count := [ 9, 2, 14, 11 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "p edge \n ", count :\ = [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 14, 11 ], mathmode := "Text", name := "C", next := 438542, root :=\ ~, start := 438482, stop := 438541 ), rec( content := ", where ", count := [ \ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "", count := [ 9, 2\ , 14, 11 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "C", next := 438576, root :\ = ~, start := 438550, stop := 438575 ), rec( content := " and\n ", coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "", count := [ 9, 2,\ 14, 11 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "C", next := 438612, root :\ = ~, start := 438589, stop := 438611 ), rec( content := " are replaced by the \ relevant integers. There\n must be exactly one such line in the file, a\ nd it must occur before any\n of the following kinds of line.", count :\ = [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 14, 12 ], mathmode := "Text" , name\ := "P", next := 438781, root := ~, start := 438777, stop := 438780 ), rec( cont\ ent := "\n\n Although it is required to be present, the value of\n \ ", count := [ 9, 2, 14, 12 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "", c\ ount := [ 9, 2, 14, 12 ], mathmode := "Text", name := "PCDATA", next := 6, roo\ t := ~ ) ], count := [ 9, 2, 14, 12 ], mathmode := "Text", name := "C", next :\ = 438874, root := ~, start := 438851, stop := 438873 ), rec( content := " will\ be ignored. The correct number of edges\n will be deduced from the res\ t of the information in the file.\n ", count := [ 9, 2, 14, 12 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 11 ], name := "Ite\ m", next := 439003, root := ~, start := 438347, stop := 439002 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n A line of the form ", count\ := [ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "e ", count := [ 9, 2, \ 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2\ , 14, 14 ], mathmode := "Text", name := "C", next := 439072, root := ~, start \ := 439044, stop := 439071 ), rec( content := ", where ", count := [ 9, 2, 14, \ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "", count := [ 9, 2, 14, 14 ], mathmode\ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 1\ 4 ], mathmode := "Text", name := "C", next := 439096, root := ~, start := 4390\ 80, stop := 439095 ), rec( content := "\n and ", count := [ 9, 2, 14, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "", count := [ 9, 2, 14, 14 ], mathmode \ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 14\ ], mathmode := "Text", name := "C", next := 439125, root := ~, start := 43910\ 9, stop := 439124 ), rec( content := " are integers in the range ", count := [ \ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "[1 ..\n ]\ ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 9, 2, 14, 14 ], mathmode := "Text", name := "C", next := 43919\ 5, root := ~, start := 439152, stop := 439194 ), rec( content := ", specifies \ that there is a (symmetric) edge\n in the digraph between the vertices \ ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "", count := [ 9, 2, 14, 14\ ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ \ 9, 2, 14, 14 ], mathmode := "Text", name := "C", next := 439300, root := ~, star\ t := 439284, stop := 439299 ), rec( content := " and\n ", count := [ 9,\ 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "", count := [ 9, 2, 14, 14 ], \ mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, \ 2, 14, 14 ], mathmode := "Text", name := "C", next := 439329, root := ~, start\ := 439313, stop := 439328 ), rec( content := ". A symmetric edge only needs t\ o be defined once; an\n additional line ", count := [ 9, 2, 14, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "e ", count := [ 9, 2, 14, 14 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 14 ], mathmode :=\ "Text", name := "C", next := 439434, root := ~, start := 439406, stop := 4394\ 33 ), rec( content := ", or ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "e \n ", count := [ 9, 2, 14, 14 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 14 ], mathmode := "Text", name\ := "C", next := 439477, root := ~, start := 439439, stop := 439476 ), rec( cont\ ent := ", will be interpreted as an additional, multiple, edge.\n Loops\ are permitted.\n ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 14 ], name := "Item", next := 4395\ 75, root := ~, start := 439010, stop := 439574 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n A line of the form ", count := [ 9, 2, 14, 16 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "n ", count := [ 9, 2, 14, 16 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 16 ], mathmode := "\ Text", name := "C", next := 439648, root := ~, start := 439616, stop := 439647\ ), rec( content := ", where\n ", count := [ 9, 2, 14, 16 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "", count := [ 9, 2, 14, 16 ], mathmode := "Text", name\ := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 16 ], mathmode :=\ "Text", name := "C", next := 439680, root := ~, start := 439664, stop := 4396\ 79 ), rec( content := " is an integer in the range ", count := [ 9, 2, 14, 16 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 ..\n ]", count := [ 9, 2, 1\ 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2,\ 14, 16 ], mathmode := "Text", name := "C", next := 439751, root := ~, start :\ = 439708, stop := 439750 ), rec( content := " and ", count := [ 9, 2, 14, 16 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "", count := [ 9, 2, 14, 16 ], mathmode := "Te\ xt", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 16 ], math\ mode := "Text", name := "C", next := 439776, root := ~, start := 439756, stop \ := 439775 ), rec( content := " is an integer,\n signifies that the vert\ ex ", count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "", count :=\ [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ \ ) ], count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "C", next := 43984\ 2, root := ~, start := 439826, stop := 439841 ), rec( content := " has the lab\ el\n ", count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "", c\ ount := [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", next := 6, roo\ t := ~ ) ], count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "C", next :\ = 439885, root := ~, start := 439865, stop := 439884 ), rec( content := " in t\ he digraph. If a label is not specified for a\n vertex, then ", count :\ = [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "ReadDIMACSDigraph", count := \ [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 9, 2, 14, 16 ], mathmode := "Text", name := "C", next := 439981, root := ~ , star\ t := 439957, stop := 439980 ), rec( content := " will assign the label\n \ ", count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "1", count := [ \ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 9, 2, 14, 16 ], mathmode := "Text", name := "C", next := 440020, root := ~, star\ t := 440012, stop := 440019 ), rec( content := ", according to the DIMACS spec\ ification.\n ", count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 9, 2, 14, 16 ], name := "Item", next := 4400\ 74, root := ~, start := 439582, stop := 440073 ) ], count := [ 9, 2, 14, 8 ], mathmode := "Text", name := "List", next := 440086, root := ~, start := 438184, stop := 440085 ), rec( content := "\n\n \ A detailed definition of the DIMACS format can be found at\n ", count := [ 9, 2, 14, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://mat.gsia.cmu.edu/COLOR/general/ccformat.ps", count := [ 9, 2, 14, 18 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 18 ], mathmode := "Text", name := "URL", next := 440215, root := ~, start := 440155, stop := 440214 ), rec( content := ", in Sec\ tion\n 2.1. Note that optional descriptor lines, as described in Section 2.\ 1, will\n be ignored.\n ", count := [ 9, 2, 14, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 14, 19 ], mathmode := "Text", name := "P", next := 440332, root := ~, start := 440328, stop := 440331 ), rec( content := "\n\n\n \ ", count := [ 9, 2, 14, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2], [1, 3, 4], [2, 4], [2, 3]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(),\n> \ \"/tst/out/dimacs.dimacs\");;\ngap> WriteDIMACSDigrap\ h(filename, gr);;\ngap> ReadDIMACSDigraph(filename);\n", count := [ 9, 2, 14, 20 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 20 ], mathmode := "Text", name := "Example", next := 440694, root := ~, start := 440339, stop := 440693 ), rec( content := "\n ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "Description", next := 440711, root := ~, start := 437332, stop := 440710 ) ], count := [ 9, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 440725, root := ~, start := 437207, stop := 440724 ), rec( content := "\n\n ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 440739, root := ~, start := 407893, stop := 440738 ) , rec( content := "\n", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 440750, root := ~, start := 396976, stop := 440749 ), rec( content := "\n ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " visualising, io ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "Body", next := 440784, root := ~, start := 3968, stop := 440783 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Grape to Digraphs Command M\ ap\n ", count := [ "A", 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 440855, root := ~, start := 440799, stop := 440854 ), rec( content := "\n Below is a table of ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.ac.\ uk/~leonard/grape/", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 0, 0, 3 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ "A", 0, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " commands with the ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " counterparts.\n The sections in this c\ hapter correspond to the chapters in the ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.ac.\ uk/~leonard/grape/", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 0, 0, 3 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ "A", 0, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := "\n manual.\n\n ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 1, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to cons\ truct and modify graphs\n ", count := [ "A", 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 441100, root := ~, start := 441029, stop := 441099 ), rec( content := "\n ", count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 3 ], name := "E", next := 441238, root := ~, start := 441128, stop := 441237 ), rec( content := "\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "Alt", next := 441249, root := ~, start := 441105, stop := 441248 ), rec( content := "\n\n ", count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 4 ], name := "HorLine", next := 441295, root := ~, start := 441285, stop := 441294 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 1, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 1, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 5 ], name := "Item", next := 441364, root := ~, start := 441316, stop := 441363 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 7 ], name := "Item", next := 441424, root := ~, start := 441373, stop := 441423 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 1, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 9 ], name := "Alt", next := 441505, root := ~, start := 441450, stop := 441504 ), rec( content := "\n \ ", count := [ "A", 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 9 ], name := "Item", next := 441521, root := ~, start := 441433, stop := 441520 ) ], count := [ "A", 1, 0, 4 ], name := "Row", next := 441534, root := ~, start := 441302, stop := 441533 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 10 ], name := "HorLine", next := 441551, root := ~, start := 441541, stop := 441550 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raph", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "C", next := 441602, root := ~, start := 441590, stop := 441601 ), rec( content := "\n \ ", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 11 ], name := "Item", next := 441618, root := ~, start := 441573, stop := 441617 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ " ), content := 0, count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "Ref", next := 441665, root := ~, start := 441644, stop := 441664 ), rec( content := "\n \ ", count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 13 ], name := "Item", next := 441681, root := ~, start := 441627, stop := 441680 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Used with the arguments of ", count := [ "A", 1, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Graph", count := [ "A", 1, 0, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "C", next := 4417\ 74, root := ~, start := 441762, stop := 441773 ), rec( HTML := true, content := "\ in ", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , ", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Oper := "Digraph" ), content := 0, count := [ "A", 1, 0, 15 ], mathmode := \ "Text", name := "Ref", next := 441820, root := ~, start := 441787, stop := 4418\ 19 ), rec( HTML := true, content := " will return an equivalent digraph. ", count := [ "A", 1, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0 , count := [ "A", 1, 0, 16 ], mathmode := \ "Text", name := "P", next := 441860, root := ~, start := 441856, stop := 44185\ 9 ), rec( HTML := true, content := "\n ", count := [ "A", 1, 0, 16 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], name := "Alt", next := 441877, root := ~, start := 441707, stop := 441876 ), rec( content := "\n \ ", count := [ "A", 1, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], name := "Item", next := 441893, root := ~, start := 441690, stop := 441892 ) ], count := [ "A", 1, 0, 10 ], name := "Row", next := 441906, root := ~, start := 441559, stop := 441905 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 17 ], name := "HorLine", next := 441923, root := ~, start := 441913, stop := 441922 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ dgeOrbitsGraph", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "C", next := 441984, root := ~, start := 441962, stop := 441983 ), rec( content := "\n \ ", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 18 ], name := "Item", next := 442000, root := ~, start := 441945, stop := 441999 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EdgeOrb\ itsDigraph" ), content := 0, count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "Ref", next := 442057, root := ~, start := 442026, stop := 442056 ), rec( content := "\n \ ", count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 20 ], name := "Item", next := 442073, root := ~, start := 442009, stop := 442072 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "EdgeOrbitsDigraph" ), content := 0 , count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "Ref", next := 442172, root := ~, start := 442141, stop := 4421\ 71 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "EdgeOrbitsDigraph", count := [ "A", 1, 0, 22 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "C", next := 442227, root := ~, start := 442203, stop := 44222\ 6 ), rec( HTML := true, content := " in ", count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 22 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 22 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 22 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 22 ], name := "Alt", next := 442256, root := ~, start := 442099, stop := 442255 ), rec( content := "\n \ ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], name := "Item", next := 442272, root := ~, start := 442082, stop := 442271 ) ], count := [ "A", 1, 0, 17 ], name := "Row", next := 442285, root := ~, start := 441931, stop := 442284 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 23 ], name := "HorLine", next := 442302, root := ~, start := 442292, stop := 442301 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ ullGraph", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "C", next := 442357, root := ~, start := 442341, stop := 442356 ), rec( content := "\n \ ", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 24 ], name := "Item", next := 442373, root := ~, start := 442324, stop := 442372 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "NullDig\ raph" ), content := 0, count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "Ref", next := 442424, root := ~, start := 442399, stop := 442423 ), rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n or\n ", count := [ "A", 1, 0, 26 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EmptyDigrap\ h" ), content := 0, count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "Ref\ ", next := 442506, root := ~, start := 442480, stop := 442505 ), rec( HTML := true , content := "\ \n ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 26 ], name := "Alt", next := 442523, root := ~, start := 442435, stop := 442522 ), rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 26 ], name := "Item", next := 442539, root := ~, start := 442382, stop := 442538 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 28 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 28 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 28 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 28 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the arguments of ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "NullDigraph" ), content := 0 , count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "Ref", next := 442682, root := ~, start := 442643, stop := 4426\ 81 ), rec( HTML := true, content := " and ", count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EmptyD\ igraph" ), content := 0, count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "Ref\ ", next := 442713, root := ~, start := 442687, stop := 442712 ), rec( HTML := true , content := "\ are just\n non-negative integers. Since digraphs in ", count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 1, 0, 28 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ are immutable,\n there is no reason (although it is possible) to c\ reate a null digraph\n and then add edges, as you might be used to \ doing in ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 28 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 28 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 28 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 28 ], name := "Alt", next := 442974, root := ~, start := 442565, stop := 442973 ), rec( content := "\n \ ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], name := "Item", next := 442990, root := ~, start := 442548, stop := 442989 ) ], count := [ "A", 1, 0, 23 ], name := "Row", next := 443003, root := ~, start := 442310, stop := 443002 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 29 ], name := "HorLine", next := 443020, root := ~, start := 443010, stop := 443019 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ompleteGraph", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "C", next := 443079, root := ~, start := 443059, stop := 443078 ), rec( content := "\n \ ", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 30 ], name := "Item", next := 443095, root := ~, start := 443042, stop := 443094 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Complet\ eDigraph" ), content := 0, count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "Ref", next := 443150, root := ~, start := 443121, stop := 443149 ), rec( content := "\n \ ", count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 32 ], name := "Item", next := 443166, root := ~, start := 443104, stop := 443165 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 34 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 34 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 34 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 34 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the argument to ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "CompleteDigraph" ), content := 0 , count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "Ref", next := 443311, root := ~, start := 443268, stop := 4433\ 10 ), rec( HTML := true, content := " is simply a non-negative integer\n \ indicating the number of vertices. By default a complete digraph in\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ does not have any loops. If you want a complete digraph\n with loo\ ps at every vertex, then use ", count := [ "A", 1, 0, 34 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "DigraphAddAllLoops" ), content := 0 , count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "Ref", next := 443599, root := ~, start := 443553, stop := 4435\ 98 ), rec( HTML := true, content := ".\n ", count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], name := "Alt", next := 443617, root := ~, start := 443192, stop := 443616 ), rec( content := "\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], name := "Item", next := 443633, root := ~, start := 443175, stop := 443632 ) ], count := [ "A", 1, 0, 29 ], name := "Row", next := 443646, root := ~, start := 443028, stop := 443645 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 35 ], name := "HorLine", next := 443663, root := ~, start := 443653, stop := 443662 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ ohnsonGraph", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "C", next := 443721, root := ~, start := 443702, stop := 443720 ), rec( content := "\n \ ", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 36 ], name := "Item", next := 443737, root := ~, start := 443685, stop := 443736 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Johnson\ Digraph" ), content := 0, count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 443791, root := ~, start := 443763, stop := 443790 ), rec( content := "\n \ ", count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 38 ], name := "Item", next := 443807, root := ~, start := 443746, stop := 443806 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 40 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 40 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 40 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 40 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "JohnsonGraph", count :=\ [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], mathmode := \ "Text", name := "C", next := 443903, root := ~, start := 443884, stop := 44390\ 2 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 1, 0, 40 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Johnso\ nDigraph" ), content := 0, count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "Ref\ ", next := 443961, root := ~, start := 443933, stop := 443960 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 40 ], name := "Alt", next := 443979, root := ~, start := 443833, stop := 443978 ), rec( content := "\n \ ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], name := "Item", next := 443995, root := ~, start := 443816, stop := 443994 ) ], count := [ "A", 1, 0, 35 ], name := "Row", next := 444008, root := ~, start := 443671, stop := 444007 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 41 ], name := "HorLine", next := 444025, root := ~, start := 444015, stop := 444024 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ayleyGraph", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "C", next := 444082, root := ~, start := 444064, stop := 444081 ), rec( content := "\n \ ", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 42 ], name := "Item", next := 444098, root := ~, start := 444047, stop := 444097 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CayleyD\ igraph" ), content := 0, count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "Ref", next := 444151, root := ~, start := 444124, stop := 444150 ), rec( content := "\n \ ", count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 44 ], name := "Item", next := 444167, root := ~, start := 444107, stop := 444166 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "CayleyDigraph" ), content := 0 , count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "Ref", next := 444264, root := ~, start := 444237, stop := 4442\ 63 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "CayleyGraph", count := [ "A", 1, 0, 46 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "C", next := 444315, root := ~, start := 444297, stop := 44431\ 4 ), rec( HTML := true, content := " in ", count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 46 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 46 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 46 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 46 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 46 ], name := "Alt", next := 444344, root := ~, start := 444193, stop := 444343 ), rec( content := "\n \ ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], name := "Item", next := 444360, root := ~, start := 444176, stop := 444359 ) ], count := [ "A", 1, 0, 41 ], name := "Row", next := 444373, root := ~, start := 444033, stop := 444372 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 47 ], name := "HorLine", next := 444390, root := ~, start := 444380, stop := 444389 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ ddEdgeOrbit", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "C", next := 444448, root := ~, start := 444429, stop := 444447 ), rec( content := "\n \ ", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 48 ], name := "Item", next := 444464, root := ~, start := 444412, stop := 444463 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ AddEdgeOrbit" ), content := 0, count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "Ref", next := 444523, root := ~, start := 444490, stop := 444522 ), rec( content := "\n \ ", count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 50 ], name := "Item", next := 444539, root := ~, start := 444473, stop := 444538 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 52 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 52 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 52 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 52 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 52 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , digraphs in ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ are immutable,\n and this operation returns a new digraph, i.e. it\ does not modify the\n digraph given as an argument.\n ", count := [ "A", 1, 0, 52 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], name := "Alt", next := 444803, root := ~, start := 444565, stop := 444802 ), rec( content := "\n \ ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], name := "Item", next := 444819, root := ~, start := 444548, stop := 444818 ) ], count := [ "A", 1, 0, 47 ], name := "Row", next := 444832, root := ~, start := 444398, stop := 444831 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 53 ], name := "HorLine", next := 444849, root := ~, start := 444839, stop := 444848 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ emoveEdgeOrbit", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "C", next := 444910, root := ~, start := 444888, stop := 444909 ), rec( content := "\n \ ", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 54 ], name := "Item", next := 444926, root := ~, start := 444871, stop := 444925 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ RemoveEdgeOrbit" ), content := 0, count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "Ref", next := 444988, root := ~, start := 444952, stop := 444987 ), rec( content := "\n \ ", count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 56 ], name := "Item", next := 445004, root := ~, start := 444935, stop := 445003 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 58 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 58 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 58 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 58 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 58 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , digraphs in ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ are immutable,\n and this operation returns a new digraph, i.e. it\ does not modify the\n digraph given as an argument.\n ", count := [ "A", 1, 0, 58 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], name := "Alt", next := 445268, root := ~, start := 445030, stop := 445267 ), rec( content := "\n \ ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], name := "Item", next := 445284, root := ~, start := 445013, stop := 445283 ) ], count := [ "A", 1, 0, 53 ], name := "Row", next := 445297, root := ~, start := 444857, stop := 445296 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 59 ], name := "HorLine", next := 445314, root := ~, start := 445304, stop := 445313 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ ssignVertexNames", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "C", next := 445377, root := ~, start := 445353, stop := 445376 ), rec( content := "\n \ ", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 60 ], name := "Item", next := 445393, root := ~, start := 445336, stop := 445392 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "SetDigr\ aphVertexLabels" ), content := 0, count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "Ref", next := 445455, root := ~, start := 445419, stop := 445454 ), rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n and\n ", count := [ "A", 1, 0, 62 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "SetDigraph\ VertexLabel" ), content := 0, count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "Ref\ ", next := 445547, root := ~, start := 445512, stop := 445546 ), rec( HTML := true , content := "\ \n ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 62 ], name := "Alt", next := 445564, root := ~, start := 445466, stop := 445563 ), rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 62 ], name := "Item", next := 445580, root := ~, start := 445402, stop := 445579 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Use ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "SetDigraphVertexLabel" ), content := 0 , count := [ "A", 1, 0, 64 ], mathmode := \ "Text", name := "Ref", next := 445676, root := ~, start := 445641, stop := 4456\ 75 ), rec( HTML := true, content := " to set an individual\n vertex\ label and ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( Oper := "SetDigraphVertexLabels" ), content := 0, count := [ "A", 1, 0, 64 ], mathmode := \ "Text", name := "Ref", next := 445763, root := ~, start := 445727, stop := 4457\ 62 ), rec( HTML := true, content := " to set all of\n the vertex la\ bels.\n ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 1, 0, 64 ], name := "Alt", next := 445825, root := ~, start := 445606, stop := 445824 ), rec( content := "\n \ ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 64 ], name := "Item", next := 445841, root := ~, start := 445589, stop := 445840 ) ], count := [ "A", 1, 0, 59 ], name := "Row", next := 445854, root := ~, start := 445322, stop := 445853 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 65 ], name := "HorLine", next := 445871, root := ~, start := 445861, stop := 445870 ) ], count := [ "A", 1, 0, 4 ], mathmode := "Text", name := "Table", next := 445884, root := ~, start := 441255, stop := 445883 ), rec( content := "\n ", count := [ "A", 1, 0, 66 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 1 ], mathmode := "Text", name := "Section", next := 445897, root := ~, start := 441015, stop := 445896 ), rec( content := "\n\n ", count := [ "A", 1, 0, 66 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 2, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to insp\ ect graphs, vertices and edges\n ", count := [ "A", 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 445993, root := ~, start := 445915, stop := 445992 ), rec( content := "\n ", count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 3 ], name := "E", next := 446131, root := ~, start := 446021, stop := 446130 ), rec( content := "\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 446142, root := ~, start := 445998, stop := 446141 ), rec( content := "\n\n ", count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 4 ], name := "HorLine", next := 446188, root := ~, start := 446178, stop := 446187 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 2, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 2, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 5 ], name := "Item", next := 446258, root := ~, start := 446210, stop := 446257 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 7 ], name := "Item", next := 446318, root := ~, start := 446267, stop := 446317 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 2, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 9 ], name := "Alt", next := 446399, root := ~, start := 446344, stop := 446398 ), rec( content := "\n \ ", count := [ "A", 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 9 ], name := "Item", next := 446415, root := ~, start := 446327, stop := 446414 ) ], count := [ "A", 2, 0, 4 ], name := "Row", next := 446428, root := ~, start := 446196, stop := 446427 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 10 ], name := "HorLine", next := 446445, root := ~, start := 446435, stop := 446444 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sGraph", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "C", next := 446498, root := ~, start := 446484, stop := 446497 ), rec( content := "\n \ ", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 11 ], name := "Item", next := 446514, root := ~, start := 446467, stop := 446513 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsDigra\ ph" ), content := 0, count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "Ref", next := 446563, root := ~, start := 446540, stop := 446562 ), rec( content := "\n \ ", count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 13 ], name := "Item", next := 446579, root := ~, start := 446523, stop := 446578 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Filt := "IsDigraph" ), content := 0, count := [ "A", 2, 0, 15 ], mathmode := \ "Text", name := "Ref", next := 446658, root := ~, start := 446635, stop := 4466\ 57 ), rec( HTML := true, content := " is a category and not a function like\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsGraph", count := [ "A\ ", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], mathmode := \ "Text", name := "C", next := 446723, root := ~, start := 446709, stop := 44672\ 2 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 15 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], name := "Alt", next := 446741, root := ~, start := 446605, stop := 446740 ), rec( content := "\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], name := "Item", next := 446757, root := ~, start := 446588, stop := 446756 ) ], count := [ "A", 2, 0, 10 ], name := "Row", next := 446770, root := ~, start := 446453, stop := 446769 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 16 ], name := "HorLine", next := 446787, root := ~, start := 446777, stop := 446786 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ rderGraph", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "C", next := 446843, root := ~, start := 446826, stop := 446842 ), rec( content := "\n \ ", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 17 ], name := "Item", next := 446859, root := ~, start := 446809, stop := 446858 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ NrVertices" ), content := 0, count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "Ref", next := 446916, root := ~, start := 446885, stop := 446915 ), rec( content := "\n \ ", count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 19 ], name := "Item", next := 446932, root := ~, start := 446868, stop := 446931 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphNrVertices" ), content := 0 , count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "Ref", next := 447033, root := ~, start := 447002, stop := 4470\ 32 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "OrderGraph", count := [ "A", 2, 0, 21 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "C", next := 447083, root := ~, start := 447066, stop := 44708\ 2 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 21 ], name := "Alt", next := 447112, root := ~, start := 446958, stop := 447111 ), rec( content := "\n \ ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], name := "Item", next := 447128, root := ~, start := 446941, stop := 447127 ) ], count := [ "A", 2, 0, 16 ], name := "Row", next := 447141, root := ~, start := 446795, stop := 447140 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 22 ], name := "HorLine", next := 447158, root := ~, start := 447148, stop := 447157 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sVertex(", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "graph", count := [ "A", 2, 0, 23 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], mathmode := \ "Text", name := "A", next := 447221, root := ~, start := 447209, stop := 44722\ 0 ), rec( content := ", ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "A", next := 4472\ 31, root := ~, start := 447223, stop := 447230 ), rec( content := ")", count := [ "A", 2, 0, 23 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "C", next := 447236, root := ~, start := 447197, stop := 447235 ), rec( content := "\n \ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], name := "Item", next := 447252, root := ~, start := 447180, stop := 447251 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "v", count := [ "A", 2, 0, 25 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "A", next := 447289, root := ~, start := 447281, stop := 44728\ 8 ), rec( content := " in DigraphVertices(", count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "A", next := 4473\ 23, root := ~, start := 447309, stop := 447322 ), rec( content := ")", count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "C", next := 447328, root := ~, start := 447278, stop := 447327 ), rec( content := "\n \ ", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], name := "Item", next := 447344, root := ~, start := 447261, stop := 447343 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphVertices" ), content := 0 , count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "Ref", next := 447443, root := ~, start := 447414, stop := 4474\ 42 ), rec( HTML := true, content := " can be used to return\n the l\ ist of vertices of a digraph, or if ", count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "v", count := [ "A", 2, 0, 27 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "A", next := 4475\ 27, root := ~, start := 447519, stop := 447526 ), rec( HTML := true, content := "\ is known to be\n a positive integer, then you can simply do\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( HTML := \ true, content := "v", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], mathmode := "Text", name \ := "A", next := 447621, root := ~, start := 447613, stop := 447620 ), rec( HTML\ := true, content := " <= DigraphNrVertices(", count := [ "A", 2, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( HTML := true, content := "digraph", count := [ "A", 2, 0, 27 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ] , math\ mode := "Text", name := "A", next := 447660, root := ~, start := 447646, stop \ := 447659 ), rec( HTML := true, content := ")", count := [ "A", 2, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "C", next := 447665, root := ~, start := 447610, stop := 44766\ 4 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], name := "Alt", next := 447682, root := ~, start := 447370, stop := 447681 ), rec( content := "\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], name := "Item", next := 447698, root := ~, start := 447353, stop := 447697 ) ], count := [ "A", 2, 0, 22 ], name := "Row", next := 447711, root := ~, start := 447166, stop := 447710 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 28 ], name := "HorLine", next := 447728, root := ~, start := 447718, stop := 447727 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexName", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "C", next := 447784, root := ~, start := 447767, stop := 447783 ), rec( content := "\n \ ", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 29 ], name := "Item", next := 447800, root := ~, start := 447750, stop := 447799 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabel" ), content := 0, count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "Ref", next := 447858, root := ~, start := 447826, stop := 447857 ), rec( content := "\n \ ", count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 31 ], name := "Item", next := 447874, root := ~, start := 447809, stop := 447873 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Unlike ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "VertexName", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "C", next := 4479\ 54, root := ~, start := 447937, stop := 447953 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the return value of\n ", count := [ "A", 2, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digrap\ hVertexLabel" ), content := 0, count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "Ref\ ", next := 448031, root := ~, start := 447999, stop := 448030 ), rec( HTML := true , content := "\ is mutable.\n ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], name := "Alt", next := 448060, root := ~, start := 447900, stop := 448059 ), rec( content := "\n \ ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], name := "Item", next := 448076, root := ~, start := 447883, stop := 448075 ) ], count := [ "A", 2, 0, 28 ], name := "Row", next := 448089, root := ~, start := 447736, stop := 448088 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 34 ], name := "HorLine", next := 448106, root := ~, start := 448096, stop := 448105 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexNames", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "C", next := 448163, root := ~, start := 448145, stop := 448162 ), rec( content := "\n \ ", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 35 ], name := "Item", next := 448179, root := ~, start := 448128, stop := 448178 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "Ref", next := 448238, root := ~, start := 448205, stop := 448237 ), rec( content := "\n \ ", count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 37 ], name := "Item", next := 448254, root := ~, start := 448188, stop := 448253 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Unlike ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "VertexNames", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "C", next := 4483\ 35, root := ~, start := 448317, stop := 448334 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 39 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 39 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 39 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the return value of\n ", count := [ "A", 2, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digrap\ hVertexLabels" ), content := 0, count := [ "A", 2, 0, 39 ], mathmode := "Text" , name := "Ref\ ", next := 448413, root := ~, start := 448380, stop := 448412 ), rec( HTML := true , content := "\ is mutable.\n ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], name := "Alt", next := 448442, root := ~, start := 448280, stop := 448441 ), rec( content := "\n \ ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], name := "Item", next := 448458, root := ~, start := 448263, stop := 448457 ) ], count := [ "A", 2, 0, 34 ], name := "Row", next := 448471, root := ~, start := 448114, stop := 448470 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 40 ], name := "HorLine", next := 448488, root := ~, start := 448478, stop := 448487 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertices", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "C", next := 448542, root := ~, start := 448527, stop := 448541 ), rec( content := "\n \ ", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 41 ], name := "Item", next := 448558, root := ~, start := 448510, stop := 448557 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "Ref", next := 448613, root := ~, start := 448584, stop := 448612 ), rec( content := "\n \ ", count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 43 ], name := "Item", next := 448629, root := ~, start := 448567, stop := 448628 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The vertices of a digraph in ", count := [ "A", 2, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 45 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ is always the range ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "[1\n .. n]", count := [ "A", 2, 0, 45 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "\ Text", name := "C", next := 448774, root := ~, start := 448745, stop := 448773\ ), rec( HTML := true, content := " where ", count := [ "A", 2, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "n", count := [ "A", 2, 0, 45 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "C", next := 4487\ 89, root := ~, start := 448781, stop := 448788 ), rec( HTML := true, content := "\ is the number of vertices of the\n digraph. The operation ", count := [ "A", 2, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digrap\ hVertices" ), content := 0, count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "Ref\ ", next := 448888, root := ~, start := 448859, stop := 448887 ), rec( HTML := true , content := "\ behaves the\n same as ", count := [ "A", 2, 0, 45 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Vertices", count := [ "A", 2, 0, 45 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "T\ ext", name := "C", next := 448936, root := ~, start := 448921, stop := 448935 ), rec( HTML := true , content := "\ in ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 45 ], name := "Alt", next := 448965, root := ~, start := 448655, stop := 448964 ), rec( content := "\n \ ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], name := "Item", next := 448981, root := ~, start := 448638, stop := 448980 ) ], count := [ "A", 2, 0, 40 ], name := "Row", next := 448994, root := ~, start := 448496, stop := 448993 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 46 ], name := "HorLine", next := 449011, root := ~, start := 449001, stop := 449010 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexDegree", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "C", next := 449069, root := ~, start := 449050, stop := 449068 ), rec( content := "\n \ ", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 47 ], name := "Item", next := 449085, root := ~, start := 449033, stop := 449084 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "Ref", next := 449142, root := ~, start := 449111, stop := 449141 ), rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also: ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "InDegreeOfVertex" ), content := 0 , count := [ "A", 2, 0, 49 ], mathmode := \ "Text", name := "Ref", next := 449224, root := ~, start := 449194, stop := 4492\ 23 ), rec( HTML := true, content := ")\n ", count := [ "A", 2, 0, 49 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 49 ], name := "Alt", next := 449242, root := ~, start := 449153, stop := 449241 ), rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 49 ], name := "Item", next := 449258, root := ~, start := 449094, stop := 449257 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "OutDegreeOfVertex" ), content := 0 , count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "Ref", next := 449359, root := ~, start := 449328, stop := 4493\ 58 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "VertexDegree", count := [ "A", 2, 0, 51 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "C", next := 449411, root := ~, start := 449392, stop := 44941\ 0 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 51 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 51 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 51 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 51 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 51 ], name := "Alt", next := 449440, root := ~, start := 449284, stop := 449439 ), rec( content := "\n \ ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], name := "Item", next := 449456, root := ~, start := 449267, stop := 449455 ) ], count := [ "A", 2, 0, 46 ], name := "Row", next := 449469, root := ~, start := 449019, stop := 449468 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 52 ], name := "HorLine", next := 449486, root := ~, start := 449476, stop := 449485 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexDegrees", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "C", next := 449545, root := ~, start := 449525, stop := 449544 ), rec( content := "\n \ ", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 53 ], name := "Item", next := 449561, root := ~, start := 449508, stop := 449560 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDegr\ eeSet" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref", next := 449613, root := ~, start := 449587, stop := 449612 ), rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also:\n ", count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDeg\ reeSequence" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref\ ", next := 449708, root := ~, start := 449677, stop := 449707 ), rec( HTML := true , content := "\ ,\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "OutDegrees" ), content := 0 , count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "Ref", next := 449746, root := ~, start := 449722, stop := 4497\ 45 ), rec( HTML := true, content := ",\n ", count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "InDegr\ ees" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref\ ", next := 449783, root := ~, start := 449760, stop := 449782 ), rec( HTML := true , content := "\ ,\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "InDegreeSequence" ), content := 0 , count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "Ref", next := 449827, root := ~, start := 449797, stop := 4498\ 26 ), rec( HTML := true, content := ",\n ", count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "InDegr\ eeSet" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref\ ", next := 449866, root := ~, start := 449841, stop := 449865 ), rec( HTML := true , content := "\ )\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 55 ], name := "Alt", next := 449884, root := ~, start := 449624, stop := 449883 ), rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 55 ], name := "Item", next := 449900, root := ~, start := 449570, stop := 449899 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "OutDegreeSet" ), content := 0 , count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "Ref", next := 449996, root := ~, start := 449970, stop := 4499\ 95 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "VertexDegrees", count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "C", next := 450049, root := ~, start := 450029, stop := 45004\ 8 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 57 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 57 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 57 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 57 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 57 ], name := "Alt", next := 450078, root := ~, start := 449926, stop := 450077 ), rec( content := "\n \ ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], name := "Item", next := 450094, root := ~, start := 449909, stop := 450093 ) ], count := [ "A", 2, 0, 52 ], name := "Row", next := 450107, root := ~, start := 449494, stop := 450106 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 58 ], name := "HorLine", next := 450124, root := ~, start := 450114, stop := 450123 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sLoopy", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "C", next := 450177, root := ~, start := 450163, stop := 450176 ), rec( content := "\n \ ", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 59 ], name := "Item", next := 450193, root := ~, start := 450146, stop := 450192 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "Ref", next := 450248, root := ~, start := 450219, stop := 450247 ), rec( content := "\n \ ", count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 61 ], name := "Item", next := 450264, root := ~, start := 450202, stop := 450263 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "DigraphHasLoops" ), content := 0 , count := [ "A", 2, 0, 63 ], mathmode := \ "Text", name := "Ref", next := 450363, root := ~, start := 450334, stop := 4503\ 62 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "IsLoopy", count := [ "A", 2, 0, 63 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "C", next := 4504\ 10, root := ~, start := 450396, stop := 450409 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 63 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 63 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 63 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 63 ], name := "Alt", next := 450439, root := ~, start := 450290, stop := 450438 ), rec( content := "\n \ ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 63 ], name := "Item", next := 450455, root := ~, start := 450273, stop := 450454 ) ], count := [ "A", 2, 0, 58 ], name := "Row", next := 450468, root := ~, start := 450132, stop := 450467 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 64 ], name := "HorLine", next := 450485, root := ~, start := 450475, stop := 450484 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sSimpleGraph", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "C", next := 450544, root := ~, start := 450524, stop := 450543 ), rec( content := "\n \ ", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 65 ], name := "Item", next := 450560, root := ~, start := 450507, stop := 450559 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Prop := "IsSymmetricDigraph" ), content := 0 , count := [ "A", 2, 0, 67 ], mathmode := \ "Text", name := "Ref", next := 450648, root := ~, start := 450616, stop := 4506\ 47 ), rec( HTML := true, content := "\n and not ", count := [ "A", 2, 0, 67 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digrap\ hHasLoops" ), content := 0, count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "Ref\ ", next := 450699, root := ~, start := 450670, stop := 450698 ), rec( HTML := true , content := "\ \n ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 67 ], name := "Alt", next := 450716, root := ~, start := 450586, stop := 450715 ), rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\\ n ", count := [ "A", 2, 0, 67 ], name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Prop := "IsSymmetricDigraph" ), content := 0, count := [ "A", 2, 0, 67 ], name := "Ref\ ", next := 450788, root := ~, start := 450756, stop := 450787 ), rec( content := "\ \n ", count := [ "A", 2, 0, 67 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "Alt", next := 450805, root := ~, start := 450727, stop := 450804 ), rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 67 ], name := "Item", next := 450821, root := ~, start := 450569, stop := 450820 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 69 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 69 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 69 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 69 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsSimpleGraph", count :\ = [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "C", next := 450918, root := ~, start := 450898, stop := 45091\ 7 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymm\ etricDigraph" ), content := 0, count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "Ref\ ", next := 450980, root := ~, start := 450948, stop := 450979 ), rec( HTML := true , content := "\ \n and not ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "DigraphHasLoops" ), content := 0 , count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "Ref", next := 451031, root := ~, start := 451002, stop := 4510\ 30 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], name := "Alt", next := 451049, root := ~, start := 450847, stop := 451048 ), rec( content := "\n \ ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], name := "Item", next := 451065, root := ~, start := 450830, stop := 451064 ) ], count := [ "A", 2, 0, 64 ], name := "Row", next := 451078, root := ~, start := 450493, stop := 451077 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 70 ], name := "HorLine", next := 451095, root := ~, start := 451085, stop := 451094 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacency", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "C", next := 451150, root := ~, start := 451134, stop := 451149 ), rec( content := "\n \ ", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 71 ], name := "Item", next := 451166, root := ~, start := 451117, stop := 451165 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutNeig\ hboursOfVertex" ), content := 0, count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "Ref", next := 451227, root := ~, start := 451192, stop := 451226 ), rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "InNeighboursOfVertex" ), content := 0 , count := [ "A", 2, 0, 73 ], mathmode := \ "Text", name := "Ref", next := 451312, root := ~, start := 451278, stop := 4513\ 11 ), rec( HTML := true, content := ")\n ", count := [ "A", 2, 0, 73 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 73 ], name := "Alt", next := 451330, root := ~, start := 451238, stop := 451329 ), rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 73 ], name := "Item", next := 451346, root := ~, start := 451175, stop := 451345 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "OutNeighboursOfVertex" ), content := 0 , count := [ "A", 2, 0, 75 ], mathmode := \ "Text", name := "Ref", next := 451451, root := ~, start := 451416, stop := 4514\ 50 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Adjacency", count := [ "A", 2, 0, 75 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], mathmode := "\ Text", name := "C", next := 451500, root := ~, start := 451484, stop := 451499\ ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 75 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 75 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 75 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 75 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 75 ], name := "Alt", next := 451529, root := ~, start := 451372, stop := 451528 ), rec( content := "\n \ ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], name := "Item", next := 451545, root := ~, start := 451355, stop := 451544 ) ], count := [ "A", 2, 0, 70 ], name := "Row", next := 451558, root := ~, start := 451103, stop := 451557 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 76 ], name := "HorLine", next := 451575, root := ~, start := 451565, stop := 451574 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sEdge", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "C", next := 451627, root := ~, start := 451614, stop := 451626 ), rec( content := "\n \ ", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 77 ], name := "Item", next := 451643, root := ~, start := 451597, stop := 451642 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graph and list", Oper := "IsDigraphEdge" ), content := 0, count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "Ref", next := 451725, root := ~, start := 451669, stop := 451724 ), rec( content := "\n \ ", count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 79 ], name := "Item", next := 451741, root := ~, start := 451652, stop := 451740 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The arguments of\n ", count := [ "A", 2, 0, 81 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for d\ igraph and list", Oper := "IsDigraphEdge" ), content := 0, count := [ "A", 2, 0, 81 ], mathmode := \ "Text", name := "Ref", next := 451882, root := ~, start := 451826, stop := 4518\ 81 ), rec( HTML := true, content := "\n can be a digraph and an edg\ e, or a digraph and two vertices. In the\n first form the operation\ \n ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Label := "for digraph and list", Oper := \ "IsDigraphEdge" ), content := 0, count := [ "A", 2, 0, 81 ], mathmode := "Text\ ", name := "Ref", next := 452068, root := ~, start := 452012, stop := 452067 ) , rec( HTML := true , content := "\ \n behaves the same as ", count := [ "A", 2, 0, 81 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "IsEdge", count := [ "A", 2, 0, 81 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "C", next := 4521\ 14, root := ~, start := 452101, stop := 452113 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 81 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 81 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 81 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 81 ], name := "Alt", next := 452143, root := ~, start := 451767, stop := 452142 ), rec( content := "\n \ ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 81 ], name := "Item", next := 452159, root := ~, start := 451750, stop := 452158 ) ], count := [ "A", 2, 0, 76 ], name := "Row", next := 452172, root := ~, start := 451583, stop := 452171 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 82 ], name := "HorLine", next := 452189, root := ~, start := 452179, stop := 452188 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ irectedEdges", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "C", next := 452248, root := ~, start := 452228, stop := 452247 ), rec( content := "\n \ ", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 83 ], name := "Item", next := 452264, root := ~, start := 452211, stop := 452263 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Edges" ), content := 0, count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "Ref", next := 452316, root := ~, start := 452290, stop := 452315 ), rec( content := "\n \ ", count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 85 ], name := "Item", next := 452332, root := ~, start := 452273, stop := 452331 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphEdges" ), content := 0 , count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "Ref", next := 452428, root := ~, start := 452402, stop := 4524\ 27 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "DirectedEdges", count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "C", next := 452481, root := ~, start := 452461, stop := 45248\ 0 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 87 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 87 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 87 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 87 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 87 ], name := "Alt", next := 452510, root := ~, start := 452358, stop := 452509 ), rec( content := "\n \ ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], name := "Item", next := 452526, root := ~, start := 452341, stop := 452525 ) ], count := [ "A", 2, 0, 82 ], name := "Row", next := 452539, root := ~, start := 452197, stop := 452538 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 88 ], name := "HorLine", next := 452556, root := ~, start := 452546, stop := 452555 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedEdges", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "C", next := 452617, root := ~, start := 452595, stop := 452616 ), rec( content := "\n \ ", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 89 ], name := "Item", next := 452633, root := ~, start := 452578, stop := 452632 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 2, 0, 91 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 91 ], name := "Item", next := 452679, root := ~, start := 452642, stop := 452678 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 2, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 93 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 2, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], name := "Alt", next := 452838, root := ~, start := 452705, stop := 452837 ), rec( content := "\n \ ", count := [ "A", 2, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], name := "Item", next := 452854, root := ~, start := 452688, stop := 452853 ) ], count := [ "A", 2, 0, 88 ], name := "Row", next := 452867, root := ~, start := 452564, stop := 452866 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 94 ], name := "HorLine", next := 452884, root := ~, start := 452874, stop := 452883 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istance", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "C", next := 452938, root := ~, start := 452923, stop := 452937 ), rec( content := "\n \ ", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 95 ], name := "Item", next := 452954, root := ~, start := 452906, stop := 452953 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and two vertices", Oper := "DigraphShortestDistance" ), content := 0, count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "Ref", next := 453071, root := ~, start := 452980, stop := 453070 ), rec( content := "\n \ ", count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 97 ], name := "Item", next := 453087, root := ~, start := 452963, stop := 453086 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Label := "for a digraph and two ver\ tices", Oper := "DigraphShortestDistance" ), content := 0, count := [ "A", 2, 0, 99 ], mathmode := \ "Text", name := "Ref", next := 453245, root := ~, start := 453157, stop := 4532\ 44 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 99 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Distance", count := [ "A", 2, 0, 99 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 99 ], mathmode := "T\ ext", name := "C", next := 453293, root := ~, start := 453278, stop := 453292 ), rec( HTML := true , content := "\ in ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 99 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 99 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 99 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 99 ], name := "Alt", next := 453322, root := ~, start := 453113, stop := 453321 ), rec( content := "\n \ ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 99 ], name := "Item", next := 453338, root := ~, start := 453096, stop := 453337 ) ], count := [ "A", 2, 0, 94 ], name := "Row", next := 453351, root := ~, start := 452892, stop := 453350 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 100 ], name := "HorLine", next := 453368, root := ~, start := 453358, stop := 453367 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ iameter", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 101 ] , mathmode := "Text", name := "C", next := 453422, root := ~, start := 453407, stop := 453421 ), rec( content := "\n \ ", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 101 ], name := "Item", next := 453438, root := ~, start := 453390, stop := 453437 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Diameter" ), content := 0, count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "Ref", next := 453493, root := ~, start := 453464, stop := 453492 ), rec( content := "\n \ ", count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 103 ], name := "Item", next := 453509, root := ~, start := 453447, stop := 453508 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 105 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphDiameter" ), content := 0 , count := [ "A", 2, 0, 105 ] , mathmode := \ "Text", name := "Ref", next := 453608, root := ~, start := 453579, stop := 4536\ 07 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 105 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Diameter", count := [ "A", 2, 0, 105 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], mathmode := \ "Text", name := "C", next := 453656, root := ~, start := 453641, stop := 45365\ 5 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 105 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 105 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 105 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 105 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 105 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 105 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 105 ], name := "Alt", next := 453685, root := ~, start := 453535, stop := 453684 ), rec( content := "\n \ ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], name := "Item", next := 453701, root := ~, start := 453518, stop := 453700 ) ], count := [ "A", 2, 0, 100 ], name := "Row", next := 453714, root := ~, start := 453376, stop := 453713 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 106 ], name := "HorLine", next := 453731, root := ~, start := 453721, stop := 453730 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ irth", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 107 ] , mathmode := "Text", name := "C", next := 453782, root := ~, start := 453770, stop := 453781 ), rec( content := "\n \ ", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 107 ], name := "Item", next := 453798, root := ~, start := 453753, stop := 453797 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ UndirectedGirth" ), content := 0, count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "Ref", next := 453860, root := ~, start := 453824, stop := 453859 ), rec( content := "\n \ ", count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 109 ], name := "Item", next := 453876, root := ~, start := 453807, stop := 453875 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 111 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphUndirectedGirth" ), content := 0 , count := [ "A", 2, 0, 111 ] , mathmode := \ "Text", name := "Ref", next := 453982, root := ~, start := 453946, stop := 4539\ 81 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 111 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Girth", count := [ "A", 2, 0, 111 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "C", next := 4540\ 27, root := ~, start := 454015, stop := 454026 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 111 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 111 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 111 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 111 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ except that in the case that the\n argument is a forest, ", count := [ "A", 2, 0, 111 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "infinity", count := [ "A", 2, 0, 111 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := \ "Text", name := "K", next := 454121, root := ~, start := 454106, stop := 45412\ 0 ), rec( HTML := true, content := " is returned by ", count := [ "A", 2, 0, 111 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 111 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ \n whereas ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "-1", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "C", next := 4541\ 77, root := ~, start := 454168, stop := 454176 ), rec( HTML := true, content := "\ is returned by ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 111 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 111 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 111 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 111 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 111 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 111 ], name := "Alt", next := 454218, root := ~, start := 453902, stop := 454217 ), rec( content := "\n \ ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], name := "Item", next := 454234, root := ~, start := 453885, stop := 454233 ) ], count := [ "A", 2, 0, 106 ], name := "Row", next := 454247, root := ~, start := 453739, stop := 454246 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 112 ], name := "HorLine", next := 454264, root := ~, start := 454254, stop := 454263 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sConnectedGraph", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "C", next := 454326, root := ~, start := 454303, stop := 454325 ), rec( content := "\n \ ", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 113 ], name := "Item", next := 454342, root := ~, start := 454286, stop := 454341 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsStron\ glyConnectedDigraph" ), content := 0, count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "Ref", next := 454408, root := ~, start := 454368, stop := 454407 ), rec( content := "\n \ ", count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 115 ], name := "Item", next := 454424, root := ~, start := 454351, stop := 454423 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 117 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsStronglyConnectedDigraph\ " ), content := 0, count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "Ref\ ", next := 454534, root := ~, start := 454494, stop := 454533 ), rec( HTML := true , content := "\ behaves the\n same as ", count := [ "A", 2, 0, 117 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "IsConnectedGraph", count := [ "A", 2, 0, 117 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], mathmode := \ "Text", name := "C", next := 454590, root := ~, start := 454567, stop := 45458\ 9 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 117 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 117 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 117 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 117 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 117 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 117 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 117 ], name := "Alt", next := 454619, root := ~, start := 454450, stop := 454618 ), rec( content := "\n \ ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], name := "Item", next := 454635, root := ~, start := 454433, stop := 454634 ) ], count := [ "A", 2, 0, 112 ], name := "Row", next := 454648, root := ~, start := 454272, stop := 454647 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 118 ], name := "HorLine", next := 454665, root := ~, start := 454655, stop := 454664 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBipartite", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 2, 0, 119 ] , mathmode := "Text", name := "C", next := 454722, root := ~, start := 454704, stop := 454721 ), rec( content := "\n \ ", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 119 ], name := "Item", next := 454738, root := ~, start := 454687, stop := 454737 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "Ref", next := 454796, root := ~, start := 454764, stop := 454795 ), rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n and ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsSymmetricDigraph" ), content := 0 , count := [ "A", 2, 0, 121 ] , mathmode := \ "Text", name := "Ref", next := 454873, root := ~, start := 454841, stop := 4548\ 72 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 121 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 121 ], name := "Alt", next := 454890, root := ~, start := 454807, stop := 454889 ), rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 121 ], name := "Item", next := 454906, root := ~, start := 454747, stop := 454905 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 123 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 123 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 123 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 123 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 123 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ function ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsBipartite", count := [ \ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ] , mathmode := \ "Text", name := "C", next := 455001, root := ~, start := 454983, stop := 45500\ 0 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 2, 0, 123 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipa\ rtiteDigraph" ), content := 0, count := [ "A", 2, 0, 123 ], mathmode := "Text" , name := "Ref\ ", next := 455063, root := ~, start := 455031, stop := 455062 ), rec( HTML := true , content := "\ and ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Prop := "IsSymmetricDigraph" ), content := 0, count := [ "A", 2, 0, 123 ], mathmode := \ "Text", name := "Ref", next := 455114, root := ~, start := 455068, stop := 4551\ 13 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 123 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], name := "Alt", next := 455132, root := ~, start := 454932, stop := 455131 ), rec( content := "\n \ ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], name := "Item", next := 455148, root := ~, start := 454915, stop := 455147 ) ], count := [ "A", 2, 0, 118 ], name := "Row", next := 455161, root := ~, start := 454673, stop := 455160 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 124 ], name := "HorLine", next := 455178, root := ~, start := 455168, stop := 455177 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sNullGraph", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 2, 0, 125 ] , mathmode := "Text", name := "C", next := 455235, root := ~, start := 455217, stop := 455234 ), rec( content := "\n \ ", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 125 ], name := "Item", next := 455251, root := ~, start := 455200, stop := 455250 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsNullD\ igraph" ), content := 0, count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "Ref", next := 455304, root := ~, start := 455277, stop := 455303 ), rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n or ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsEmptyDigraph" ), content := 0 , count := [ "A", 2, 0, 127 ] , mathmode := \ "Text", name := "Ref", next := 455376, root := ~, start := 455348, stop := 4553\ 75 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 127 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 127 ], name := "Alt", next := 455393, root := ~, start := 455315, stop := 455392 ), rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 127 ], name := "Item", next := 455409, root := ~, start := 455260, stop := 455408 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 129 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 129 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 129 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 129 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 129 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ function ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsNullGraph", count := [ \ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ] , mathmode := \ "Text", name := "C", next := 455504, root := ~, start := 455486, stop := 45550\ 3 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 2, 0, 129 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsEmpt\ yDigraph" ), content := 0, count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "Ref\ ", next := 455562, root := ~, start := 455534, stop := 455561 ), rec( HTML := true , content := "\ and its synonym ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsNullDigraph" ), content := 0 , count := [ "A", 2, 0, 129 ] , mathmode := \ "Text", name := "Ref", next := 455620, root := ~, start := 455579, stop := 4556\ 19 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 129 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], name := "Alt", next := 455638, root := ~, start := 455435, stop := 455637 ), rec( content := "\n \ ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], name := "Item", next := 455654, root := ~, start := 455418, stop := 455653 ) ], count := [ "A", 2, 0, 124 ], name := "Row", next := 455667, root := ~, start := 455186, stop := 455666 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 130 ], name := "HorLine", next := 455684, root := ~, start := 455674, stop := 455683 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCompleteGraph", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "C", next := 455745, root := ~, start := 455723, stop := 455744 ), rec( content := "\n \ ", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 131 ], name := "Item", next := 455761, root := ~, start := 455706, stop := 455760 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsCompl\ eteDigraph" ), content := 0, count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "Ref", next := 455818, root := ~, start := 455787, stop := 455817 ), rec( content := "\n \ ", count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 133 ], name := "Item", next := 455834, root := ~, start := 455770, stop := 455833 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 135 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 135 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 135 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 135 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 135 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ function ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsCompleteGraph", count\ := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ] , mathmode := \ "Text", name := "C", next := 455933, root := ~, start := 455911, stop := 45593\ 2 ), rec( HTML := true, content := " is roughly\n equivalent to\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( Prop := "IsCompleteDigraph" ), content := 0, count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "Ref", next := 456014, root := ~, start := 455983, stop := 4560\ 13 ), rec( HTML := true, content := " except that in ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 135 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ \n a complete digraph has ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "no loops", count := [ "A", 2, 0, 135 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "E", next := 456091, root := ~, start := 456076, stop := 45609\ 0 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], name := "Alt", next := 456109, root := ~, start := 455860, stop := 456108 ), rec( content := "\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], name := "Item", next := 456125, root := ~, start := 455843, stop := 456124 ) ], count := [ "A", 2, 0, 130 ], name := "Row", next := 456138, root := ~, start := 455692, stop := 456137 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 136 ], name := "HorLine", next := 456155, root := ~, start := 456145, stop := 456154 ) ], count := [ "A", 2, 0, 4 ], mathmode := "Text", name := "Table", next := 456168, root := ~, start := 446148, stop := 456167 ), rec( content := "\n ", count := [ "A", 2, 0, 137 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 1 ], mathmode := "Text", name := "Section", next := 456181, root := ~, start := 445901, stop := 456180 ), rec( content := "\n\n ", count := [ "A", 2, 0, 137 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 3, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to dete\ rmine regularity properties of graphs\n ", count := [ "A", 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 456284, root := ~, start := 456199, stop := 456283 ), rec( content := "\n ", count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 3 ], name := "E", next := 456422, root := ~, start := 456312, stop := 456421 ), rec( content := "\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "Alt", next := 456433, root := ~, start := 456289, stop := 456432 ), rec( content := "\n\n ", count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 4 ], name := "HorLine", next := 456479, root := ~, start := 456469, stop := 456478 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 3, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 3, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 5 ], name := "Item", next := 456549, root := ~, start := 456501, stop := 456548 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 7 ], name := "Item", next := 456609, root := ~, start := 456558, stop := 456608 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 3, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 9 ], name := "Alt", next := 456690, root := ~, start := 456635, stop := 456689 ), rec( content := "\n \ ", count := [ "A", 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 9 ], name := "Item", next := 456706, root := ~, start := 456618, stop := 456705 ) ], count := [ "A", 3, 0, 4 ], name := "Row", next := 456719, root := ~, start := 456487, stop := 456718 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 10 ], name := "HorLine", next := 456736, root := ~, start := 456726, stop := 456735 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sRegularGraph", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "C", next := 456796, root := ~, start := 456775, stop := 456795 ), rec( content := "\n \ ", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 11 ], name := "Item", next := 456812, root := ~, start := 456758, stop := 456811 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutRe\ gularDigraph" ), content := 0, count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "Ref", next := 456871, root := ~, start := 456838, stop := 456870 ), rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also\n ", count := [ "A", 3, 0, 13 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsInRe\ gularDigraph" ), content := 0, count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "Ref\ ", next := 456966, root := ~, start := 456934, stop := 456965 ), rec( HTML := true , content := "\ \n and\n ", count := [ "A", 3, 0, 13 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRegularD\ igraph" ), content := 0, count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "Ref\ ", next := 457025, root := ~, start := 456995, stop := 457024 ), rec( HTML := true , content := "\ )\n ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 13 ], name := "Alt", next := 457043, root := ~, start := 456882, stop := 457042 ), rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 13 ], name := "Item", next := 457059, root := ~, start := 456821, stop := 457058 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsRegularGraph", count \ := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], mathmode := \ "Text", name := "C", next := 457157, root := ~, start := 457136, stop := 45715\ 6 ), rec( HTML := true, content := " is\n equivalent to ", count := [ "A", 3, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutR\ egularDigraph" ), content := 0, count := [ "A", 3, 0, 15 ], mathmode := "Text" , name := "Ref\ ", next := 457220, root := ~, start := 457187, stop := 457219 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 15 ], name := "Alt", next := 457238, root := ~, start := 457085, stop := 457237 ), rec( content := "\n \ ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], name := "Item", next := 457254, root := ~, start := 457068, stop := 457253 ) ], count := [ "A", 3, 0, 10 ], name := "Row", next := 457267, root := ~, start := 456744, stop := 457266 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 16 ], name := "HorLine", next := 457284, root := ~, start := 457274, stop := 457283 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ocalParameters", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "C", next := 457345, root := ~, start := 457323, stop := 457344 ), rec( content := "\n \ ", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 17 ], name := "Item", next := 457361, root := ~, start := 457306, stop := 457360 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 19 ], name := "Item", next := 457407, root := ~, start := 457370, stop := 457406 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "LocalParameters", count\ := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], mathmode := \ "Text", name := "C", next := 457505, root := ~, start := 457483, stop := 45750\ 4 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], name := "Alt", next := 457618, root := ~, start := 457432, stop := 457617 ), rec( content := "\n \ ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], name := "Item", next := 457634, root := ~, start := 457416, stop := 457633 ) ], count := [ "A", 3, 0, 16 ], name := "Row", next := 457647, root := ~, start := 457292, stop := 457646 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 22 ], name := "HorLine", next := 457664, root := ~, start := 457654, stop := 457663 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ lobalParameters", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "C", next := 457726, root := ~, start := 457703, stop := 457725 ), rec( content := "\n \ ", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 23 ], name := "Item", next := 457742, root := ~, start := 457686, stop := 457741 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 25 ], name := "Item", next := 457788, root := ~, start := 457751, stop := 457787 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 27 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "GlobalParameters", coun\ t := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], mathmode := \ "Text", name := "C", next := 457887, root := ~, start := 457864, stop := 45788\ 6 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], name := "Alt", next := 458000, root := ~, start := 457813, stop := 457999 ), rec( content := "\n \ ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], name := "Item", next := 458016, root := ~, start := 457797, stop := 458015 ) ], count := [ "A", 3, 0, 22 ], name := "Row", next := 458029, root := ~, start := 457672, stop := 458028 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 28 ], name := "HorLine", next := 458046, root := ~, start := 458036, stop := 458045 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDistanceRegular", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "C", next := 458109, root := ~, start := 458085, stop := 458108 ), rec( content := "\n \ ", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 29 ], name := "Item", next := 458125, root := ~, start := 458068, stop := 458124 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsDista\ nceRegularDigraph" ), content := 0, count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "Ref", next := 458189, root := ~, start := 458151, stop := 458188 ), rec( content := "\n \ ", count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 31 ], name := "Item", next := 458205, root := ~, start := 458134, stop := 458204 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 33 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsDistanceRegular", coun\ t := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "C", next := 458306, root := ~, start := 458282, stop := 45830\ 5 ), rec( HTML := true, content := " is\n equivalent to ", count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsDist\ anceRegularDigraph" ), content := 0, count := [ "A", 3, 0, 33 ], mathmode := "\ Text", name := "Ref", next := 458374, root := ~, start := 458336, stop := 4583\ 73 ), rec( HTML := true, content := ".\n ", count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], name := "Alt", next := 458392, root := ~, start := 458231, stop := 458391 ), rec( content := "\n \ ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], name := "Item", next := 458408, root := ~, start := 458214, stop := 458407 ) ], count := [ "A", 3, 0, 28 ], name := "Row", next := 458421, root := ~, start := 458054, stop := 458420 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 34 ], name := "HorLine", next := 458438, root := ~, start := 458428, stop := 458437 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ollapsedAdjacencyMat", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "C", next := 458505, root := ~, start := 458477, stop := 458504 ), rec( content := "\n \ ", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 35 ], name := "Item", next := 458521, root := ~, start := 458460, stop := 458520 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 37 ], name := "Item", next := 458567, root := ~, start := 458530, stop := 458566 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 39 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 39 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 39 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 39 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "CollapsedAdjacencyMat", coun\ t := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], mathmode := \ "Text", name := "C", next := 458671, root := ~, start := 458643, stop := 45867\ 0 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], name := "Alt", next := 458784, root := ~, start := 458592, stop := 458783 ), rec( content := "\n \ ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], name := "Item", next := 458800, root := ~, start := 458576, stop := 458799 ) ], count := [ "A", 3, 0, 34 ], name := "Row", next := 458813, root := ~, start := 458446, stop := 458812 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 40 ], name := "HorLine", next := 458830, root := ~, start := 458820, stop := 458829 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ rbitalGraphColadjMats", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "C", next := 458898, root := ~, start := 458869, stop := 458897 ), rec( content := "\n \ ", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 41 ], name := "Item", next := 458914, root := ~, start := 458852, stop := 458913 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 43 ], name := "Item", next := 458960, root := ~, start := 458923, stop := 458959 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 45 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "OrbitalGraphColadjMats" , coun\ t := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], mathmode := \ "Text", name := "C", next := 459065, root := ~, start := 459036, stop := 45906\ 4 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], name := "Alt", next := 459178, root := ~, start := 458985, stop := 459177 ), rec( content := "\n \ ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], name := "Item", next := 459194, root := ~, start := 458969, stop := 459193 ) ], count := [ "A", 3, 0, 40 ], name := "Row", next := 459207, root := ~, start := 458838, stop := 459206 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 46 ], name := "HorLine", next := 459224, root := ~, start := 459214, stop := 459223 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexTransitiveDRGs", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "C", next := 459290, root := ~, start := 459263, stop := 459289 ), rec( content := "\n \ ", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 47 ], name := "Item", next := 459306, root := ~, start := 459246, stop := 459305 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 49 ], name := "Item", next := 459352, root := ~, start := 459315, stop := 459351 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 51 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 51 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 51 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 51 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "VertexTransitiveDRGs", coun\ t := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], mathmode := \ "Text", name := "C", next := 459455, root := ~, start := 459428, stop := 45945\ 4 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], name := "Alt", next := 459568, root := ~, start := 459377, stop := 459567 ), rec( content := "\n \ ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], name := "Item", next := 459584, root := ~, start := 459361, stop := 459583 ) ], count := [ "A", 3, 0, 46 ], name := "Row", next := 459597, root := ~, start := 459232, stop := 459596 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 52 ], name := "HorLine", next := 459614, root := ~, start := 459604, stop := 459613 ) ], count := [ "A", 3, 0, 4 ], mathmode := "Text", name := "Table", next := 459628, root := ~, start := 456439, stop := 459627 ), rec( content := "\n\n ", count := [ "A", 3, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 1 ], mathmode := "Text", name := "Section", next := 459642, root := ~, start := 456185, stop := 459641 ), rec( content := "\n\n ", count := [ "A", 3, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 4, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Some special vert\ ex subsets of a graph\n ", count := [ "A", 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 459729, root := ~, start := 459660, stop := 459728 ), rec( content := "\n ", count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 3 ], name := "E", next := 459867, root := ~, start := 459757, stop := 459866 ), rec( content := "\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "Alt", next := 459878, root := ~, start := 459734, stop := 459877 ), rec( content := "\n\n ", count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 4 ], name := "HorLine", next := 459924, root := ~, start := 459914, stop := 459923 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 4, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 4, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 5 ], name := "Item", next := 459994, root := ~, start := 459946, stop := 459993 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 7 ], name := "Item", next := 460054, root := ~, start := 460003, stop := 460053 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 4, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 9 ], name := "Alt", next := 460135, root := ~, start := 460080, stop := 460134 ), rec( content := "\n \ ", count := [ "A", 4, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 9 ], name := "Item", next := 460151, root := ~, start := 460063, stop := 460150 ) ], count := [ "A", 4, 0, 4 ], name := "Row", next := 460164, root := ~, start := 459932, stop := 460163 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 10 ], name := "HorLine", next := 460181, root := ~, start := 460171, stop := 460180 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ onnectedComponent", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "C", next := 460245, root := ~, start := 460220, stop := 460244 ), rec( content := "\n \ ", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 11 ], name := "Item", next := 460261, root := ~, start := 460203, stop := 460260 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ConnectedComponent" ), content := 0, count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "Ref", next := 460326, root := ~, start := 460287, stop := 460325 ), rec( content := "\n \ ", count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 13 ], name := "Item", next := 460342, root := ~, start := 460270, stop := 460341 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n When applied to a simple di/graph and a vertex,\n ", count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "ConnectedComponent", count := [ "A", 4, 0, 15 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "C", next := 460483, root := ~, start := 460458, stop := 46048\ 2 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphCon\ nectedComponent" ), content := 0, count := [ "A", 4, 0, 15 ], mathmode := "Tex\ t", name := "Ref", next := 460563, root := ~, start := 460524, stop := 460562 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "DigraphConnectedComponent"\ ), content := 0, count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "Ref\ ", next := 460616, root := ~, start := 460577, stop := 460615 ), rec( HTML := true , content := "\ can also be applied to\n non-simple graphs (in the language of ", count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ ).\n ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 4, 0, 15 ], name := "Alt", next := 460716, root := ~, start := 460368, stop := 460715 ), rec( content := "\n \ ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], name := "Item", next := 460732, root := ~, start := 460351, stop := 460731 ) ], count := [ "A", 4, 0, 10 ], name := "Row", next := 460745, root := ~, start := 460189, stop := 460744 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 16 ], name := "HorLine", next := 460762, root := ~, start := 460752, stop := 460761 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ onnectedComponents", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "C", next := 460827, root := ~, start := 460801, stop := 460826 ), rec( content := "\n \ ", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 17 ], name := "Item", next := 460843, root := ~, start := 460784, stop := 460842 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "Ref", next := 460909, root := ~, start := 460869, stop := 460908 ), rec( content := "\n \ ", count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 19 ], name := "Item", next := 460925, root := ~, start := 460852, stop := 460924 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n When applied to a simple di/graph,\n ", count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "ConnectedComponents", count := [ "A", 4, 0, 21 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "C", next := 461054, root := ~, start := 461028, stop := 46105\ 3 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 21 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphCon\ nectedComponents" ), content := 0, count := [ "A", 4, 0, 21 ], mathmode := "Te\ xt", name := "Ref", next := 461135, root := ~, start := 461095, stop := 461134\ ), rec( HTML := true, content := ".\n ", count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digrap\ hConnectedComponents" ), content := 0, count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "Ref", next := 461189, root := ~, start := 461149, stop := 4611\ 88 ), rec( HTML := true, content := " can also be applied to\n non-\ simple graphs (in the language of ", count := [ "A", 4, 0, 21 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ ).\n ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 4, 0, 21 ], name := "Alt", next := 461289, root := ~, start := 460951, stop := 461288 ), rec( content := "\n \ ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], name := "Item", next := 461305, root := ~, start := 460934, stop := 461304 ) ], count := [ "A", 4, 0, 16 ], name := "Row", next := 461318, root := ~, start := 460770, stop := 461317 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 22 ], name := "HorLine", next := 461335, root := ~, start := 461325, stop := 461334 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ icomponents", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "C", next := 461393, root := ~, start := 461374, stop := 461392 ), rec( content := "\n \ ", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 23 ], name := "Item", next := 461409, root := ~, start := 461357, stop := 461408 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Bicomponents" ), content := 0, count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "Ref", next := 461468, root := ~, start := 461435, stop := 461467 ), rec( content := "\n \ ", count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 25 ], name := "Item", next := 461484, root := ~, start := 461418, stop := 461483 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n When applied to a simple bipartite di/graph,\n ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Bicomponents", count := [ "A", 4, 0, 27 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "C", next := 461616, root := ~, start := 461597, stop := 46161\ 5 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 27 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphBic\ omponents" ), content := 0, count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Ref\ ", next := 461690, root := ~, start := 461657, stop := 461689 ), rec( HTML := true , content := "\ .\n The definition of ", count := [ "A", 4, 0, 27 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipart\ iteDigraph" ), content := 0, count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Ref\ ", next := 461754, root := ~, start := 461722, stop := 461753 ), rec( HTML := true , content := "\ does not require\n its argument to be simple (in the language of " , count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ ), i.e. the\n bipartite graphs in ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ are a proper subset of the bipartite\n digraphs in ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 4, 0, 27 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 27 ], name := "Alt", next := 461978, root := ~, start := 461510, stop := 461977 ), rec( content := "\n \ ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], name := "Item", next := 461994, root := ~, start := 461493, stop := 461993 ) ], count := [ "A", 4, 0, 22 ], name := "Row", next := 462007, root := ~, start := 461343, stop := 462006 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 28 ], name := "HorLine", next := 462024, root := ~, start := 462014, stop := 462023 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istanceSet", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "C", next := 462081, root := ~, start := 462063, stop := 462080 ), rec( content := "\n \ ", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 29 ], name := "Item", next := 462097, root := ~, start := 462046, stop := 462096 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph, a pos int, and an int", Oper := "DigraphDistanceSet" ), content := 0, count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "Ref", next := 462216, root := ~, start := 462125, stop := 462215 ), rec( content := "\n \ ", count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 31 ], name := "Item", next := 462232, root := ~, start := 462106, stop := 462231 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "DistanceSet", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "C", next := 4623\ 06, root := ~, start := 462288, stop := 462305 ), rec( HTML := true, content := "\ in ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 33 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a dig\ raph, a pos int, and an int", Oper := "DigraphDistanceSet" ), content := 0, count := [ "A", 4, 0, 33 ], mathmode := \ "Text", name := "Ref", next := 462438, root := ~, start := 462347, stop := 4624\ 37 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 4, 0, 33 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 33 ], name := "Alt", next := 462470, root := ~, start := 462258, stop := 462469 ), rec( content := "\n \ ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], name := "Item", next := 462486, root := ~, start := 462241, stop := 462485 ) ], count := [ "A", 4, 0, 28 ], name := "Row", next := 462499, root := ~, start := 462032, stop := 462498 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 34 ], name := "HorLine", next := 462516, root := ~, start := 462506, stop := 462515 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ayers", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "C", next := 462568, root := ~, start := 462555, stop := 462567 ), rec( content := "\n \ ", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 35 ], name := "Item", next := 462584, root := ~, start := 462538, stop := 462583 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Layers" ), content := 0, count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "Ref", next := 462637, root := ~, start := 462610, stop := 462636 ), rec( content := "\n \ ", count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 37 ], name := "Item", next := 462653, root := ~, start := 462593, stop := 462652 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Layers", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "C", next := 4627\ 22, root := ~, start := 462709, stop := 462721 ), rec( HTML := true, content := "\ in ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 39 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 39 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 39 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 39 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphLay\ ers" ), content := 0, count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "Ref\ ", next := 462790, root := ~, start := 462763, stop := 462789 ), rec( HTML := true , content := "\ in ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 39 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 39 ], name := "Alt", next := 462822, root := ~, start := 462679, stop := 462821 ), rec( content := "\n \ ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 39 ], name := "Item", next := 462838, root := ~, start := 462662, stop := 462837 ) ], count := [ "A", 4, 0, 34 ], name := "Row", next := 462851, root := ~, start := 462524, stop := 462850 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 40 ], name := "HorLine", next := 462868, root := ~, start := 462858, stop := 462867 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ ndependentSet", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "C", next := 462928, root := ~, start := 462907, stop := 462927 ), rec( content := "\n \ ", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 41 ], name := "Item", next := 462944, root := ~, start := 462890, stop := 462943 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ IndependentSet" ), content := 0, count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "Ref", next := 463005, root := ~, start := 462970, stop := 463004 ), rec( content := "\n \ ", count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 43 ], name := "Item", next := 463021, root := ~, start := 462953, stop := 463020 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "IndependentSet", count := [ "A", 4, 0, 45 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "C", next := 4630\ 98, root := ~, start := 463077, stop := 463097 ), rec( HTML := true, content := "\ in ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 45 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphInd\ ependentSet" ), content := 0, count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "Ref\ ", next := 463174, root := ~, start := 463139, stop := 463173 ), rec( HTML := true , content := "\ in ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 45 ], name := "Alt", next := 463206, root := ~, start := 463047, stop := 463205 ), rec( content := "\n \ ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], name := "Item", next := 463222, root := ~, start := 463030, stop := 463221 ) ], count := [ "A", 4, 0, 40 ], name := "Row", next := 463235, root := ~, start := 462876, stop := 463234 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 46 ], name := "HorLine", next := 463252, root := ~, start := 463242, stop := 463251 ) ], count := [ "A", 4, 0, 4 ], mathmode := "Text", name := "Table", next := 463266, root := ~, start := 459884, stop := 463265 ), rec( content := "\n\n ", count := [ "A", 4, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 1 ], mathmode := "Text", name := "Section", next := 463280, root := ~, start := 459646, stop := 463279 ), rec( content := "\n\n ", count := [ "A", 4, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 5, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to cons\ truct new graphs from old\n ", count := [ "A", 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 463371, root := ~, start := 463298, stop := 463370 ), rec( content := "\n ", count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 3 ], name := "E", next := 463509, root := ~, start := 463399, stop := 463508 ), rec( content := "\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "Alt", next := 463520, root := ~, start := 463376, stop := 463519 ), rec( content := "\n\n ", count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 4 ], name := "HorLine", next := 463566, root := ~, start := 463556, stop := 463565 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 5, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 5, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 5 ], name := "Item", next := 463636, root := ~, start := 463588, stop := 463635 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 7 ], name := "Item", next := 463696, root := ~, start := 463645, stop := 463695 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 5, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 9 ], name := "Alt", next := 463777, root := ~, start := 463722, stop := 463776 ), rec( content := "\n \ ", count := [ "A", 5, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 9 ], name := "Item", next := 463793, root := ~, start := 463705, stop := 463792 ) ], count := [ "A", 5, 0, 4 ], name := "Row", next := 463806, root := ~, start := 463574, stop := 463805 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 10 ], name := "HorLine", next := 463823, root := ~, start := 463813, stop := 463822 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nducedSubgraph", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "C", next := 463884, root := ~, start := 463862, stop := 463883 ), rec( content := "\n \ ", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 11 ], name := "Item", next := 463900, root := ~, start := 463845, stop := 463899 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Induced\ Subdigraph" ), content := 0, count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "Ref", next := 463957, root := ~, start := 463926, stop := 463956 ), rec( content := "\n \ ", count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 13 ], name := "Item", next := 463973, root := ~, start := 463909, stop := 463972 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Oper := "InducedSubdigraph" ), content := 0 , count := [ "A", 5, 0, 15 ], mathmode := \ "Text", name := "Ref", next := 464060, root := ~, start := 464029, stop := 4640\ 59 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ does not allow a third\n argument ", count := [ "A", 5, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "G", count := [ "A", 5, 0, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "C", next := 4641\ 27, root := ~, start := 464119, stop := 464126 ), rec( HTML := true, content := "\ specifying a group of automorphisms on the\n resultant graph. This\ may be included in a future release.\n ", count := [ "A", 5, 0, 15 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], name := "Alt", next := 464258, root := ~, start := 463999, stop := 464257 ), rec( content := "\n \ ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], name := "Item", next := 464274, root := ~, start := 463982, stop := 464273 ) ], count := [ "A", 5, 0, 10 ], name := "Row", next := 464287, root := ~, start := 463831, stop := 464286 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 16 ], name := "HorLine", next := 464304, root := ~, start := 464294, stop := 464303 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istanceSetInduced", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "C", next := 464368, root := ~, start := 464343, stop := 464367 ), rec( content := "\n \ ", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 17 ], name := "Item", next := 464384, root := ~, start := 464326, stop := 464383 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 19 ], name := "Item", next := 464430, root := ~, start := 464393, stop := 464429 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 21 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], name := "Alt", next := 464589, root := ~, start := 464456, stop := 464588 ), rec( content := "\n \ ", count := [ "A", 5, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], name := "Item", next := 464605, root := ~, start := 464439, stop := 464604 ) ], count := [ "A", 5, 0, 16 ], name := "Row", next := 464618, root := ~, start := 464312, stop := 464617 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 22 ], name := "HorLine", next := 464635, root := ~, start := 464625, stop := 464634 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istanceGraph", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "C", next := 464694, root := ~, start := 464674, stop := 464693 ), rec( content := "\n \ ", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 23 ], name := "Item", next := 464710, root := ~, start := 464657, stop := 464709 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graph and list", Oper := "DistanceDigraph" ), content := 0, count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "Ref", next := 464794, root := ~, start := 464736, stop := 464793 ), rec( content := "\n \ ", count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 25 ], name := "Item", next := 464810, root := ~, start := 464719, stop := 464809 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "DistanceGraph", count := [ "A", 5, 0, 27 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "C", next := 4648\ 86, root := ~, start := 464866, stop := 464885 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 27 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for digra\ ph and list", Oper := "DistanceDigraph" ), content := 0, count := [ "A", 5, 0, 27 ], mathmode := \ "Text", name := "Ref", next := 464985, root := ~, start := 464927, stop := 4649\ 84 ), rec( HTML := true, content := " in\n ", count := [ "A", 5, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 27 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 27 ], name := "Alt", next := 465029, root := ~, start := 464836, stop := 465028 ), rec( content := "\n \ ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], name := "Item", next := 465045, root := ~, start := 464819, stop := 465044 ) ], count := [ "A", 5, 0, 22 ], name := "Row", next := 465058, root := ~, start := 464643, stop := 465057 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 28 ], name := "HorLine", next := 465075, root := ~, start := 465065, stop := 465074 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ omplementGraph", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "C", next := 465136, root := ~, start := 465114, stop := 465135 ), rec( content := "\n \ ", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 29 ], name := "Item", next := 465152, root := ~, start := 465097, stop := 465151 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Dual" ), content := 0, count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "Ref", next := 465203, root := ~, start := 465178, stop := 465202 ), rec( content := "\n \ ", count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 31 ], name := "Item", next := 465219, root := ~, start := 465161, stop := 465218 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Oper := "DigraphDual" ), content := 0 , count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "Ref", next := 465300, root := ~, start := 465275, stop := 4652\ 99 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 33 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ is equivalent to\n ", count := [ "A", 5, 0, 33 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "ComplementGraph", count := [ "A", 5, 0, 33 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "C", next := 465366, root := ~, start := 465344, stop := 46536\ 5 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 5, 0, 33 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ with the optional argument\n ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "comploops", count := [ "A", 5, 0, 33 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := "\ Text", name := "A", next := 465433, root := ~, start := 465417, stop := 465432\ ), rec( HTML := true, content := " set to ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "true", count := [ "A", 5, 0, 33 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "K", next := 4654\ 52, root := ~, start := 465441, stop := 465451 ), rec( HTML := true, content := "\ (loops/nonloops are\n complemented).\n ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], name := "Alt", next := 465516, root := ~, start := 465245, stop := 465515 ), rec( content := "\n \ ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], name := "Item", next := 465532, root := ~, start := 465228, stop := 465531 ) ], count := [ "A", 5, 0, 28 ], name := "Row", next := 465545, root := ~, start := 465083, stop := 465544 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 34 ], name := "HorLine", next := 465562, root := ~, start := 465552, stop := 465561 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ ointGraph", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "C", next := 465618, root := ~, start := 465601, stop := 465617 ), rec( content := "\n \ ", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 35 ], name := "Item", next := 465634, root := ~, start := 465584, stop := 465633 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 37 ], name := "Item", next := 465680, root := ~, start := 465643, stop := 465679 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 39 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], name := "Alt", next := 465839, root := ~, start := 465706, stop := 465838 ), rec( content := "\n \ ", count := [ "A", 5, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], name := "Item", next := 465855, root := ~, start := 465689, stop := 465854 ) ], count := [ "A", 5, 0, 34 ], name := "Row", next := 465868, root := ~, start := 465570, stop := 465867 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 40 ], name := "HorLine", next := 465885, root := ~, start := 465875, stop := 465884 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ dgeGraph", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "C", next := 465940, root := ~, start := 465924, stop := 465939 ), rec( content := "\n \ ", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 41 ], name := "Item", next := 465956, root := ~, start := 465907, stop := 465955 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EdgeUnd\ irectedDigraph" ), content := 0, count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "Ref", next := 466017, root := ~, start := 465982, stop := 466016 ), rec( content := "\n \ ", count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 43 ], name := "Item", next := 466033, root := ~, start := 465965, stop := 466032 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "EdgeGraph", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "C", next := 4661\ 05, root := ~, start := 466089, stop := 466104 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 45 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EdgeUndire\ ctedDigraph" ), content := 0, count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "Ref\ ", next := 466181, root := ~, start := 466146, stop := 466180 ), rec( HTML := true , content := "\ in ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 45 ], name := "Alt", next := 466213, root := ~, start := 466059, stop := 466212 ), rec( content := "\n \ ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], name := "Item", next := 466229, root := ~, start := 466042, stop := 466228 ) ], count := [ "A", 5, 0, 40 ], name := "Row", next := 466242, root := ~, start := 465893, stop := 466241 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 46 ], name := "HorLine", next := 466259, root := ~, start := 466249, stop := 466258 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ witchedGraph", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "C", next := 466318, root := ~, start := 466298, stop := 466317 ), rec( content := "\n \ ", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 47 ], name := "Item", next := 466334, root := ~, start := 466281, stop := 466333 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 49 ], name := "Item", next := 466380, root := ~, start := 466343, stop := 466379 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 51 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], name := "Alt", next := 466539, root := ~, start := 466406, stop := 466538 ), rec( content := "\n \ ", count := [ "A", 5, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], name := "Item", next := 466555, root := ~, start := 466389, stop := 466554 ) ], count := [ "A", 5, 0, 46 ], name := "Row", next := 466568, root := ~, start := 466267, stop := 466567 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 52 ], name := "HorLine", next := 466585, root := ~, start := 466575, stop := 466584 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ nderlyingGraph", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "C", next := 466646, root := ~, start := 466624, stop := 466645 ), rec( content := "\n \ ", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 53 ], name := "Item", next := 466662, root := ~, start := 466607, stop := 466661 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ SymmetricClosure" ), content := 0, count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "Ref", next := 466725, root := ~, start := 466688, stop := 466724 ), rec( content := "\n \ ", count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 55 ], name := "Item", next := 466741, root := ~, start := 466671, stop := 466740 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "UnderlyingGraph", count := [ "A", 5, 0, 57 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "C", next := 4668\ 19, root := ~, start := 466797, stop := 466818 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 57 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 57 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 57 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 57 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphSym\ metricClosure" ), content := 0, count := [ "A", 5, 0, 57 ], mathmode := "Text" , name := "Ref\ ", next := 466897, root := ~, start := 466860, stop := 466896 ), rec( HTML := true , content := "\ in ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 57 ], name := "Alt", next := 466929, root := ~, start := 466767, stop := 466928 ), rec( content := "\n \ ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], name := "Item", next := 466945, root := ~, start := 466750, stop := 466944 ) ], count := [ "A", 5, 0, 52 ], name := "Row", next := 466958, root := ~, start := 466593, stop := 466957 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 58 ], name := "HorLine", next := 466975, root := ~, start := 466965, stop := 466974 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ uotientGraph", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "C", next := 467034, root := ~, start := 467014, stop := 467033 ), rec( content := "\n \ ", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 59 ], name := "Item", next := 467050, root := ~, start := 466997, stop := 467049 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Quotien\ tDigraph" ), content := 0, count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "Ref", next := 467105, root := ~, start := 467076, stop := 467104 ), rec( content := "\n \ ", count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 61 ], name := "Item", next := 467121, root := ~, start := 467059, stop := 467120 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Oper := "QuotientDigraph" ), content := 0 , count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "Ref", next := 467206, root := ~, start := 467177, stop := 4672\ 05 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 63 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ takes a partition\n ", count := [ "A", 5, 0, 63 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "p", count := [ "A", 5, 0, 63 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "C", next := 4672\ 59, root := ~, start := 467251, stop := 467258 ), rec( HTML := true, content := "\ of the vertices of the digraph, rather than a set of\n generating \ pairs. Hence, the equivalence might not be\n group-invariant. Als\ o note that in ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ multiple edges may be\n created, whereas in ", count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 5, 0, 63 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 63 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 63 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 63 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ they are effectively combined into a\n single edge.\n ", count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], name := "Alt", next := 467579, root := ~, start := 467147, stop := 467578 ), rec( content := "\n \ ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], name := "Item", next := 467595, root := ~, start := 467130, stop := 467594 ) ], count := [ "A", 5, 0, 58 ], name := "Row", next := 467608, root := ~, start := 466983, stop := 467607 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 64 ], name := "HorLine", next := 467625, root := ~, start := 467615, stop := 467624 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ipartiteDouble", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "C", next := 467686, root := ~, start := 467664, stop := 467685 ), rec( content := "\n \ ", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 65 ], name := "Item", next := 467702, root := ~, start := 467647, stop := 467701 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Biparti\ teDoubleDigraph" ), content := 0, count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "Ref", next := 467764, root := ~, start := 467728, stop := 467763 ), rec( content := "\n \ ", count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 67 ], name := "Item", next := 467780, root := ~, start := 467711, stop := 467779 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "BipartiteDouble", count := [ "A", 5, 0, 69 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "C", next := 4678\ 58, root := ~, start := 467836, stop := 467857 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 69 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 69 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 69 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 69 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "BipartiteD\ oubleDigraph" ), content := 0, count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "Ref\ ", next := 467935, root := ~, start := 467899, stop := 467934 ), rec( HTML := true , content := "\ in ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 69 ], name := "Alt", next := 467967, root := ~, start := 467806, stop := 467966 ), rec( content := "\n \ ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], name := "Item", next := 467983, root := ~, start := 467789, stop := 467982 ) ], count := [ "A", 5, 0, 64 ], name := "Row", next := 467996, root := ~, start := 467633, stop := 467995 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 70 ], name := "HorLine", next := 468013, root := ~, start := 468003, stop := 468012 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eodesicsGraph", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "C", next := 468073, root := ~, start := 468052, stop := 468072 ), rec( content := "\n \ ", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 71 ], name := "Item", next := 468089, root := ~, start := 468035, stop := 468088 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 73 ], name := "Item", next := 468135, root := ~, start := 468098, stop := 468134 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 75 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], name := "Alt", next := 468294, root := ~, start := 468161, stop := 468293 ), rec( content := "\n \ ", count := [ "A", 5, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], name := "Item", next := 468310, root := ~, start := 468144, stop := 468309 ) ], count := [ "A", 5, 0, 70 ], name := "Row", next := 468323, root := ~, start := 468021, stop := 468322 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 76 ], name := "HorLine", next := 468340, root := ~, start := 468330, stop := 468339 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ollapsedIndependentOrbitsGraph", count := [ "A", 5, 0, 77 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "C", next := 468417, root := ~, start := 468379, stop := 468416 ), rec( content := "\n \ ", count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 77 ], name := "Item", next := 468433, root := ~, start := 468362, stop := 468432 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 79 ], name := "Item", next := 468479, root := ~, start := 468442, stop := 468478 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 81 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 81 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this feature\n might be implemented in the future.\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], name := "Alt", next := 468667, root := ~, start := 468505, stop := 468666 ), rec( content := "\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], name := "Item", next := 468683, root := ~, start := 468488, stop := 468682 ) ], count := [ "A", 5, 0, 76 ], name := "Row", next := 468696, root := ~, start := 468348, stop := 468695 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 82 ], name := "HorLine", next := 468713, root := ~, start := 468703, stop := 468712 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ollapsedCompleteOrbitsGraph", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "C", next := 468787, root := ~, start := 468752, stop := 468786 ), rec( content := "\n \ ", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 83 ], name := "Item", next := 468803, root := ~, start := 468735, stop := 468802 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 85 ], name := "Item", next := 468849, root := ~, start := 468812, stop := 468848 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 87 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this feature\n might be implemented in the future.\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], name := "Alt", next := 469037, root := ~, start := 468875, stop := 469036 ), rec( content := "\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], name := "Item", next := 469053, root := ~, start := 468858, stop := 469052 ) ], count := [ "A", 5, 0, 82 ], name := "Row", next := 469066, root := ~, start := 468721, stop := 469065 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 88 ], name := "HorLine", next := 469083, root := ~, start := 469073, stop := 469082 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ ewGroupGraph", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "C", next := 469142, root := ~, start := 469122, stop := 469141 ), rec( content := "\n \ ", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 89 ], name := "Item", next := 469158, root := ~, start := 469105, stop := 469157 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 91 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 91 ], name := "Item", next := 469204, root := ~, start := 469167, stop := 469203 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 93 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ . However, the same\n results would be possible by calling ", count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digrap\ hCopy" ), content := 0, count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "Ref\ ", next := 469399, root := ~, start := 469374, stop := 469398 ), rec( HTML := true , content := "\ on\n the digraph to produce a copy ", count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "gr", count := [ "A", 5, 0, 93 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "C", next := 4694\ 54, root := ~, start := 469445, stop := 469453 ), rec( HTML := true, content := "\ , and then applying\n ", count := [ "A", 5, 0, 93 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "SetDigraphGroup(gr, G)", count := [ "A", 5, 0, 93 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "C", next := 469515, root := ~, start := 469486, stop := 46951\ 4 ), rec( HTML := true, content := ".\n ", count := [ "A", 5, 0, 93 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], name := "Alt", next := 469533, root := ~, start := 469230, stop := 469532 ), rec( content := "\n \ ", count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], name := "Item", next := 469549, root := ~, start := 469213, stop := 469548 ) ], count := [ "A", 5, 0, 88 ], name := "Row", next := 469562, root := ~, start := 469091, stop := 469561 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 94 ], name := "HorLine", next := 469579, root := ~, start := 469569, stop := 469578 ) ], count := [ "A", 5, 0, 4 ], mathmode := "Text", name := "Table", next := 469593, root := ~, start := 463526, stop := 469592 ), rec( content := "\n\n ", count := [ "A", 5, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 1 ], mathmode := "Text", name := "Section", next := 469607, root := ~, start := 463284, stop := 469606 ), rec( content := "\n\n ", count := [ "A", 5, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 6, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Vertex-Colouring \ and Complete Subgraphs\n ", count := [ "A", 6, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 2 ], mathmode := "Text", name := "Heading", next := 469695, root := ~, start := 469625, stop := 469694 ), rec( content := "\n ", count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 3 ], name := "E", next := 469833, root := ~, start := 469723, stop := 469832 ), rec( content := "\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "Alt", next := 469844, root := ~, start := 469700, stop := 469843 ), rec( content := "\n\n ", count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 4 ], name := "HorLine", next := 469890, root := ~, start := 469880, stop := 469889 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 6, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 6, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 5 ], name := "Item", next := 469960, root := ~, start := 469912, stop := 469959 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 7 ], name := "Item", next := 470020, root := ~, start := 469969, stop := 470019 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 6, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 9 ], name := "Alt", next := 470101, root := ~, start := 470046, stop := 470100 ), rec( content := "\n \ ", count := [ "A", 6, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 9 ], name := "Item", next := 470117, root := ~, start := 470029, stop := 470116 ) ], count := [ "A", 6, 0, 4 ], name := "Row", next := 470130, root := ~, start := 469898, stop := 470129 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 10 ], name := "HorLine", next := 470147, root := ~, start := 470137, stop := 470146 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexColouring", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "C", next := 470208, root := ~, start := 470186, stop := 470207 ), rec( content := "\n \ ", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 11 ], name := "Item", next := 470224, root := ~, start := 470169, stop := 470223 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ GreedyColouring", Label := "for a digraph" ), content := 0, count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "Ref", next := 470308, root := ~, start := 470250, stop := 470307 ), rec( content := "\n \ ", count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 13 ], name := "Item", next := 470324, root := ~, start := 470233, stop := 470323 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "VertexColouring", count := [ "A", 6, 0, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "C", next := 4704\ 02, root := ~, start := 470380, stop := 470401 ), rec( HTML := true, content := "\ in ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 6, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 6, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGreedyColouring", Label :=\ "for a digraph" ), content := 0, count := [ "A", 6, 0, 15 ], mathmode := "Tex\ t", name := "Ref", next := 470501, root := ~, start := 470431, stop := 470500 ), rec( HTML := true , content := "\ in ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ ,\n except it returns a transformation rather than a list of vertex\ \n colors.\n ", count := [ "A", 6, 0, 15 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], name := "Alt", next := 470629, root := ~, start := 470350, stop := 470628 ), rec( content := "\n \ ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], name := "Item", next := 470645, root := ~, start := 470333, stop := 470644 ) ], count := [ "A", 6, 0, 10 ], name := "Row", next := 470658, root := ~, start := 470155, stop := 470657 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 16 ], name := "HorLine", next := 470675, root := ~, start := 470665, stop := 470674 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ompleteSubgraphs", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "C", next := 470738, root := ~, start := 470714, stop := 470737 ), rec( content := "\n \ ", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 17 ], name := "Item", next := 470754, root := ~, start := 470697, stop := 470753 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ Cliques" ), content := 0, count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "Ref", next := 470808, root := ~, start := 470780, stop := 470807 ), rec( content := "\n \ ", count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 19 ], name := "Item", next := 470824, root := ~, start := 470763, stop := 470823 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The functionality of ", count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "CompleteSubgraphs", count := [ "A", 6, 0, 21 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "C", next := 470925, root := ~, start := 470901, stop := 47092\ 4 ), rec( HTML := true, content := " in ", count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 6, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 6, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 6, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is provided\n by ", count := [ "A", 6, 0, 21 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Func := "DigraphCliques" ), content := 0 , count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "Ref", next := 470992, root := ~, start := 470964, stop := 4709\ 91 ), rec( HTML := true, content := " and ", count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digrap\ hMaximalCliques" ), content := 0, count := [ "A", 6, 0, 21 ], mathmode := "Tex\ t", name := "Ref", next := 471032, root := ~, start := 470997, stop := 471031 ), rec( HTML := true , content := "\ \n in ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 6, 0, 21 ], name := "Alt", next := 471076, root := ~, start := 470850, stop := 471075 ), rec( content := "\n \ ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], name := "Item", next := 471092, root := ~, start := 470833, stop := 471091 ) ], count := [ "A", 6, 0, 16 ], name := "Row", next := 471105, root := ~, start := 470683, stop := 471104 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 22 ], name := "HorLine", next := 471122, root := ~, start := 471112, stop := 471121 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ompleteSubgraphsOfGivenSize", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "C", next := 471196, root := ~, start := 471161, stop := 471195 ), rec( content := "\n \ ", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 23 ], name := "Item", next := 471212, root := ~, start := 471144, stop := 471211 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ Cliques" ), content := 0, count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "Ref", next := 471266, root := ~, start := 471238, stop := 471265 ), rec( content := "\n \ ", count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 25 ], name := "Item", next := 471282, root := ~, start := 471221, stop := 471281 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Much of the functionality of ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "CompleteSubgraphsOfGivenSize", count := [ "A", 6, 0, 27 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "C", next := 471402, root := ~, start := 471367, stop := 47140\ 1 ), rec( HTML := true, content := " in\n ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 6, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 6, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 6, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is provided by ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Func := "DigraphCliques" ), content := 0 , count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "Ref", next := 471469, root := ~, start := 471441, stop := 4714\ 68 ), rec( HTML := true, content := " and ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digrap\ hMaximalCliques" ), content := 0, count := [ "A", 6, 0, 27 ], mathmode := "Tex\ t", name := "Ref", next := 471523, root := ~, start := 471474, stop := 471522 ), rec( HTML := true , content := "\ in ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , except that the\n functionality relating to vertex weights and pa\ rtial colourings is\n not implemented.\n ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], name := "Alt", next := 471679, root := ~, start := 471308, stop := 471678 ), rec( content := "\n \ ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], name := "Item", next := 471695, root := ~, start := 471291, stop := 471694 ) ], count := [ "A", 6, 0, 22 ], name := "Row", next := 471708, root := ~, start := 471130, stop := 471707 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 28 ], name := "HorLine", next := 471725, root := ~, start := 471715, stop := 471724 ) ], count := [ "A", 6, 0, 4 ], mathmode := "Text", name := "Table", next := 471739, root := ~, start := 469850, stop := 471738 ), rec( content := "\n ", count := [ "A", 6, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 1 ], mathmode := "Text", name := "Section", next := 471752, root := ~, start := 469611, stop := 471751 ), rec( content := "\n\n ", count := [ "A", 6, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 7, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Automorphism grou\ ps and isomorphism testing for graphs\n ", count := [ "A", 7, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 2 ], mathmode := "Text", name := "Heading", next := 471855, root := ~, start := 471770, stop := 471854 ), rec( content := "\n ", count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 3 ], name := "E", next := 471993, root := ~, start := 471883, stop := 471992 ), rec( content := "\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "Alt", next := 472004, root := ~, start := 471860, stop := 472003 ), rec( content := "\n\n ", count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 4 ], name := "HorLine", next := 472050, root := ~, start := 472040, stop := 472049 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 7, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 7, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 5 ], name := "Item", next := 472120, root := ~, start := 472072, stop := 472119 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 7 ], name := "Item", next := 472180, root := ~, start := 472129, stop := 472179 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 7, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 9 ], name := "Alt", next := 472261, root := ~, start := 472206, stop := 472260 ), rec( content := "\n \ ", count := [ "A", 7, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 9 ], name := "Item", next := 472277, root := ~, start := 472189, stop := 472276 ) ], count := [ "A", 7, 0, 4 ], name := "Row", next := 472290, root := ~, start := 472058, stop := 472289 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 10 ], name := "HorLine", next := 472307, root := ~, start := 472297, stop := 472306 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ utGroupGraph", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "C", next := 472366, root := ~, start := 472346, stop := 472365 ), rec( content := "\n \ ", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 11 ], name := "Item", next := 472382, root := ~, start := 472329, stop := 472381 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "Ref", next := 472461, root := ~, start := 472408, stop := 472460 ), rec( content := "\n \ ", count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 13 ], name := "Item", next := 472477, root := ~, start := 472391, stop := 472476 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "AutGroupGraph", count := [ "A", 7, 0, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "C", next := 4725\ 53, root := ~, start := 472533, stop := 472552 ), rec( HTML := true, content := "\ in ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 7, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 7, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 7, 0, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automorphi\ smGroup", Label := "for a digraph" ), content := 0, count := [ "A", 7, 0, 15 ] , mathmode := \ "Text", name := "Ref", next := 472647, root := ~, start := 472594, stop := 4726\ 46 ), rec( HTML := true, content := " in ", count := [ "A", 7, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 7, 0, 15 ], name := "Alt", next := 472679, root := ~, start := 472503, stop := 472678 ), rec( content := "\n \ ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], name := "Item", next := 472695, root := ~, start := 472486, stop := 472694 ) ], count := [ "A", 7, 0, 10 ], name := "Row", next := 472708, root := ~, start := 472315, stop := 472707 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 16 ], name := "HorLine", next := 472725, root := ~, start := 472715, stop := 472724 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphIsomorphism", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "C", next := 472787, root := ~, start := 472764, stop := 472786 ), rec( content := "\n \ ", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 17 ], name := "Item", next := 472803, root := ~, start := 472747, stop := 472802 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "Ref", next := 472883, root := ~, start := 472829, stop := 472882 ), rec( content := "\n \ ", count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 19 ], name := "Item", next := 472899, root := ~, start := 472812, stop := 472898 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "GraphIsomorphism", count := [ "A", 7, 0, 21 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "C", next := 4729\ 78, root := ~, start := 472955, stop := 472977 ), rec( HTML := true, content := "\ in ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 7, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 7, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Label := "for digraphs", Oper := "I\ somorphismDigraphs" ), content := 0, count := [ "A", 7, 0, 21 ], mathmode := "\ Text", name := "Ref", next := 473073, root := ~, start := 473007, stop := 4730\ 72 ), rec( HTML := true, content := " in ", count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 21 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n Note however that ", count := [ "A", 7, 0, 21 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "GraphIsomorphism", count := [ "A", 7, 0, 21 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "C", next := 473142, root := ~, start := 473119, stop := 47314\ 1 ), rec( HTML := true, content := "'s optional argument\n ", count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "firstunbindcanon", count := [ "A", 7, 0, 21 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "A", next := 473198, root := ~, start := 473175, stop := 47319\ 7 ), rec( HTML := true, content := " is not applicable, since the canonical\n \ labellings of a digraph are not portable between sessions.\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], name := "Alt", next := 473325, root := ~, start := 472925, stop := 473324 ), rec( content := "\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], name := "Item", next := 473341, root := ~, start := 472908, stop := 473340 ) ], count := [ "A", 7, 0, 16 ], name := "Row", next := 473354, root := ~, start := 472733, stop := 473353 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 22 ], name := "HorLine", next := 473371, root := ~, start := 473361, stop := 473370 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sIsomorphicGraph", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "C", next := 473434, root := ~, start := 473410, stop := 473433 ), rec( content := "\n \ ", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 23 ], name := "Item", next := 473450, root := ~, start := 473393, stop := 473449 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsIsomorphicDigraph" ), content := 0, count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "Ref", next := 473530, root := ~, start := 473476, stop := 473529 ), rec( content := "\n \ ", count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 25 ], name := "Item", next := 473546, root := ~, start := 473459, stop := 473545 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "IsIsomorphicGraph", count := [ "A", 7, 0, 27 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "C", next := 4736\ 26, root := ~, start := 473602, stop := 473625 ), rec( HTML := true, content := "\ in ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 7, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 7, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Label := "for digraphs", Oper := "I\ sIsomorphicDigraph" ), content := 0, count := [ "A", 7, 0, 27 ], mathmode := "\ Text", name := "Ref", next := 473721, root := ~, start := 473655, stop := 4737\ 20 ), rec( HTML := true, content := " in ", count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 27 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n Note however that ", count := [ "A", 7, 0, 27 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "IsIsomorphicGraph", count := [ "A", 7, 0, 27 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "C", next := 473791, root := ~, start := 473767, stop := 47379\ 0 ), rec( HTML := true, content := "'s optional argument\n ", count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "firstunbindcanon", count := [ "A", 7, 0, 27 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "A", next := 473847, root := ~, start := 473824, stop := 47384\ 6 ), rec( HTML := true, content := " is not applicable, since the canonical\n \ labellings of a digraph are not portable between sessions.\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], name := "Alt", next := 473974, root := ~, start := 473572, stop := 473973 ), rec( content := "\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], name := "Item", next := 473990, root := ~, start := 473555, stop := 473989 ) ], count := [ "A", 7, 0, 22 ], name := "Row", next := 474003, root := ~, start := 473379, stop := 474002 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 28 ], name := "HorLine", next := 474020, root := ~, start := 474010, stop := 474019 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphIsomorphismClassRepresentatives", count := [ "A", 7, 0, 29 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "C", next := 474102, root := ~, start := 474059, stop := 474101 ), rec( content := "\n \ ", count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 29 ], name := "Item", next := 474118, root := ~, start := 474042, stop := 474117 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 7, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 31 ], name := "Item", next := 474164, root := ~, start := 474127, stop := 474163 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 7, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 33 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this feature\n might be implemented in the future.\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], name := "Alt", next := 474352, root := ~, start := 474190, stop := 474351 ), rec( content := "\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], name := "Item", next := 474368, root := ~, start := 474173, stop := 474367 ) ], count := [ "A", 7, 0, 28 ], name := "Row", next := 474381, root := ~, start := 474028, stop := 474380 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 34 ], name := "HorLine", next := 474398, root := ~, start := 474388, stop := 474397 ) ], count := [ "A", 7, 0, 4 ], mathmode := "Text", name := "Table", next := 474412, root := ~, start := 472010, stop := 474411 ), rec( content := "\n ", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 1 ], mathmode := "Text", name := "Section", next := 474425, root := ~, start := 471756, stop := 474424 ), rec( content := "\n", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 1 ], mathmode := "Text", name := "Appendix", next := 474437, root := ~, start := 440786, stop := 474436 ), rec( content := " Grape to Digraphs command map ", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( attributes := rec( Databases := "digraphs" ), content := 0, count := [ "Bib", 0, 0, 1 ], mathmode := "Text", name := "Bibliography", next := 474517, root := ~, start := 474480, stop := 474516 ), rec( attributes := rec( ), content := 0, count := [ "Ind", 0, 0, 1 ], mathmode := "Text", name := "TheIndex", next := 474529, root := ~, start := 474518, stop := 474528 ) ], count := [ 0, 0, 0, 1 ], name := "Book", next := 474538, root := ~, start := 1961, stop := 474537 ) ], count := [ 0, 0, 0, 1 ], indent := "", indextext := "< (for digraphs) 4.1
\n= \ (for digraphs) 4.1
\n<\ code class=\"func\">AdjacencyMatrix 5.2-1
\nAdjacencyMatrixMutableCopy 5.2-1 \nAdjacencyMatrixUpperTriangleLineDecoder 9.2-10 \nArticulati\ onPoints 5.3-14 \nAsBinaryRelation 3.2-1 \nAsDigraph 3.2-2 \nAsGraph 3.2-4 \ \nAsMonoid 5.5-1 \nAsSemigroup, for a filter\ and a digraph 5.5-1 \ \n    for a filter, semilattice digraph, and two lists 5.5-2 \nAsTransformation 3.2-5\ \nAutomorphismGroup, for a digraph 7.2-2 \n   \ ; for a digraph and a homogeneous list 7.2-5 \n    for a digraph, homogeneo\ us list, and list 7.2-6 \nBipartiteDoubleDigraph 3.3-37 \nBlissAutomorphi\ smGroup, for a digraph 7.2\ -3 \n    for a digraph and homogenous list 7.2-3 \n   \ for a digraph, homogenous list, and list 7.2-3 \nBlissCanonicalDigraph 7.2-9 \nBlissCanonicalLabelling, for a digraph 7.2-7 \n    for a digr\ aph and a list 7.2-8 \ \nBooleanAdjacencyMatrix 5.2-3 \nBooleanAdjacencyMa\ trixMutableCopy 5.2-3 \ \nBridges 5.3-15 \nCayleyDigraph 3.1-12 \nChainDigraph 3.5-1\ \nCharacteristicPolynomial 5.2-2 \nChromati\ cNumber 7.3-17 \ \nCliqueNumber 8.1-5 \nCliquesFinder 8.1-2 \n\ CompleteBipartiteDigraph 3\ .5-3 \nCompleteDigraph 3.5-2 \nCompleteM\ ultipartiteDigraph 3.5-4 \nCycleDigraph 3.5-5 \nDegreeMatrix 5.2-16 \nDigraph 3.1-7 \n    for a group, list, function, and function \ 3.1-7 \n  &nb\ sp; for a list and function \ 3.1-7 \nDigraph6String 9.2-3 \nDigraphAd\ dAllLoops 3.3-38 \nDigraphAddAllLoopsAttr 3.3-38 \nDigraphAddEdge<\ /code>, for a digraph and an edge 3.3-16 \n    for a digraph, source, and range \ 3.3-16 \nDigraphAddEdgeOrbit 3.3-17 \nDigraphAddEdges 3.3-18 \nDig\ raphAddVertex 3.3-14 \ \nDigraphAddVertices, for a digraph and a li\ st of labels 3.3-15 \n\     for a digraph and an integer 3.3-15 \nDigraphAdjacencyF\ unction 5.2-4 \ \nDigraphAllSimpleCircuits 5.3-23 \nDigraphBicompon\ ents 5.3-13 \n<\ code class=\"func\">DigraphByAdjacencyMatrix 3.1-8 \nDigraphByEdges 3.1-9 \nDigraphByInNeighbors 3.1-11 \nDigraphByInNeighbours 3.1-11 \nDigraphByOutNeighboursType 3.1-6 \nDigraphCartesianProduct, \ for a list of digraphs 3.3-30\ \n    for a positive number of digraphs 3.3-30 \nD\ igraphCartesianProductProjections 3.3-32 \nDigraphClique 8.1-3 \nD\ igraphCliques 8.1-4 <\ br />\nDigraphCliquesReps 8.1-4 \nDigraphClosure 3.3-40 \nDigraphColouring, for a digraph and a number of colours <\ a href=\"chap7_mj.html#X86D0A74B83D6B9A7\">7.3-14 \nDigraphConnectedComponent 5.3-10 \nDigraphConnectedComponents 5.3-9 \nDigraphCopy 3.3-\ 1 \nDigraphCopySameMutability 3.3-1 \nDi\ graphCore 7.3-18 \nDigraphDegeneracy 5.3-26 \nDigraphDegeneracyOrd\ ering 5.3-27 \n\ DigraphDiameter 5.3-1 \nDigraphDijkstra, for\ a source 5.3-30 \n&nb\ sp;   for a source and target 5.3-30 \nDigraphDirectProduct, for a list of digraphs 3.3-31\ \n    for a positive number of digraphs 3.3-31 \nDigraphDirectProductProjections 3.3-33 \nDigraphDisjointUnion,\ for a list of digraphs 3.3-27 \n    for an\n arbitrary number of digraphs <\ a href=\"chap3_mj.html#X814F1DFC83DB273F\">3.3-27 \nDigraphDistanceSet, for a digraph, a pos int, and a list 5.3-5 \n   &nbs\ p;for a digraph, a pos int, and an int 5.3-5 \nDigraphDual 3.3-10 \nDigra\ phDualAttr 3.3-10 \nDigraphEdgeLabel 5.1-10 \nDigraphEdgeLabels 5.1-11 \nDigraphEdges \ 5.1-3 \nDigraphEdgeUnion, for a list of\ digraphs 3.3-28 \n&nb\ sp;   for a positive number of digraphs 3.3-28 \nDigraphEmbeddin\ g 7.3-8 \nDigraphEpimorphism 7.3-6 \nDigraphFamily 3.1-6 \nD\ igraphFile 9.2-4 \nDigraphFloydWarshall 5.3-18 \nDigraphFromDigrap\ h6String 9.2-2 \ \nDigraphFromDiSparse6String 9.2-2 \nDigraphFromGra\ ph6String 9.2-2 \nDigraphFromPlainTextString 9.2-12 \nDigraphFromS\ parse6String 9.2-2 \nDigraphGirth 5.3-6 \nDigraphGreedyColouring, for a digraph 7.3-15 \ \n    for a digraph and vertex order 7.3-15 \n    for \ a digraph and vertex order function 7.3-15 \nDigraphGroup 7.2-10 \nDigrap\ hHasLoops 6.1-1 \nDigraphHomomorphism 7.3-2 \nDigraphImmutableCopy\ 3.3-1 \nDigraphImmutableCopyIfImmutable 3.3-2 \nDigraphImmutableC\ opyIfMutable 3.3-2 \nDigraphIndependentSet 8.2-2 \nDigraphIndepend\ entSets 8.2-3 \ \nDigraphIndependentSetsReps 8.2-3 \nDigraphInEdges\ 5.1-12 \nDigraphJoin, for a list of digraphs 3.3-29 \n    for a po\ sitive number of digraphs 3.3-29<\ /a> \nDigraphLayers 5.3-25 \nDigraphLongestD\ istanceFromVertex 5.3-4 \nDigraphLongestSimpleCircuit 5.3-24 \nDig\ raphLoops 5.2-14 \nDigraphMaximalClique 8.1-3 \nDigraphMaximalCliq\ ues 8.1-4 \nDigraphMaximalCliquesAttr 8.1-4 \nDigraphMaximalClique\ sReps 8.1-4 \n<\ code class=\"func\">DigraphMaximalCliquesRepsAttr 8.1-4
\nDigraphMaximal\ IndependentSet 8.2-2 \
\nDigraphMaximalIndependentSets 8.2-3
\nDigra\ phMaximalIndependentSetsAttr 8.2-3
\nDigraphMaximalIndependentSetsReps 8.2-3 \nDigraphMaximalIndependentSetsRepsAttr 8.2-3 \nDigraphMaxima\ lMatching 5.1-16 \nDigraphMaximumMatching 5.1-17 \nDigraphMonomorp\ hism 7.3-4 \nDigraphMutableCopy 3.3-1
\nDigraphMutableCopyIfImmuta\ ble 3.3-2
\nDigraphMutableCopyIfMutable 3.3-2
\nDigraphMycielskian\ 3.3-41
\nDigraphMycielskianAttr 3.3-41
\nDigraphNrConnectedCompon\ ents 5.3-9
\nDigraphNrEdges 5.1-4
\nDigraphNrStronglyConnectedComp\ onents 5.3-11
\ \nDigraphNrVertices 5.1-2
\nDigraphOddGirth \ 5.3-7
\nDigraphOrbitReps 7.\ 2-12
\nDigraphOrbits 7.2-11
\nDigraphOut\ Edges 5.1-13
\n\ DigraphPath 5.3-20
\nDigraphPeriod 5.3-17
\nD\ igraphPlainTextLineDecoder 9.2-8
\nDigraphPlainTextLineEncoder <\ a href=\"chap9_mj.html#X7AAA35867B5D8BF4\">9.2-8
\nDigraphRange 5.2-5
\nDigraphReflexiveTransitiveClosure 3.3-12
\nDigraphReflexiveTransitiveClosureAttr 3.3-12
\nDigraphReflexiveTransi\ tiveReduction 3.3-13 \
\nDigraphReflexiveTransitiveReductionAttr <\ a href=\"chap3_mj.html#X82AD17517E273600\">3.3-13
\nDigraphRemoveAllMultipleEdges 3.3-25
\nDigraphRemoveAllMultipleEdg\ esAttr 3.3-25
\ \nDigraphRemoveEdge, for a digraph and an edge 3.3-21
\n  &nbs\ p; for a digraph, source, and range 3.3-21
\nDigraphRemoveEdgeOrbit\ 3.3-22
\nDigraphRemoveEdges 3.3-23
\nDigraphRemoveLoops 3.3-24
\n\ DigraphRemoveLoopsAttr 3.3\ -24
\nDigraphRemoveVertex 3.3-19
\nDigra\ phRemoveVertices 3.3-20
\nDigraphReverse 3.3-9
\nDigraphReverseAtt\ r 3.3-9
\nDigraphReverseEdge, for a digraph and an edge 3.3-26
\n   &nbs\ p;for a digraph, source, and range 3.3-26
\nDigraphReverseEdges, for a \ digraph and a list of edges 3.3-2\ 6
\nDigraphs package overview 1.
\nDi\ graphSchreierVector 7.2-13\
\nDigraphShortestDistance, for a digra\ ph and a list 5.3-2
\n\     for a digraph and two vertices 5.3-2
\n    for a digraph\ , a list, and a list 5.3-2 <\ br />\nDigraphShortestDistances 5.3-3
\nDigraphShor\ testPath 5.3-21
\nDigraphSinks 5.1-5
\nDigraphsMakeDoc 2.4-1
\nDigraphSource 5.2-5\
\nDigraphSources 5.1-6
\nDigraphsRespectsCo\ louring 7.3-12
\ \nDigraphStabilizer 7.2-14
\nDigraphsTestExtreme2.5-3 \nDigraphsTestInstall 2.5-1 \nDigraphsTestStandard 2.5-2 \nDigraphStronglyConnectedComponent 5.3-12 \nDigraphStronglyConnectedCo\ mponents 5.3-11 \nDigraphsUseBliss 7.2-1 \nDigraphsUseNauty\ 7.2-1 \nDigraphSymmetricClosure 3.3-11 \nDigraphSymmetricClosureAttr 3.3-11 \nDigraphTopologicalSort 5.1-7 \nDigraphTransitiveClosure3.3-12 \nDigraphTransitiveClosureAttr 3.3-12 \nDigraphTransitiveReduc\ tion 3.3-13 \n<\ code class=\"func\">DigraphTransitiveReductionAttr 3.3-13 \nDigraphUndir\ ectedGirth 5.3-8 \nDigraphVertexLabel 5.1-8 \nDigraphVertexLabels<\ /code> 5.1-9 \nDigraphVertices 5.1-1 \nDigraphWelshPowellOrder 7.3-16 \nDiSparse6String 9.2-3\ \nDistanceDigraph, for digraph and int\ 3.3-39 \n  \   for digraph and list \ 3.3-39 \nDotDigraph 9.1-2 \nDotHighlight\ edDigraph 9.1-6 \nDotPartialOrderDigraph 9.1-4 \nDotPreorderDigrap\ h 9.1-5 \nDotQuasiorderDigraph 9.1-5 \nDotSymmetricDigraph \ 9.1-3 \nDotVertexLabelledDigraph 9.1-2 \nDoubleDigraph 3.3-36 \nEdg\ eDigraph 3.3-34 \nEdgeOrbitsDigraph 3.1-10 \nEdgeUndirectedDigraph\ 3.3-35 \nEmbeddingsDigraphs 7.3-9 \nEmbeddingsDigraphsRepresentat\ ives 7.3-9 \nEmptyDigraph 3.5-6
\nEpimorphismsDigraphs 7.3-7
\nEpimorphismsDigraphsRepresentatives 7.3-7
\nGeneralisedPetersenGraph<\ /code> 3.5-9 \nGeneratorsOfCayleyDigraph 5.4-2 \nGeneratorsOfEndomorphism\ Monoid 7.3-13 \ \nGeneratorsOfEndomorphismMonoidAttr 7.3-13 \nGraph\ 3.2-3 \nGraph6String 9.2-3 \nGroupOfCayleyDigraph 5.4-1 \n\ HamiltonianPath 5.3-28\ \nHomomorphismDigraphsFinder 7.3-1 \nHomomo\ rphismsDigraphs 7.3-3 \ \nHomomorphismsDigraphsRepresentatives 7.3-3 \nInDegreeOfVertex 5.2-12\ \nInDegrees 5.2-9 \nInDegreeSequence 5.2-9 \nInDegreeSet 5.\ 2-9 \nInducedSubdigraph 3.3-3 \nInNeighb\ ors 5.2-7 \nInNeighborsMutableCopy 5.2-7 \nInNeighborsOfVertex 5.2-13 \nInNeighbours 5\ .2-7 \nInNeighboursMutableCopy 5.2-7 \nI\ nNeighboursOfVertex 5.2-13\ \nIsAcyclicDigraph 6.3-1 \nIsAntiSymmet\ ricDigraph 6.1-2 \nIsAntisymmetricDigraph 6.1-2 \nIsAperiodicDigra\ ph 6.3-7 \nIsBiconnectedDigraph 6.3-4 \nIsBipartiteDigraph \ 6.1-3 \nIsBridgelessDigraph 6.3-5 \nIsCayleyDigraph 3.1-4 \nIsChain\ Digraph 6.3-2 \ \nIsClique 8.1-1 \nIsCompleteBipartiteDigraph 6.1-4 \nIsCompleteDigraph 6.1-5 \nIsCompleteMultipartiteDigraph \ 6.1-6 \nIsConnectedDigraph \ 6.3-3 \nIsCycleDigraph 6.3-12 \nIsDigrap\ h 3.1-1 \nIsDigraphAutomorphism 7.2-20 \n    for a digraph and a tr\ ansformation or permutation 7.2-2\ 0 \nIsDigraphColouring 7.2-21 \n    for a\ transformation 7.2-21 \nIsDigraphCore 6.5-1 \nIsDigraphEdge, for \ digraph and list 5.1-14 \n    for digraph and two pos ints 5.1-14 \nIsDigraphEmbe\ dding 7.3-11 \n\     for digraphs and a permutation or transformation 7.3-11 \nIsDigraphEndomorphism 7\ .3-10 \n    for digraphs and a permutation or t\ ransformation 7.3-10 \ \nIsDigraphEpimorphism 7.3-10 \n    for digraphs and\ a permutation or transformation \ 7.3-10 \nIsDigraphHomomorphism 7.3-10 \n   &nb\ sp;for digraphs and a permutation or transformation 7.3-10 \nIsDigraphIsomorphi\ sm 7.2-20 \n&nb\ sp;   for digraphs and transformation or permutation 7.2-20 \nI\ sDigraphMonomorphism 7.3-1\ 0 \n    for digraphs and a permutation or trans\ formation 7.3-10 \nIsDigraphWithAdjacencyFunction 3.1-5
\nIsDirectedTree<\ /code> 6.3-8 \nIsDistanceRegularDigraph 6.2-4 \nIsEdgeTransitive \ 6.5-2 \nIsEmptyDigraph 6.1-7\ \nIsEquivalenceDigraph 6.1-8 \nIsEuleri\ anDigraph 6.3-10 \nIsFunctionalDigraph 6.1-9 \nIsHamiltonianDigrap\ h 6.3-11 \nIsImmutableDigraph 3.1-3
\nIsIndependentSet 8.2-1
\nIsInRegularDigraph 6.2-\ 1
\nIsIsomorphicDigraph, for digraphs \ 7.2-15
\n  &n\ bsp; for digraphs and homogeneous lists 7.2-16
\nIsJoinSemilatticeDigraph<\ /code> 6.1-17 \nIsLatticeDigraph 6.1-17 \nIsMatching 5.1-15 \nIsMax\ imalClique 8.1-1 \nIsMaximalIndependentSet 8.2-1 \nIsMaximalMatchi\ ng 5.1-15 \nIsMaximumMatching 5.1-15
\nIsMeetSemilatticeDigraph 6.1-17 \nIsMultiDigraph 6.1-10 \nIsMutableDigraph 3.1-2 \nIs\ NullDigraph 6.1-7 \nIsomorphismDigraphs, for digraphs 7.2-17 \n    \ for digraphs and homogeneous lists 7.2-18 \nIsOuterPlanarDigraph 6.4-2 \nIsOutRegularDigraph 6.2-2\ \nIsPartialOrderDigraph 6.1-16 \nIsPerf\ ectMatching 5.1-15 \nIsPlanarDigraph 6.4-1 \nIsPreorderDigraph 6.1-15 \nIsQuasiorderDigraph 6.1-15 \nIsReachable 5.3-19 \nIsRe\ flexiveDigraph 6.1-11 \ \nIsRegularDigraph 6.2-3 \nIsStronglyConnect\ edDigraph 6.3-6 \nIsSubdigraph 4.1-1 \nIsSymmetricDigraph \ 6.1-12 \nIsTournament 6.1-13\ \nIsTransitiveDigraph 6.1-14 \nIsUndire\ ctedForest 6.3-9 \nIsUndirectedSpanningForest 4.1-2 \nIsUndirected\ SpanningTree 4.1-2 \nIsUndirectedTree 6.3-9 \nIsVertexTransitive 6.5-3 \nIteratorFromDigraphFile 9.2-7 \nIteratorOfPaths 5.3-22 \nJohnsonDigraph 3.5-7 \nKuratowskiOuterPlanarSubdigraph 5.6-2 \nKuratowskiPlanarSubdigraph 5.6-1 \nLaplacianMatrix 5.2-17 \nLine\ Digraph 3.3-34 \ \nLineUndirectedDigraph 3.3-35 \nMaximalAntiSymmetr\ icSubdigraph 3.3-6 \nMaximalAntiSymmetricSubdigraphAttr 3.3-6 \nMa\ ximalSymmetricSubdigraph 3\ .3-5 \nMaximalSymmetricSubdigraphAttr \ 3.3-5 \nMaximalSymmetricSubdigraphWithoutLoops 3.3-5 \nMaximalSymmetricSub\ digraphWithoutLoopsAttr 3.\ 3-5 \nMonomorphismsDigraphs 7.3-5 \nMono\ morphismsDigraphsRepresentatives 7.3-5 \nNautyAutomorphismGroup \ 7.2-4 \nNautyCanonicalDigraph 7.2-9 \nNautyCanonicalLabelling, for\ a digraph 7.2-7 \n&nb\ sp;   for a digraph and a list 7.2-8 \nNrSpanningTrees <\ a href=\"chap5_mj.html#X82F30D5681466BC6\">5.3-29 \nNullDigraph 3.5-6 \nOnDigraphs, for a digraph and a perm <\ a href=\"chap7_mj.html#X808972017C486F1F\">7.1-1 \n  &nbs\ p; for a digraph and a transformation 7.1-1 \nOnMultiDigraphs 7.1-2 \n   &\ nbsp;for a digraph, perm, and perm 7.1-2 \nOutDegreeOfVertex 5.2-10 \nO\ utDegrees 5.2-8 \nOutDegreeSequence 5.2-8 \nOutDegreeSet <\ a href=\"chap5_mj.html#X7F5ACE807D1BC2E2\">5.2-8 \nOuterPlanarEmbedding \ 5.6-4 \nOutNeighbors 5.2-6 \nOutNeighbor\ sMutableCopy 5.2-6 \nOutNeighborsOfVertex 5.2-11 \nOutNeighbours 5.2-6 \nOutNeighboursMutableCopy 5.2-6 \nOutNeighboursOfVertex 5.2-11 \nPartialOrderDigraphJoinOfVertices 5.2-15 \nPartialOrderDigrap\ hMeetOfVertices 5.2-15\ \nPetersenGraph 3.5-8 \nPlainTextString 9.2-12 \nPlanarEmbedding 5.6-3 \nPrintString 9.2-1 \nQuotientD\ igraph 3.3-8 \n\ RandomDigraph 3.4-1 \nRandomLattice 3.4-4 \nR\ andomMultiDigraph 3.4-2 \nRandomTournament 3.4-3 \nReadDigraphs 9.2-5 \nReadDIMACSDigraph 9.2-14 \nReadPlainTextDigraph 9.2-13 \nReducedDigraph 3.3-4 \nReducedDigraphAttr 3.3-4 \nRepresentativ\ eOutNeighbours 7.2-19 \ \nSemigroupOfCayleyDigraph 5.4-1 \nSetDigrap\ hEdgeLabel 5.1-10 \nSetDigraphEdgeLabels, for a digraph and a fun\ ction 5.1-11 \n &\ nbsp;  for a digraph and a list of lists 5.1-11 \nSetDigraphVertexLab\ el 5.1-8 \nSetDigraphVertexLabels 5.1-9 \nSparse6String 9.2-3 \nSplash 9.1-1 \nString 9.2-1 \nStrongOrientation 5.3-16 \nStrongOrientationAttr 5.3\ -16 \nSubdigraphHomeomorphicToK23 5.6-5 \nSubdigraphHomeomorphicToK33 5.6-5 \nSubdigraphHomeomorphicToK4\ 5.6-5 \nTCodeDecoder 9.2-\ 11 \nTournamentLineDecoder 9.2-9 \nUndir\ ectedSpanningForest 3.3-7<\ /a> \nUndirectedSpanningForestAttr 3.3-7 \nU\ ndirectedSpanningTree 3.3-\ 7 \nUndirectedSpanningTreeAttr 3.3-7 \nW\ riteDigraphs 9.2-6 \nWriteDIMACSDigraph 9.2-14 \nWritePlainTextDig\ raph 9.2-13 \n" , input := "Digraphs\">\n http://www.maths.qmul.ac.uk/~leonard/grape/Grape\">\n http://www.tcs.tkk.fi/Software/bliss/bliss\">\n https://github.com/graph-algorithms/edge-addition-planarity-suite<\ /URL>edge-addition-planarity-suite\">\n h\ ttp://pallini.di.uniroma1.it/nauty\">\n https://github.com/gap-packages/NautyTracesInte\ rfaceNautyTracesInterface<\ /Alt>\">\n \n http://gap-\ packages.github.io/io/\nIO\">\n \n http://http:\ //gap-packages.github.io/orb/\n Orb<\ /Package>\">\n https://github.com/gap-packages/datastructuresdatastructures\">\n \n\n\n\n\n\n\n\n\n\n ]>\n\n\n\n &Digraphs\ ;\n Version &VERSION;\n\n Jan De Beule\n \ jdebeule@cage.ugent.be\n http://homepages.vub.a\ c.be/~jdbeule\n \n\n Julius Jonušas\n julius.jonusas@tuwien.ac.at\n http://julius.jonusas.\ work\n \n\n James D. Mitchell\n jdm3@s\ t-and.ac.uk\n https://jdbm.me\n \n\n\ Michael Torpey\n mct25@st-andrews.ac.uk\n https://mtorpey.github.io\n \n\n Wilf A. Wi\ lson\n gap@wilf-wilson.net\n http://wilf.me\n \n\n Stuart Burrell\n\n Reinis \ Cirpons\n\n Luke Elliott\n\n Max Horn\n\n Christopher Jefferson\n\n Markus Pfeiffer<\ /Author>\n\n Chris Russell\n\n Finn Smith\ \n\n Murray White\n\n ©right; ©RIGHTYEAR\ S; by\n Jan De Beule, Julius Jonušas, James D. Mitchell, Michael\n \ Torpey, Wilf A. Wilson et al.\n\n &Digraphs; is free software; you can \ redistribute it and/or modify it\n under the terms of the \n http://www.fsf.org/licenses/gpl.html as pu\ blished by the Free Software\n Foundation; either version 3 of the License,\ or (at your option) any later\n version.\n \n\n \n\ The &Digraphs; package is a &GAP; package containing methods for graphs,\n\ digraphs, and multidigraphs.\n \n\n \n W\ e would like to thank Christopher Jefferson for his help in including \n &b\ liss; in &Digraphs;.\n\n This package's methods for computing digraph homom\ orphisms are based on work\n by Max Neunhöffer, and independently Artu\ r Schäfer.\n \n\n\n\n\n\ \n\n\n \n \n The &\ Digraphs; package\n \n &Digraphs; package \ overview\n\n Introduction\n\n This is the \ manual for version &VERSION; of the &Digraphs; package.\n This package was de\ veloped at the University of St Andrews by:\n \n Jan De Beule,\ \n Julius Jonušas,\n James D. Mitchell,\n Michael C. Torpey, and\n Wilf A. Wilson.\n \n Additional contributions were made by:\n \n S\ tuart Burrell,\n Reinis Cirpons,\n Luke Elliot\ t,\n Max Horn,\n Christopher Jefferson,\ \n Markus Pfeiffer,\n Chris Russell,\n Finn Smith, and\n Maria Tsalakou,\n Murray W\ hyte.\n \n\n The &Digraphs; package contains a variety of meth\ ods for efficiently creating\n and storing mutable and immutable digraphs and\ computing information about\n them. Full explanations of all the function\ s contained in the package are\n provided below. \n\n If the &Grape; p\ ackage is available, it will be loaded automatically.\n Digraphs created with\ the &Digraphs; package can be converted to &Grape;\n graphs with , and conversely &Grape; graphs can be\n converted to &Digraphs; o\ bjects with . &Grape; is not\n required for &Digraphs\ ; to run. \n\n The &bliss; tool is included in this \ package. It\n is an open-source tool for computing automorphism groups and c\ anonical forms\n of graphs, written by Tommi Junttila and Petteri Kaski. Seve\ ral of the methods\n in the &Digraphs; package rely on &bliss;. If the &Nauty\ TracesInterface;\n package for GAP is available then it is also possible to u\ se &nauty; for computing automorphism groups and can\ onical forms\n in &Digraphs;. See Section \n for more details. \n \n\n From version 1.0.0 of \ this package, digraphs can be either mutable or immutable.\n Mutable digraphs\ can be changed in-place by many of the methods in the\n package, which avoid\ s unnecessary copying. Immutable digraphs cannot be\n changed in-place, but t\ heir advantage is that the value of an attribute of\n an immutable digraph is\ only ever computed once. Mutable digraphs can be\n converted into immutable \ digraphs in-place using . \ One of the motivations for introducing mutable\n digraphs in version 1.0.0 wa\ s that in practice the authors often wanted\n to create a digraph and immedia\ tely modify it (removing certain edges,\n loops, and so on). Before version 1\ .0.0, this involved copying the digraph\n several times, with each copy being\ discarded almost immediately.\n From version 1.0.0, this unnecessary copying\ can be eliminated by\n first creating a mutable digraph, then changing it in\ -place, and finally\n converting the mutable digraph to an immutable one in-\ place\n (if desirable).\n\n Defin\ itions\n\n For the purposes of this package and its documentation\ , the following\n definitions apply: \n\n A digraph E=(E^0\ ,E^1,r,s), also known as a directed\n graph, consists of a set\ of vertices E^0 and a set of edges\n E^1 together with funct\ ions s, r: E^1 \\to E^0, called the\n source and range\ , respectively.\n\n The source and range of an edge is respectively the val\ ues of s, r at\n that edge.\n\n An edge is called a loop i\ f its source and range are the same.\n\n A digraph is called a multidigr\ aph if there exist two or more edges\n with the same source and the sam\ e range. \n\n A directed walk on a digraph is a sequence of alte\ rnating vertices\n and edges (v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)\ such that each edge\n e_i has source v_i and range v_{i+1\ }. A directed\n path is a directed walk where no vertex (and h\ ence no edge) is\n repeated. A directed circuit is a directed walk \ where v_1 =\n v_n, and a directed cycle is a directed circu\ it where where no\n vertex is repeated, except for v_1 = v_n. \ \n\n The length of a directed walk (v_1, e_1, v_2, e_2, ..., e_{n\ -1},\n v_n) is equal to n-1, the number of edges it contains. \ A\n directed walk (or path) (v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)\ is\n sometimes called a directed walk (or path) from vertex v_1 \ to\n vertex v_n. A directed walk of zero length, i.e. a seque\ nce\n (v) for some vertex v, is called trivial. A triv\ ial\n directed walk is considered to be both a circuit and a cycle, as is t\ he\n empty directed walk (). A simple circuit is another name\ for a\n non-trivial and non-empty directed cycle. \n\n \n\ \n\n \n \n Installing &Digraphs;\n\n \n For those in a hurry\n In this s\ ection we give a brief description of how to start using\n &Digraphs;.\ \n\n It is assumed that you have a working copy of &GAP; with version numbe\ r\n &GAPVERS; or higher. The most up-to-date version of &GAP; and instructi\ ons\n on how to install it can be obtained from the main &GAP; webpage\n \ http://www.gap-system.org. \n\n The following is a summary o\ f the steps that should lead to a\n successful installation of &Digraphs;:\ \n\n \n\n \n ensure that the &IO; package version &\ IOVERS; or higher is available.\n &IO; must be compiled before &Digraph\ s; can be loaded.\n \n\n \n ensure that the &Orb\ ; package version &ORBVERS; or higher is available.\n &Orb; has better \ performance when compiled, but although compilation is\n recommended, i\ t is not required to be compiled for &Digraphs; to be\n loaded.\n \ \n\n ensure that the &datastructures; package version\n \ &DATASTRUCTURESVERS; or higher is available. \n \n\n \ \n This step is optional: certain functions in &Digraphs; requir\ e\n the &Grape; package to be available; see Section\n for full details. To use these\n functions\ make sure\n that the &Grape; package version &GRAPEVERS; or higher is \ available.\n If &Grape; is not available, then &Digraphs; can be used a\ s normal with\n the exception that the functions listed in Subsection\n\ will not work.\n \n \ \n \n This step is optional: certain functions in\ &Digraphs; require\n the &NautyTracesInterface; package to be availa\ ble.\n \n If you want to make use of the\ se functions, please ensure that the\n &NautyTracesInterface; package v\ ersion &NAUTYTRACESINTERFACEVERS; or\n higher is available. If &NautyT\ racesInterface; is not available,\n then &Digraphs; can be used as norm\ al with the exception that \n functions whose names contain Nauty<\ /Q> will not work.\n \n\n \n download the packag\ e archive &ARCHIVENAME;.tar.gz from\n http://gap-packages.github.io/Digraphs/.\n \ \n\n \n unzip and untar the file, this should create a direc\ tory called\n &ARCHIVENAME;.\n \n\n \n \ locate the pkg directory of your &GAP; directory, which contains\n\ the directories lib, doc and so on. Move the directory\n\ &ARCHIVENAME; into the pkg directory.\n \n\n\ \n it is necessary to compile the &Digraphs; package.\n \ Inside the pkg/&ARCHIVENAME; directory, type\n \n\n Further information about this \ step can be found in Section\n .\n \n\n \n start &GAP; in the usual way (i.\ e. type gap at the command line).\n \n\n \n \ type LoadPackage(\"digraphs\");\n \n \n\n I\ f you want to check that the package is working correctly, you\n should run\ some of the tests described in Section\n .\n \n\n \n \n\n \n Optional package dependencies\n\n \n\n The\ &Digraphs; package is written in &GAP; and C code and requires the\n &IO; \ package. The &IO; package is used to read and write transformations,\n par\ tial permutations, and bipartitions to a file. \n\n \n\n \n The Grape package\n\n \ The &Grape; package must be available for the following operations to be\n \ available:\n\n \n with a dig\ raph argument\n with a digraph arg\ ument\n with a &Grape; graph argum\ ent\n \n\n If &Grape; is not available, then &Digraphs\ ; can be used as normal with\n the exception that the functions above wil\ l not work.\n \n \n\n \n \n\n \n Compiling the kernel module\n The &Digraphs; package has a &GAP; kernel component in\n C which sho\ uld be compiled. This component contains certain low-level\n functions requ\ ired by &Digraphs;.\n \n\n It is not possible to use the &Digraphs; \ package without compiling it. \n\n To compile the kernel component insid\ e the pkg/&ARCHIVENAME;\n directory, type\n \n\n \n\n If you installed the package in a\ nother 'pkg' directory than the standard 'pkg'\n directory in your &GAP; in\ stallation, then you have to do two things. Firstly\n during compilation yo\ u have to use the option '--with-gaproot=PATH' of the\n 'configure' script \ where 'PATH' is a path to the main GAP root directory (if\n not given the d\ efault '../..' is assumed). \n\n If you installed &GAP; on several archi\ tectures, you must execute the\n configure/make step for each of the archit\ ectures. You can either do this\n immediately after configuring and compili\ ng GAP itself on this architecture, or\n alternatively set the environment\ \n variable 'CONFIGNAME' to the name of the configuration you used when com\ piling\n GAP before running './configure'. Note however that your compiler\ choice and\n flags (environment variables 'CC' and 'CFLAGS') need to be ch\ osen to match the\n setup of the original GAP compilation. For example you \ have to specify 32-bit\n or 64-bit mode correctly!\n \n\n \n \n\ \n \n Rebuilding \ the documentation\n The &Digraphs; package comes complete with pd\ f, html, and text versions of\n the documentation. However, you might find \ it necessary, at some point, to\n rebuild the documentation. To rebuild the\ documentation, please use the\n function .\ \n\n \n \n \ Nothing\n \n This function should be called with no\ argument to compile the &Digraphs;\n documentation.\n \n\n\n \n\n \n \n\n \n Testing your installation\n\n In this secti\ on we describe how to test that &Digraphs; is working as\n intended. To te\ st that &Digraphs; is installed correctly use\n or for more extensive tests use\n . \n\n If something goes wrong, then please review the instructions\ in Section and ensure that &Digra\ phs; has been\n properly installed. If you continue having problems, pleas\ e use the https://github.com/gap-packages/D\ igraphs/issues\n to report the issues you are having.\n\n \n \n true \ or false.\n \n This function can be called w\ ithout arguments to test your installation of\n &Digraphs; is working corre\ ctly. These tests should take no more than a\n few seconds to complete. To\ test more comprehensively that\n &Digraphs; is working correctly, use .\n \n\n\n \n \n true\ or false.\n \n This function can be cal\ led without arguments to test all of the methods\n included in &Digraphs;.\ \n These tests should take less than a minute to complete. \n\n To q\ uickly test that &Digraphs; is installed correctly use . For a more thorough test, use .\n \n\n\n \n \n Nothing.\n \n This function should be called with no argument. It executes a se\ ries of\n very demanding tests, which measure the performance of a variety \ of\n functions on large examples. These tests take a long time to\n com\ plete, at least several minutes. \n\n For these tests to complete, the \ digraphs library digraphs-lib must\n be downloaded and placed in the\ digraphs directory in a subfolder\n named digraphs-lib.\n \ This library can be found on the &Digraphs; website.\n \n\n\n\n \n\n \n \n\n\n\n \n Creating digraphs\n\n I\ n this chapter we describe how to create digraphs.\n\n \ Creating digraphs\n \n \n \n Every digraph in &Digraphs; belongs to\ the category\n IsDigraph. Some basic attributes and operations f\ or digraphs\n are , , and\n .\n \n\n\n \n \n \ \n IsMutableDigraph is a synonym for and . A mutable dig\ raph may be changed\n in-place by methods in the &Digraphs; package, and is\ not attribute-storing\n – see .\n \n\n A mutable digraph may be converted into an imm\ utable attribute-storing\n digraph by calling on the\n digraph.\n \n\n\n \ \n \n \n IsImmutableDigraph is a subcategory of .\n Digraphs that lie in IsImmutableDigraph are immutable a\ nd\n attribute-storing. In particular, they lie in\n . \n\n A mutable digraph may be con\ verted to an immutable\n digraph that lies in the category IsImmutableDi\ graph by calling\n on th\ e digraph. \n\n The operation can be \ used\n to construct a mutable copy of an immutable digraph. It is however n\ ot\n possible to convert an immutable digraph into a mutable digraph in-pla\ ce.\n \n\n\n \n \n \n IsCayleyDigraph is\ a subcategory of IsDigraph. Digraphs that\n are Cayley digraphs of \ a group and that are constructed by the operation\n are constructed in this category, and are\n always immutable.\n <\ /Description>\n\n\n \n \n \n IsDigraphWithAd\ jacencyFunction is a subcategory of IsDigraph.\n Digraphs that a\ re created using an adjacency function are constructed\n in this cat\ egory.\n \n\n\n \n \n \n \ \n The type of all digraphs is DigraphByOutNeighboursType.\n The \ family of all digraphs is DigraphFamily.\n \n\n\n \n \n \n \n A digraph.\n \n If the optional f\ irst argument filt is present, then this should\n specify the catego\ ry or representation the digraph being created will\n belong to. For exampl\ e, if filt is ,\n then the digraph b\ eing created will be mutable, if filt is , then the digraph will be immutable. \n If the optional first arg\ ument filt is not present, then \ is used by default.\n \n for a list (i.e. an adjacency li\ st)\n \n if obj is a list of lists of positive\ integers in\n the range from 1 to Length(obj), th\ en this function\n returns the digraph with vertices\n E ^ 0 \ = [1 .. Length(obj)],\n and edges corresponding to th\ e entries of obj. \n\n More precisely, there is an edge from\ vertex i to j if and\n only if j is in obj<\ /A>[i]; the source of this edge is\n i and the range is j\ . If j occurs in\n obj[i] with multiplicity k, then there are k\n edges from i to j.\n \ \n\n for three lists\n \n if \ obj is a duplicate-free list, and source and\n range \ are lists of equal length consisting of positive integers\n in the list\ [1 .. Length(obj)], then this function\n returns a digra\ ph with vertices\n E ^ 0 = [1 .. Length(obj)], and\ \n Length(source) edges. For each i in\n <\ C>[1 .. Length(source)] there exists an edge with source\n v\ ertex source[i] and range vertex range[i].\n See and .\n \n\n \ The vertices of the digraph will be labelled by the elements of\n obj.\n \n\n for an integer, and two lists\n\ \n if obj is an integer, and source and\n \ range are lists of equal length consisting of positive integers\n \ in the list [1 .. obj], then this function\n return\ s a digraph with vertices\n E ^ 0 = [1 .. obj], an\ d\n Length(source) edges. For each i in\n \ [1 .. Length(source)] there exists an edge with source\n \ vertex source[i] and range vertex range[i].\n See and .\n \n\n \ for a list and a function\n \n if list is a list and func is a function taking 2\n arguments that are\ elements of list, and func returns\n true or f\ alse, then this operation creates a digraph with\n vertices [1 .\ . Length(list)] and an edge from vertex\n i to vertex\ j if and only if\n func(list[i], list[j\ ]) returns true.\n \n\n for a group, a list,\ and two functions\n \n The arguments will be G, l\ ist, act, adj. \n\n Let G be a group acting on the objec\ ts in list via\n the action act, and let adj be a \ function taking\n two objects from list as arguments and returni\ ng true or\n false. The function adj will describe\ the adjacency\n between objects from list, which is invariant u\ nder the action of\n G. This variant of the constructor returns \ a digraph with\n vertices the objects of list and directed edges\ [x, y]\n when f(x, y) is true. \n\n Th\ e action of the group G on the objects in list is stored\n \ in the attribute , and is used to speed up\n \ operations like .\n \n\n <\ Mark>for a Grape package graph\n \n if obj is \ a &Grape; package graph (i.e. a record for which the\n function IsGr\ aph returns true), then this function returns\n a digraph is\ omorphic to obj.\n \n\n for a binary relation\n \n if obj is a binary relation on the points <\ C>[1 .. n] for\n some posititve integer n, then this functio\ n returns the digraph\n defined by obj. Specifically, this funct\ ion returns a digraph\n which has n vertices, and which has an e\ dge with source i\n and range j if and only if [i,j] is a pair in\n the binary relation obj.\n \n <\ /List>\n\n gr := Digraph([\n> [2, 5, 8, 10], [2, 3\ , 4, 2, 5, 6, 8, 9, 10], [1],\n> [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10]\ , [1, 4],\n> [1, 5, 9], [1, 2, 7, 8], [3, 5]]);\n\ngap> gr := Digraph([\"a\", \"b\", \"c\"], [\"a\"], [\"\ b\"]);\n\ngap> gr := Digraph(5, [1,\ 2, 2, 4, 1, 1], [2, 3, 5, 5, 1, 1]);\n\ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> fun\ ction(x, y) return Intersection(x, y) = []; end);;\ngap> Digraph(Petersen);\n<\ immutable digraph with 10 vertices, 30 edges>\ngap> gr := Digraph([1 .. 10], R\ eturnTrue);\n]]>\n\n \ The next example illustrates the uses of the fourth and fifth variants of\ \n this constructor. The resulting digraph is a strongly regular graph, a\ nd\n it is actually the point graph of the van Lint-Schrijver partial\n \ geometry, . The algebraic description\n is take\ n from the seminal paper of Calderbank and Kantor .\n f := GF(3 ^ 4);\nGF(3^4)\ngap> gamma := First(f, \ x -> Order(x) = 5);\nZ(3^4)^64\ngap> L := Union([Zero(f)], List(Group(gamma)))\ ;\n[ 0*Z(3), Z(3)^0, Z(3^4)^16, Z(3^4)^32, Z(3^4)^48, Z(3^4)^64 ]\ngap> omega \ := Union(List(L, x -> List(Difference(L, [x]), y -> x - y)));\n[ Z(3)^0, Z(3),\ Z(3^4)^5, Z(3^4)^7, Z(3^4)^8, Z(3^4)^13, Z(3^4)^15, \n Z(3^4)^16, Z(3^4)^21,\ Z(3^4)^23, Z(3^4)^24, Z(3^4)^29, Z(3^4)^31, \n Z(3^4)^32, Z(3^4)^37, Z(3^4)^\ 39, Z(3^4)^45, Z(3^4)^47, Z(3^4)^48, \n Z(3^4)^53, Z(3^4)^55, Z(3^4)^56, Z(3^\ 4)^61, Z(3^4)^63, Z(3^4)^64, \n Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^77, Z\ (3^4)^79 ]\ngap> adj := function(x, y)\n> return x - y in omega;\n> end;\nfu\ nction( x, y ) ... end\ngap> digraph := Digraph(AsList(f), adj);\n\ngap> group := Group(Z(3));;\ngap> act :=\ \\*;\n\ngap> digraph := Digraph(group, List(f), act, adj);\n\ \n]]>\n \n\n\n \n \n A digraph.\n \n\ If the optional first argument filt is present, then this should\n \ specify the category or representation the digraph being created will\n \ belong to. For example, if filt is ,\n \ then the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If t\ he optional first argument filt is not present, then is used by default. \n\n If list is the adj\ acency matrix of a digraph in the sense of\n , then this operation returns the digraph\n which is defined by list. \n\n Alternatively, if list is a square boolean matrix, then \ this\n operation returns the digraph with Length(list)\ \n vertices which has the edge [i,j] if and only if\n list\ [i][j] is true. \n\n DigraphByAd\ jacencyMatrix([\n> [0, 1, 0, 2, 0],\n> [1, 1, 1, 0, 1],\n> [0, 3, 2, 1, 1],\n>\ [0, 0, 1, 0, 1],\n> [2, 0, 0, 0, 0]]);\n\ngap> D := DigraphByAdjacencyMatrix([\n> [true, false, true],\n>\ [false, false, true],\n> [false, true, false]]);\n\ngap> OutNeighbours(D);\n[ [ 1, 3 ], [ 3 ], [ 2 ] ]\ngap> D \ := DigraphByAdjacencyMatrix(IsMutableDigraph, \n> [[true, false, true],\n> [f\ alse, false, true],\n> [false, true, false]]);\n\n]]>\n \n\n\n \n \n A\ digraph.\n \n If the optional first argument fi\ lt is present, then this should\n specify the category or representatio\ n the digraph being created will\n belong to. For example, if filt i\ s ,\n then the digraph being created will b\ e mutable, if filt is , then the\ digraph will be immutable. \n If the optional first argument filt i\ s not present, then is used by default\ . \n\n If list is list of pairs of positive integers, then this f\ unction\n returns the digraph with the minimum number of vertices m \ such that\n its list equal list. \n\n If the optional second a\ rgument n is a positive integer with\n n >= m (with m defined as above),\n then this function returns the digraph with \ n vertices and list\n list. \n\n See .\n\n DigraphByEdges(\n> [[1, 3], [2, 1], \ [2, 3], [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]]);\n\ngap> DigraphByEdges(\n> [[1, 3], [2, 1\ ], [2, 3], [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);\n\ \ngap> DigraphByEdges(IsMutableD\ igraph, \n> [[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4\ ], [5, 6], [6, 6]], 12);\n\n]]>\n \n\n\n \n \n \n An immutable digr\ aph.\n \n \n If G is a permutation group, <\ A>edges is an edge or list of\n edges, and n is a non-negative i\ nteger such that G fixes\n [1 .. n] setwise, then\n \ this operation returns an immutable digraph with n vertices and the un\ ion\n of the orbits of the edges in edges under the action of the\ \n permutation group G. An edge in this context is simply a pair of\ \n positive integers.\n \n\n If the optional third argument n<\ /A> is not present, then the largest\n moved point of the permutation group\ G is used by default. \n\n digraph := E\ dgeOrbitsDigraph(Group((1, 3), (1, 2)(3, 4)),\n> \ [[1, 2], [4, 5]], 5);\n\ngap> \ OutNeighbours(digraph);\n[ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4, 5 ], [ 1, 3, 5 ],\ [ ] ]\ngap> RepresentativeOutNeighbours(digraph);\n[ [ 2, 4, 5 ], [ ] ]\n]]\ >\n \n\n\n \n \n \n A digraph.\n \n If the optional first argument filt is present, then this shou\ ld\n specify the category or representation the digraph being created will\n \ belong to. For example, if filt is ,\n\ then the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the op\ tional first argument filt is not present, then is used by default. \n\n If list is a list of lists \ of positive integers list the range\n [1 .. Length(list)], then\ this function returns the\n digraph with vertices E^0=[1 .. Length\ (list)], and\n edges corresponding to the entries of list. \ More precisely, there is an\n edge with source vertex i and range vert\ ex j if i is list\n list[j]. \n\n If i occurs list list[j] with multiplicity k,\n then there \ are k multiple edges from i to j. \n\n See .\n\n D := DigraphByInNeighbour\ s([\n> [2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10],\n> [1], [3, 5, 7, 8, 10], [2,\ 5, 7], [3, 6, 7, 9, 10], [1, 4],\n> [1, 5, 9], [1, 2, 7, 8], [3, 5]]);\n\ngap> D := DigraphByInNeighbours([[2\ , 3, 2], [1], [1, 2, 3]]);\n\ \ngap> D := DigraphByInNeighbours(IsMutableDigraph, \n> \ [[2, 3, 2], [1], [1, 2, 3]]);\n\n]]>\n \n\n\n \n\ \n An immutable di\ graph.\n \n Let G be any group and let ge\ ns be a list of elements of\n G. This operation returns an immut\ able digraph that corresponds to\n the Cayley graph of G with respec\ t\n gens. The vertices are the elements of G. There exists an\ edge\n from the vertex u to the vertex v if and only if ther\ e exists\n a generator g in gens such that x * g = y. \ \n\n \n\n If the optional second argument gens is not present, then\ the\n generators of G are used by default. \n\n The digraph c\ reated by this operation belongs to the category , the group G can be recovered from the\n digraph using , and the generators\n gens can be \ obtained using . \n\n Note that\ this function can only return an immutable digraph.\n\n G := DihedralGroup(8);\n\ngap> Ca\ yleyDigraph(G);\n\ngap> G := Dihe\ dralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> CayleyDigraph(G)\ ;\n\ngap> digraph := CayleyDigrap\ h(G, [()]);\n\ngap> GroupOfCayleyD\ igraph(digraph) = G;\ntrue\ngap> GeneratorsOfCayleyDigraph(digraph);\n[ () ]]]\ >\n \n\n\n \n\n Changing representations\n \n \n A binary relation.\n \ \n If digraph is a digraph with a positive number of ve\ rtices\n n, and no multiple edges, then this operation returns a bin\ ary\n relation on the points [1..n].\n The pair [i,j] is i\ n the binary relation if and only if\n [i,j] is an edge in digrap\ h.\n \n D := Digraph([[3, 2], [1, 2], [2], \ [3, 4]]);\n\ngap> AsBinaryRelation\ (D);\nBinary Relation on 4 points\n]]>\n \n\n\n \n \n A digraph, or fail.\n \n If t\ he optional first argument filt is present, then this should\n speci\ fy the category or representation the digraph being created will\n belong t\ o. For example, if filt is ,\n then \ the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optio\ nal first argument filt is not present, then is used by default. \n\n If trans is a transformati\ on, and n is a non-negative integer\n such that the restriction of <\ A>trans to [1 .. n] defines\n a transformation of [1 .\ . n], then AsDigraph\n returns the functional digraph wit\ h n vertices defined by\n trans. See .\n \n\n Specifically, the digraph returned by AsDigra\ ph has n edges:\n for each vertex x in [1 .. n\ ], there is a unique edge\n with source x; this edge has range <\ C>x^trans.\n \n\n If the optional second argument n is not supplied, then the degree of\n the transformation trans is \ used by default. If the restriction of\n trans to [1 .. n\ ] does not define a transformation of\n [1 .. n], then AsDigraph(trans, n)\n returns fail.\n\n f := Transformation([4, 3, 3, 1, 7, 9, 10, 4, 2, 3]);\nTran\ sformation( [ 4, 3, 3, 1, 7, 9, 10, 4, 2, 3 ] )\ngap> AsDigraph(f);\n\ngap> AsDigraph(f, 4);\n\ngap> AsDigraph(f, 5);\nfail\ngap> b := Binary\ RelationOnPoints(\n> [[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\nBinary Rel\ ation on 5 points\ngap> D := AsDigraph(b);\n\n]]>\n \n\n\n \n \n A &Grape; package grap\ h.\n \n If digraph is a mutable or immutable\ digraph without multiple edges, then this operation\n returns a &Grape; pa\ ckage graph that is isomorphic to digraph. \n\n If digraph is a multidigraph, then since &Grape; does not support\n multiple edges, \ the multiple edges will be reduced to a single edge in the\n result. In ord\ er words, for a multidigraph this operation will return the\n same as Gr\ aph(DigraphRemoveAllMultipleEdges(digraph)).\n\n Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> fun\ ction(x, y) return Intersection(x, y) = []; end);;\ngap> Display(Petersen);\nr\ ec(\n adjacencies := [ [ 3, 5, 8 ] ],\n group := \n Group( [ ( 1, 2, 3, 5,\ 7)( 4, 6, 8, 9,10), ( 2, 4)( 6, 9)( 7,10) \n ] ),\n isGraph := true,\n \ names := [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], \n [ 2, 4 ],\ [ 1, 5 ], [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ],\n order := 10,\n representatives \ := [ 1 ],\n schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )\ngap> Digra\ ph(Petersen);\n\ngap> Graph(last\ ) = Petersen;\ntrue]]>\n \n\n\n \n \n A &Grape; pack\ age graph.\n \n If digraph is a digraph, the\ n this method returns the same as\n , except that if <\ A>digraph is immutable, then the\n result will be stored as a mutable a\ ttribute of digraph.\n In this latter case, when AsGraph(d\ igraph) is called\n subsequently, the same &GAP; object will be \ returned as before.\n\n D := Digraph([[1, 2], [3],\ []]);\n\ngap> G := AsGraph(D);\nr\ ec( adjacencies := [ [ 1, 2 ], [ 3 ], [ ] ], group := Group(()), \n isGraph \ := true, names := [ 1 .. 3 ], order := 3, \n representatives := [ 1, 2, 3 ], \ schreierVector := [ -1, -2, -3 ] )\n]]>\n \n\n\n \n \ \n A transformation, or fail\n \n \ If digraph is a functional digraph, then AsTransformation\n r\ eturns the transformation which is defined by digraph. See . Otherwise,\n AsTransformation(di\ graph) returns fail. \n\n If digraph is a fun\ ctional digraph with n vertices, then\n AsTransformation(d\ igraph) will return the\n transformation f of degree at m\ ost n where for each 1\n \\leq i \\leq n, i ^ f is e\ qual to the unique out-neighbour of\n vertex i in digraph.\n\ \n D := Digraph([[1], [3], [2]]);\n\ngap> AsTransformation(D);\nTransformation( [ 1,\ 3, 2 ] )\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> AsTransformation(D);\nTransformation( [ 2, 3, 1 ] \ )\ngap> AsPermutation(last);\n(1,2,3)\ngap> D := Digraph([[2, 3], [], []]);\n<\ immutable digraph with 3 vertices, 2 edges>\ngap> AsTransformation(D);\nfail]]\ >\n \n\n\n \n\n New digraphs from old\n \n \n \n \n <\ Oper Name=\"DigraphCopy\" Arg=\"digraph\"/>\n A digraph.\n\ \n Each of these operations returns a new copy of digraph\ ,\n of the appropriate mutability, retaining none of the attributes or\ \n properties of digraph. \n\n DigraphCopy is a synony\ m for DigraphCopySameMutability.\n D := CycleDi\ graph(10);\n\ngap> DigraphCopy(D) = \ D;\ntrue\ngap> IsIdenticalObj(DigraphCopy(D), D);\nfalse\ngap> DigraphMutableC\ opy(D);\n\n]]>\n \n\n\n \n \n \n \n \n <\ Returns>A digraph.\n \n Each of these operations re\ turns either the original argument\n digraph, or a new copy of di\ graph of the appropriate\n mutability, according to the mutability of <\ A>digraph.\n C := CycleDigraph(10);\n\ngap> D := DigraphImmutableCopyIfImmutable(C);\n\ \ngap> IsIdenticalObj(C, D);\nfa\ lse\ngap> C = D;\ntrue\ngap> D := DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap> C = \ D;\ntrue\ngap> D := DigraphMutableCopyIfMutable(C);\n\ngap> IsMutableDigraph(D);\nfalse\ngap> D := DigraphMutableC\ opyIfImmutable(C);\n\ngap> IsMutab\ leDigraph(D);\ntrue\ngap> C := CycleDigraph(IsMutableDigraph, 10);\n\ngap> D := DigraphImmutableCopyIfImmutable(\ C);\n\ngap> IsIdenticalObj(C, D);\ \ntrue\ngap> C = D;\ntrue\ngap> D := DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ng\ ap> C = D;\ntrue\ngap> D := DigraphMutableCopyIfMutable(C);\n\ngap> IsMutableDigraph(D);\ntrue\ngap> D := Digrap\ hMutableCopyIfImmutable(C);\n\ngap\ > IsIdenticalObj(C, D);\ntrue\ngap> IsMutableDigraph(D);\ntrue]]>\n \ \n\n\n \n \n A digraph.\n \n If digraph is a digraph, and verts is a subset of the\ \n vertices of digraph, then this operation returns a digraph\n c\ onstructed from digraph by retaining precisely those vertices in\n <\ A>verts, and those edges whose source and range vertices are both\n con\ tained in verts. \n\n The vertices of the induced subdigraph are\ \n [1..Length(verts)] but the original vertex labels can be\n acc\ essed via . \n\n If digraph b\ elongs to , then\n digraph is modifi\ ed in place. If digraph belongs to , a new immutable digraph containing the\n appropriate\n vertices a\ nd edges is returned. \n\n D := Digraph([[1, 1,\ 2, 3, 4, 4], [1, 3, 4], [3, 1], [1, 1]]);\n\ngap> InducedSubdigraph(D, [1, 3, 4]);\n\ngap> DigraphVertices(last);\n[ 1 .. 3 ]\ngap> D \ := DigraphMutableCopy(D);\n\ng\ ap> new := InducedSubdigraph(D, [1, 3, 4]);\n\ngap> D = new;\ntrue\n]]>\n \n\n\n \n \n\ \n A digraph.\n \n This function returns a digraph isomorphic to t\ he subdigraph of\n digraph induced by the set of non-isolated vertic\ es,\n i.e. the set of those vertices of digraph which are the source\ or\n range of some edge in digraph.\n See .\n \n\n The ordering of the remaining vertices of dig\ raph is preserved, as are\n the labels of the remaining vertices and ed\ ges;\n see and .\n This can allow one to match a vertex in the reduced digraph to\ the\n corresponding vertex in digraph.\n \n\n If digra\ ph is immutable, then a new immutable digraph is returned.\n Otherwise,\ the isolated vertices of the mutable digraph digraph are\n removed \ in-place.\n D := Digraph([[1, 2], [], [], [1, 4], \ []]);\n\ngap> R := ReducedDigraph(\ D);\n\ngap> OutNeighbours(R);\n[ [\ 1, 2 ], [ ], [ 1, 3 ] ]\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 4, 1\ ], [ 4, 4 ] ]\ngap> DigraphEdges(R);\n[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 \ ] ]\ngap> DigraphVertexLabel(R, 3);\n4\ngap> DigraphVertexLabel(R, 2);\n2\ngap\ > D := Digraph(IsMutableDigraph, [[], [3], [2]]);\n\ngap> ReducedDigraph(D);\n\ngap> D;\n\n]]>\n \ \n\n\n \n \n \n \n \n A digraph.\n \n I\ f digraph is a digraph, then MaximalSymmetricSubdigraph\n ret\ urns a symmetric digraph without multiple edges which has the same\n vertex\ set as digraph, and whose edge list is formed from\n digraph\ by ignoring the multiplicity of edges, and by ignoring\n edges [u,v] for which there does not exist an edge [v,u].\n \n\n The di\ graph returned by MaximalSymmetricSubdigraphWithoutLoops is\n the sa\ me, except that loops are removed. \n \n If digraph is immutab\ le, then a new immutable digraph is returned.\n Otherwise, the mutable digr\ aph digraph is changed in-place into such\n a digraph described abov\ e. \n\n See , , and\n for more information.\n D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsSymmetricDigraph(D) and I\ sMultiDigraph(D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], \ [ 3, 1 ] ]\ngap> S := MaximalSymmetricSubdigraph(D);\n\ngap> IsSymmetricDigraph(S) and not IsMultiDigr\ aph(S);\ntrue\ngap> OutNeighbours(S);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\ngap> D \ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> MaximalSymmetricSubdigraph(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n \n A digraph.\n \n \ If digraph is a digraph, then MaximalAntiSymmetricSubdigraph\ \n returns an anti-symmetric subdigraph of digraph formed by\n re\ taining the vertices of digraph, discarding any duplicate edges,\n a\ nd discarding any edge [i,j] of digraph where i > j\n \ and the reverse edge [j,i] is an edge of digraph.\n In other \ words, for every\n symmetric pair of edges [i,j] and [j,i] in\ digraph, where \n i and j are distinct, it discards t\ he the edge\n [\\max(i,j),\\min(i,j)].\n \n\n If digrap\ h is immutable, then a new immutable digraph is returned.\n Otherwise, \ the mutable digraph digraph is changed in-place. \n\n See for more information.\n D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsAntiSymmetricDigraph(D) and IsMultiDigrap\ h(D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\n\ gap> D := MaximalAntiSymmetricSubdigraph(D);\n\ngap> IsAntiSymmetricDigraph(D) and not IsMultiDigr\ aph(D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ]\ngap> D \ := Digraph(IsMutableDigraph, [[2], [1]]);\n\ngap> MaximalAntiSymmetricSubdigraph(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n \n \n \n A digraph, or fail.\n \n If digraph is a digraph with at least one vertex, then\n UndirectedSpanningFores\ t returns an undirected spanning forest of\n digraph, otherwise \ this attribute returns fail. See for the definition of an undirected\n spanning forest. \n\n\ If digraph is a digraph with at least one vertex and whose is connected (see ), then UndirectedSpanningTree returns\n an\ undirected spanning tree of digraph, otherwise this attribute\n ret\ urns fail. See for the\n \ definition of an undirected spanning tree. \n\n If digraph is imm\ utable, then an immutable digraph is returned.\n Otherwise, the mutable dig\ raph digraph is changed in-place into an\n undirected spanning tree \ of digraph. \n\n Note that for an immutable digraph that has know\ n undirected spanning tree,\n the attribute UndirectedSpanningTree r\ eturns the same digraph as\n the attribute UndirectedSpanningForest.\ \n\n D := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, \ 3]]);\n\ngap> UndirectedSpann\ ingTree(D);\nfail\ngap> forest := UndirectedSpanningForest(D);\n\ngap> OutNeighbours(forest);\n[ [ 2 ]\ , [ 1 ], [ 4 ], [ 3 ] ]\ngap> IsUndirectedSpanningForest(D, forest);\ntrue\nga\ p> DigraphConnectedComponents(forest).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> Dig\ raphConnectedComponents(MaximalSymmetricSubdigraph(D)).comps;\n[ [ 1, 2 ], [ 3\ , 4 ] ]\ngap> UndirectedSpanningForest(MaximalSymmetricSubdigraph(D))\n> = for\ est;\ntrue\ngap> D := CompleteDigraph(4);\n\ngap> tree := UndirectedSpanningTree(D);\n\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ng\ ap> tree = UndirectedSpanningForest(D);\ntrue\ngap> UndirectedSpanningForest(E\ mptyDigraph(0));\nfail\ngap> D := PetersenGraph(IsMutableDigraph);\n\ngap> UndirectedSpanningTree(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n A digrap\ h.\n \n If digraph is a digraph, and p is a partition of the vertices\n of digraph, then this operation r\ eturns a digraph constructed by\n amalgamating all vertices of digraph which lie in the same part of\n p.\n \n\n A partition o\ f the vertices of digraph is a list of non-empty\n disjoint lists, s\ uch that the union of all the sub-lists is equal\n to vertex set of digr\ aph. In particular, each vertex must appear\n in precisely one sub-list\ .\n \n\n The vertices of digraph in part i of p\ will become\n vertex i in the quotient, and there exists some edge \ in digraph\n with source in part i and range in part j\ if and only if there\n is an edge from i to j in the quotien\ t.\n In particular, this means that the quotient of a digraph has no multip\ le edges.\n which was a change introduced in version 1.0.0 of the &Digraphs\ ; package.\n \n\n If digraph belongs to , then\n digraph is modified in place. If digraph \ belongs to , a new immutable digraph \ with the above\n properties is returned. \n\n D := Digraph([[2, 1], [4], [1], [1, 3, 4]]);\n\ngap> DigraphVertices(D);\n[ 1 .. 4 ]\ngap> DigraphEdges(D);\n[\ [ 1, 2 ], [ 1, 1 ], [ 2, 4 ], [ 3, 1 ], [ 4, 1 ], [ 4, 3 ], \n [ 4, 4 ] ]\ng\ ap> p := [[1], [2, 4], [3]];\n[ [ 1 ], [ 2, 4 ], [ 3 ] ]\ngap> quo := Quotient\ Digraph(D, p);\n\ngap> DigraphVert\ ices(quo);\n[ 1 .. 3 ]\ngap> DigraphEdges(quo);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 1 \ ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> QuotientDigraph(EmptyDigraph(0), []);\ \n\n]]>\n \n\ \n\n \n \n \n A di\ graph.\n \n The reverse of a digraph is the digraph\ formed by reversing the orientation\n of each of its edges, i.e. for every\ edge [i, j] of a digraph, the\n reverse contains the corresponding \ edge [j, i]. \n\n DigraphReverse returns the reverse of th\ e digraph digraph. If\n digraph is immutable, then a new immu\ table digraph is returned. Otherwise,\n the mutable digraph digraph \ is changed in-place into its reverse.\n D := Digra\ ph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> DigraphReverse(D);\n\ngap> OutNeighbours(last);\n[ [ 2, 3, 4 ], [ 4, 5 ], [ 1, 2, 5 ], \ [ 4 ], [ 2, 5 ] ]\ngap> D := Digraph([[2, 4], [1], [4], [3, 4]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 4 ]\ , [ 2, 1 ], [ 3, 4 ], [ 4, 3 ], [ 4, 4 ] ]\ngap> DigraphEdges(DigraphReverse(D\ ));\n[ [ 1, 2 ], [ 2, 1 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ], [ 4, 4 ] ]\ngap> D :=\ CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> DigraphReverse(D);\n<\ mutable digraph with 3 vertices, 3 edges>\ngap> OutNeighbours(D);\n[ [ 3 ], [ \ 1 ], [ 2 ] ]\n]]>\n \n\n\n \n \n \n A digraph.\n \n \ The dual of digraph has the same vertices as digraph,\n\ and there is an edge in the dual from i to j\n whenever t\ here is no edge from i to j in digraph.\n The dual<\ /E> is sometimes called the complement. \n\n DigraphDual r\ eturns the dual of the digraph digraph. If\n digraph is an im\ mutable digraph, then a new immutable digraph is\n returned. Otherwise, the\ mutable digraph digraph is changed in-place\n into its dual.\n <\ Example> D := Digraph([[2, 3], [], [4, 6], [5], [],\n> [7, 8, 9\ ], [], [], []]);\n\ngap> DigraphDu\ al(D);\n\ngap> D := CycleDigraph(\ IsMutableDigraph, 3);\n\ngap> Digrap\ hDual(D);\n\ngap> D;\n\n]]>\n \n\n\ \n \n \ \n \n A\ digraph.\n \n If digraph is a digraph, then\ this attribute gives the minimal\n symmetric digraph which has the same ve\ rtices and contains all the edges of\n digraph. \n \n A dig\ raph is symmetric if its adjacency matrix \n is symmetric. For a digraph with multiple\n edges this means that \ there are the same number of edges from a vertex\n u to a vertex \ v as there are from v to u; \n see . \n\n If digraph is immutable, then a new immutable \ digraph is returned.\n Otherwise, the mutable digraph digraph is cha\ nged in-place into its\n symmetric closure.\n D := \ Digraph([[1, 2, 3], [2, 4], [1], [3, 4]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> IsSymmetricDigraph(D);\ntrue\ngap> List(Out\ Neighbours(D), AsSet);\n[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, 3, 4 ] ]\ng\ ap> D := Digraph([[2, 2], [1]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 1 ] ]\ng\ ap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSymmetricClosure(D);\n\ngap> D;\n\n]]>\ \n \n\n\n \n \n \n \n \n A digraph.\n \n If digra\ ph is a digraph with no multiple edges, then these\n attributes return \ the (reflexive) transitive closure of digraph. \n\n A digraph is \ reflexive if it has a loop at every vertex, and it is\n transitiv\ e if whenever [i,j] and [j,k] are edges of\n digraph, [i,k] is also an edge. The (reflexive)\n transitive closur\ e of a digraph digraph is the least\n (reflexive and) transitive\ digraph containing digraph. \n\n If digraph is immutable,\ then a new immutable digraph is returned.\n Otherwise, the mutable digraph\ digraph is changed in-place into its\n (reflexive) transitive closu\ re. \n\n Let n be the number of vertices of digraph, and l\ et\n m be the number of edges. For an arbitrary digraph, these\n \ attributes will use a version of the Floyd-Warshall algorithm, with\n comp\ lexity O(n^3).\n\n However, for a topologically sortable digraph [se\ e ], these attributes will use me\ thods\n with complexity O(m + n + m \\cdot n) when this is faster.\n\n D := DigraphFromDiSparse6String(\".H`eOWR`U\ l^\");\n\ngap> IsReflexiveDigraph(\ D) or IsTransitiveDigraph(D);\nfalse\ngap> OutNeighbours(D);\n[ [ 4, 6 ], [ 1,\ 3 ], [ ], [ 5 ], [ ], [ 7, 8, 9 ], [ ], [ ], \n [ ] ]\ngap> T := Digrap\ hTransitiveClosure(D);\n\ngap> OutNeighbours(T);\n[ [ 4, 6, 5, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ]\ , [ ], [ 5 ], \n [ ], [ 7, 8, 9 ], [ ], [ ], [ ] ]\ngap> RT := DigraphRe\ flexiveTransitiveClosure(D);\n\ngap> OutNeighbours(RT);\n[ [ 4, 6, 5, 7, 8, 9, 1 ], [ 1, 3, 4, 5, 6, 7\ , 8, 9, 2 ], [ 3 ], \n [ 5, 4 ], [ 5 ], [ 7, 8, 9, 6 ], [ 7 ], [ 8 ], [ 9 ] ]\ \ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphReflexiveTransitiveClosure(D);\n\ngap> D;\n\n]\ ]>\n \n\n\n \n \n \n \n \n A digraph.\n \n If digraph is a topologically sortable digraph\n [see ]\n with no multiple edges, then these op\ erations return the\n (reflexive) transitive reduction of digraph.\n\n The (reflexive) transitive reduction of such a digraph is the unique\ \n least subgraph such that the (reflexive) transitive closure of the\n \ subgraph is equal to the (reflexive) transitive closure of digraph\n \ [see ].\n In order words,\ it is the least subgraph of digraph which\n retains the same reacha\ bility as digraph. \n\n If digraph is immutable, then a ne\ w immutable digraph is returned.\n Otherwise, the mutable digraph digrap\ h is changed in-place into its\n (reflexive) transitive reduction. \ \n\n Let n be the number of vertices of an arbitrary digraph, and le\ t\n m be the number of edges.\n Then these operations\n us\ e methods with complexity O(m + n + m \\cdot n).\n \n D := Digraph([[1, 2, 3], [3], [3]]);;\ngap> DigraphHasLoops(\ D);\ntrue\ngap> D1 := DigraphReflexiveTransitiveReduction(D);\n\ngap> DigraphHasLoops(D1);\nfalse\ngap> OutNeigh\ bours(D1);\n[ [ 2 ], [ 3 ], [ ] ]\ngap> D2 := DigraphTransitiveReduction(D);\ \n\ngap> DigraphHasLoops(D2);\ntru\ e\ngap> OutNeighbours(D2);\n[ [ 2, 1 ], [ 3 ], [ 3 ] ]\ngap> DigraphReflexiveT\ ransitiveClosure(D)\n> = DigraphReflexiveTransitiveClosure(D1);\ntrue\ngap> D\ igraphTransitiveClosure(D)\n> = DigraphTransitiveClosure(D2);\ntrue\ngap> D :\ = Digraph(IsMutableDigraph, [[1], [1], [1, 2, 3]]);\n\ngap> DigraphReflexiveTransitiveReduction(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n A digrap\ h.\n \n The operation returns a digraph constructed\ from digraph by adding a\n single new vertex, and no new edges. \n\n If the optional second argument label is a &GAP; object,\n \ then the new vertex will be labelled label.\n \n\n If digr\ aph belongs to , then the\n vertex is a\ dded directly to digraph. If digraph belongs to\n , an immutable copy of digraph with\n the ad\ ditional vertex is returned. \n\n D := Complete\ Digraph(3);\n\ngap> new := Digraph\ AddVertex(D);\n\ngap> D = new;\nfa\ lse\ngap> DigraphVertices(new);\n[ 1 .. 4 ]\ngap> new := DigraphAddVertex(D, G\ roup([(1, 2)]));\n\ngap> DigraphVe\ rtexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]) ]\ngap> D := CompleteBipartiteDi\ graph(IsMutableDigraph, 2, 3);\n\ng\ ap> new := DigraphAddVertex(D);\n\n\ gap> D = new;\ntrue\n]]>\n \n\n\n \n \n \n \n A digraph.\n \n For a non-negative integer m, this operati\ on returns a digraph\n constructed from digraph by adding m n\ ew vertices.\n \n\n Otherwise, if labels is a list\n consi\ sting of k &GAP; objects, then this operation returns a\n digraph co\ nstructed from digraph by adding k new vertices,\n which are \ labelled according to this list.\n \n\n If digraph belongs to\ , then the\n vertices are added directly t\ o digraph, which is changed in-place.\n If digraph belongs to\ , then the\n result is returned as an im\ mutable digraph.\n \n\n D := CompleteDigrap\ h(3);\n\ngap> new := DigraphAddVer\ tices(D, 3);\n\ngap> DigraphVertic\ es(new);\n[ 1 .. 6 ]\ngap> new := DigraphAddVertices(D, [Group([(1, 2)]), \"d\ \"]);\n\ngap> DigraphVertexLabels(\ new);\n[ 1, 2, 3, Group([ (1,2) ]), \"d\" ]\ngap> DigraphAddVertices(D, 0) = D\ ;\ntrue\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> new := DigraphAddVertices(D, 4);\n\ngap> D = new;\ntrue\n]]>\ \n \n\n\n \n \n <\ Oper Name=\"DigraphAddEdge\" Arg=\"digraph, src, ran\"\n Label=\"for a digr\ aph, source, and range\"/>\n A digraph.\n \n\ If edge is a pair of vertices of digraph, or src and\ \n ran are vertices of digraph,\n then this operation retu\ rns a digraph constructed from digraph\n by adding a new edge with s\ ource edge[1] [src] and range\n edge[2] \ [ran].\n \n \n If digraph belongs to , then the edge\n is added directly to digraph. If\ digraph belongs to , then an \ immutable copy of digraph with the\n additional edge is returned. <\ P/>\n\n D1 := Digraph([[2], [3], []]);\n\ngap> DigraphEdges(D1);\n[ [ 1, 2 ], [ 2, 3\ ] ]\ngap> D2 := DigraphAddEdge(D1, [3, 1]);\n\ngap> DigraphEdges(D2);\n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D\ 3 := DigraphAddEdge(D2, [2, 3]);\n\ngap> DigraphEdges(D3);\n[ [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ]\ngap\ > D := CycleDigraph(IsMutableDigraph, 4);\n\ngap> new := DigraphAddEdge(D, [1, 3]);\n\ngap> DigraphEdges(new);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 3, 4\ ], [ 4, 1 ] ]\ngap> D = new;\ntrue\n]]>\n \n\n\n \n \n \n A new digraph.\n \n \n \ This operation returns a new digraph with the same vertices and edges as\n \ digraph and with additional edges consisting of the orbit of the\n \ edge edge under the action of the of\n \ digraph. If edge is already an edge in digraph, then\n \ digraph is returned unchanged. The argument digraph must be\n \ an immutable digraph.\n \n\n An edge is simply a pair of vertices\ of digraph.\n\n gr1 := CayleyDigraph(Dihed\ ralGroup(8));\n\ngap> gr2 := Digr\ aphAddEdgeOrbit(gr1, [1, 8]);\n\n\ gap> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 8 ], [\ 2, 6 ], \n [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], \n [\ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], \n [ 7, 8 ], [ 7, \ 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ]\ngap> DigraphEdges(gr2);\n[ [ 1,\ 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 8 ], \n [ 2, 6 ], [ 2, 3 ]\ , [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], \n [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ \ 4, 5 ], [ 5, 3 ], [ 5, 2 ], \n [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2\ ], [ 6, 7 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], \ \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := DigraphRemoveEdgeOrbit(gr2, [1, 8]);\n\ngap> gr3 = gr1;\ntrue\n]]>\n \n\n\n \n \n A digraph.\n \n If edges is a (possibly empty) list of pairs of vertices\ of\n digraph, then this operation returns a digraph constructed fro\ m\n digraph by adding the edges specified by edges. More\n \ precisely, for every edge in edges, a new edge\n will be ad\ ded with source edge[1] and range edges[2].\n \n\n If \ an edge is included in edges with multiplicity k,\n then it w\ ill be added k times.\n\n If digraph belongs to , then the edges\n are added directly to digraph.\ If digraph belongs to , then \ the result is returned as an immutable\n digraph.\n \n\n func := function(n)\n> local source, range, i;\n> source :=\ [];\n> range := [];\n> for i in [1 .. n - 2] do\n> Add(source, i);\n> \ Add(range, i + 1);\n> od;\n> return Digraph(n, source, range);\n> end;;\ng\ ap> D := func(1024);\n\ngap>\ new := DigraphAddEdges(D,\n> [[1023, 1024], [1, 1024], [1023, 1024], [1024, 1\ ]]);\n\ngap> D = new;\n\ false\ngap> D2 := DigraphMutableCopy(func(1024));\n\ngap> new := DigraphAddEdges(D2,\n> [[1023, 1024], [1, 1\ 024], [1023, 1024], [1024, 1]]);\n\ngap> D2 = new;\ntrue\n]]>\n \n\n\n \n \n A digraph.\n \n If v is a ver\ tex of digraph, then this operation returns a\n digraph constructed \ from digraph by removing vertex v,\n along with any edge whos\ e source or range vertex is v. \n\n If digraph has n vertices, then the vertices of the returned\n digraph are [1..n-1]\ , but the original labels can be\n accessed via . \n\n If digraph belongs to , then the\n vertex is removed directly from digraph. If dig\ raph belongs\n to , an immutable copy\ of digraph\n without the vertex is returned. \n\n D := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \ \"a\", \"b\", \"c\", \"c\"],\n> [\"b\", \"c\", \"a\", \"a\", \ \"c\"]);\n\ngap> DigraphVertexLabe\ ls(D);\n[ \"a\", \"b\", \"c\" ]\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], \ [ 2, 1 ], [ 3, 1 ], [ 3, 3 ] ]\ngap> new := DigraphRemoveVertex(D, 2);\n\ngap> DigraphVertexLabels(new);\n[ \"a\ \", \"c\" ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> new := DigraphRemoveVertex(D, 1);\n\ngap> DigraphVertexLabels(D);\n[ 2, 3, 4, 5 ]\ng\ ap> D = new;\ntrue\n]]>\n \n\n\n \n \n <\ Returns>A digraph.\n \n If verts is a (possi\ bly empty) duplicate-free list of vertices of\n digraph, then this o\ peration returns a digraph constructed from\n digraph by removing ev\ ery vertex in verts, along with any edge\n whose source or range ver\ tex is in verts. \n\n If digraph has n vertices, th\ en the vertices of the new\n digraph are [1 .. n-Length(verts)], but the\n original labels can be accessed via . \n\n If digraph belongs to , then the\n vertices are removed directly from digraph. If <\ A>digraph belongs\n to , an immutable\ copy of digraph\n without the vertices is returned. \n\n D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]])\ ;\n\ngap> SetDigraphVertexLabels(\ D, [\"a\", \"b\", \"c\", \"d\", \"e\"]);\ngap> new := DigraphRemoveVertices(D,\ [2, 4]);\n\ngap> DigraphVertexLab\ els(new);\n[ \"a\", \"c\", \"e\" ]\ngap> D := CycleDigraph(IsMutableDigraph, 5\ );\n\ngap> new := DigraphRemoveVerti\ ces(D, [1, 3]);\n\ngap> DigraphVertex\ Labels(D);\n[ 2, 4, 5 ]\ngap> D = new;\ntrue\n]]>\n \ \n\n\n \n \n \n A digraph.\n \n If di\ graph is a digraph with no multiple edges and edge is a\n pair o\ f vertices of digraph, or src and ran are\n vertices o\ f digraph, then this operation returns a digraph\n constructed from \ digraph by removing the edge specified by\n edge or [src, \ ran]. \n\n If digraph belongs to , then the edge is removed directly from\n digraph. If\ digraph belongs to , an immut\ able copy of digraph without\n the edge is returned.\n \n\n \ Note that if digraph belongs to ,\n\ then a new copy of digraph will be returned even if edge or \ \n [src, ran] does not define an edge of digraph. \n\n \ D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdge(D, [250000, 1]);\n\ngap> new := DigraphRemoveEdge(D\ , [25000, 2]);;\ngap> new = D;\ntrue\ngap> IsIdenticalObj(new, D);\nfalse\ngap\ > D := DigraphMutableCopy(D);;\ngap> new := DigraphRemoveEdge(D, 2500, 2);;\ng\ ap> IsIdenticalObj(new, D);\ntrue]]>\n\n \n\n\n \n \n \n A new digraph.\n \n \n \ This operation returns a new digraph with the same vertices as\n digra\ ph and with the orbit of the edge edge (under the action\n of th\ e of digraph) removed.\n If edge\ is not an edge in digraph, then digraph is\n returned unchan\ ged. The argument digraph must be an immutable\n digraph.\n \ \n\n An edge is simply a pair of vertices of digraph.\n \ gr1 := CayleyDigraph(DihedralGroup(8));\n\ngap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n\ngap> DigraphEdges(gr1);\n[ [ 1, 2 ],\ [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 8 ], [ 2, 6 ], \n [ 3, 5 ], [ 3, 4 ], [ 3\ , 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 \ ], [ 6, 5 ], [ 6, 2 ], \n [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [\ 8, 5 ] ]\ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], \ [ 2, 1 ], [ 2, 8 ], \n [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3,\ 2 ], \n [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], \n [ 5,\ 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], \n [ 7, 8 ], [ 7, 1 ]\ , [ 7, 3 ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 :\ = DigraphRemoveEdgeOrbit(gr2, [1, 8]);\n\ngap> gr3 = gr1;\ntrue\n]]>\n \n\ \n\n \n \n A digraph.\n \n If one of the foll\ owing holds:\n \n \n digraph is a digraph wi\ th no multiple edges, and\n edges is a list of pairs of vertices\ of digraph, or\n \n digraph is a digrap\ h and edges is an empty list \n \n\n then this oper\ ation returns a digraph constructed from digraph\n by removing all o\ f the edges specified by edges\n (see ).\n \n\n If digraph \ belongs to , then the edge\n is removed dir\ ectly from digraph. If digraph belongs to , the edge is removed from an immutable copy of\n dig\ raph and this new digraph is returned. \n\n Note that if edges is empty, then this operation\n will always return digraph rather \ than a copy. Also, if any element\n of edges is invalid (i.e. does n\ ot define an edge of digraph)\n then that element will simply be ign\ ored.\n\n D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdges(D, [[250000, \ 1]]);\n\ngap> D := Digra\ phMutableCopy(D);\n\ngap> \ new := DigraphRemoveEdges(D, [[1, 2], [2, 3], [3, 100]]);\n\ngap> new = D;\ntrue\n]]>\n \n\n\n \n \n \n A digraph.\n \n If digraph is\ a digraph, then this operation returns a\n digraph constructed from di\ graph by removing every loop. A loop\n is an edge with equal source and\ range. \n\n If digraph is immutable, then a new immutable digrap\ h is returned.\n Otherwise, the loops are removed from the mutable digraph \ digraph\n in-place.\n D := Digraph([[1, \ 2, 4], [1, 4], [3, 4], [1, 4, 5], [1, 5]]);\n\ngap> DigraphRemoveLoops(D);\n\ngap> D := Digraph(IsMutableDigraph, [[1, 2], [1]]);\n\ngap> DigraphRemoveLoops(D);\n\ngap> D;\n\n\ ]]>\n \n\n\n \n \n \n A digraph.\n \n If digraph is a digraph, then this operation re\ turns a\n digraph constructed from digraph by removing all multiple\ edges.\n The result is the largest subdigraph of digraph which does\ not\n contain multiple edges. \n\n If digraph is immutable, t\ hen a new immutable digraph is returned.\n Otherwise, the multiple edges of\ the mutable digraph digraph are\n removed in-place.\n <\ ![CDATA[\ngap> D1 := Digraph([[1, 2, 3, 2], [1, 1, 3], [2, 2, 2]]);\n\ngap> D2 := DigraphRemoveAllMultiple\ Edges(D1);\n\ngap> OutNeighbours(D\ 2);\n[ [ 1, 2, 3 ], [ 1, 3 ], [ 2 ] ]\ngap> D := Digraph(IsMutableDigraph, [[2\ , 2], [1]]);\n\ngap> DigraphRem\ oveAllMultipleEdges(D);\n\ngap> D;\n\ \n]]>\n \n<\ /ManSection>\n\n \n \n \n \n A digraph.\n \n If digraph is a digraph without multiple edges\ , and edges is a\n list of pairs of vertices of digraph (the \ entries of each pair\n corresponding to the source and the range of an edge\ , respectively),\n then DigraphReverseEdges returns a digraph constr\ ucted from\n digraph by reversing the orientation of every edge spec\ ified by\n edges. If only one edge is to be reversed, then\n D\ igraphReverseEdge can be used instead. In this case, the second\n argum\ ent should just be a single vertex-pair, or the second and third\n argument\ s should be the source and range of an edge respectively. \n\n Note tha\ t even though digraph cannot have multiple edges, the\n output may h\ ave multiple edges. \n\n If digraph belongs to , then the edges\n are reversed in digraph. If dig\ raph belongs to , an immutable co\ py of digraph with the\n specified edges reversed is returned. \n\ D := DigraphFromDiSparse6String(\".Tg?i@s?t_e?_qE\ sC\");\n\ngap> DigraphEdges(D);\n\ [ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 7, 19 ], [ 9, 1 ], \n [ 11, 2 ],\ [ 21, 1 ] ]\ngap> new := DigraphReverseEdge(D, [7, 19]);\n\ngap> DigraphEdges(new);\n[ [ 1, 2 ], [ 1, 7 ], [ 1\ , 8 ], [ 5, 21 ], [ 9, 1 ], [ 11, 2 ], \n [ 19, 7 ], [ 21, 1 ] ]\ngap> D2 := \ DigraphMutableCopy(new);;\ngap> new := DigraphReverseEdges(D2, [[19, 7]]);;\ng\ ap> D2 = new;\ntrue\ngap> D = new;\ntrue\n]]>\n \n\n\n \n \n \n <\ Returns>A digraph.\n \n In the first form, if D1\ , D2, etc. are digraphs, then\n DigraphDisjointUnion retu\ rns their disjoint union.\n\n In the second form, if list is a non-e\ mpty list of digraphs, then\n DigraphDisjointUnion returns the disjo\ int union of the digraphs\n contained in the list. \n\n For a disjo\ int union of digraphs, the vertex set is the disjoint union of\n the vertex\ sets, and the edge list is the disjoint union of the edge lists.\n \n\ \n More specifically, for a collection of digraphs D1, D2,\n \ ..., the disjoint union with have\n DigraphNrVertices(D\ 1) +\n DigraphNrVertices(D2) + ... vertices.\n The edges of D1 will remain unchanged, whil\ st the edges of the\n ith digraph, D[i], will be chang\ ed so that they\n belong to the vertices of the disjoint union correspondin\ g to\n D[i]. In particular, the edges of D[i] \ will\n have their source and range increased by\n DigraphNrVertices(<\ /C>D1) + ... +\n DigraphNrVertices(D[i-1]).\n\n \n\n Note that previously set will be lost. \n\n If the first digraph D1\ [list[1]] belongs to , then <\ A>D1 [list[1]] is modified in\n place to contain the appropriate\ vertices and edges. If digraph belongs\n to , a new immutable digraph containing the\n appropriate vertices \ and edges is returned.\n D1 := CycleDigraph(3);\n\ngap> OutNeighbours(D1);\n[ [ 2 ], [ 3 ],\ [ 1 ] ]\ngap> D2 := CompleteDigraph(3);\n\ngap> OutNeighbours(D2);\n[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ]\ngap> unio\ n := DigraphDisjointUnion(D1, D2);\n\ngap> OutNeighbours(union);\n[ [ 2 ], [ 3 ], [ 1 ], [ 5, 6 ], [ 4, 6 ], [ 4\ , 5 ] ]\n]]>\n \n\n\n \n \ \n \n A digraph.\n \n In the first form, if D1, D2, etc. are digraphs, then\n \ DigraphEdgeUnion returns their edge union.\n\n In the second form\ , if list is a non-empty list of digraphs, then\n DigraphEdgeUnio\ n returns the edge union of the digraphs contained\n in the list. \ \n\n The vertex set of the edge union of a collection of digraphs is the\n \ union of the vertex sets, whilst the edge list of the edge union is\ \n the concatenation of the edge lists. The number of vertices of t\ he\n edge union is equal to the maximum number of vertices of one of\ the\n digraphs, whilst the number of edges of the edge union will equal th\ e\n sum of the number of edges of each digraph. \n\n Note tha\ t previously set will be lost.\n \n\ \n If the first digraph D1 [list[1]] belongs to , then D1 [list[1]] is modified in\n \ place to contain the appropriate vertices and edges. If digraph belongs\ \n to , a new immutable digraph containin\ g the\n appropriate vertices and edges is returned.\n D := CycleDigraph(10);\n\ngap> Di\ graphEdgeUnion(D, D);\n\nga\ p> D1 := Digraph([[2], [1]]);\n\ng\ ap> D2 := Digraph([[2, 3], [2], [1]]);\n\ngap> union := DigraphEdgeUnion(D1, D2);\n\ngap> OutNeighbours(union);\n[ [ 2, 2, 3 ], [ 1, 2 ], [ 1\ ] ]\ngap> union = DigraphByEdges(\n> Concatenation(DigraphEdges(D1), DigraphE\ dges(D2)));\ntrue\n]]>\n \n\n\n \n \n \n A digraph.\n \n In the first form, if D1, D2, etc. are digraphs, then\n \ DigraphJoin returns their join.\n\n In the second form, if lis\ t is a non-empty list of digraphs,\n then DigraphJoin returns th\ e join of the digraphs contained in the\n list. \n\n The join of a c\ ollection of digraphs D1, D2, ... is\n formed by first\ taking the of the collection.\n\n In the disjoint union, if i \\neq j then there are no edges between\n vertices corresponding to digraphs D\ [i] and\n D[j] in the collection; the join is crea\ ted by including\n all such edges. \n\n For example, the join of two \ empty digraphs is a complete bipartite\n digraph. \n\n Note that prev\ iously set will be lost.\n \n\n \ If the first digraph D1 [list[1]] belongs to , then D1 [list[1]] is modified in\n place \ to contain the appropriate vertices and edges. If digraph belongs\n \ to , a new immutable digraph containing the\ \n appropriate vertices and edges is returned.\n D \ := CompleteDigraph(3);\n\ngap> IsC\ ompleteDigraph(DigraphJoin(D, D));\ntrue\ngap> D2 := CycleDigraph(3);\n\ngap> DigraphJoin(D, D2);\n\n]]>\n \n\ \n\n \n \n \n \ A digraph.\n \n In the first form, if g\ r1, gr2, etc. are digraphs, then\n DigraphCartesianProduct returns a digraph isomorphic to their\n Cartesian product.\n \n\n \ In the second form, if list is a non-empty list of digraphs,\n th\ en DigraphCartesianProduct returns a digraph isomorphic to the \n Ca\ rtesian product of the digraphs contained in the list.\n \n \n Mathe\ matically, the Cartesian product of two digraphs G, H\n is a \ digraph with vertex set Cartesian(DigraphVertices(G),\n DigraphVertices(\ H)) such that there is an edge from [u, u'] to\n [v, v'] \ iff u = v and there is an edge from u' to\n v' in H or u' = v' and there is an edge from\n u to v \ in G.\n \n\n Due to technical reasons, the digraph D r\ eturned by\n DigraphCartesianProduct has vertex set\n [1 .. D\ igraphNrVertices(G)*DigraphNrVertices(H)] instead, and the\n exact\n \ method of encoding pairs of vertices into integers is implementation\n sp\ ecific. The original vertex pair can be somewhat regained by using\n . In addition,\n are preserved: if vertex pair\n [u,u'] was en\ coded as i then the vertex label of i\n will be the pair of v\ ertex labels of u and u' i.e.\n DigraphVertexLabel(D,i) = \ [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H,u')].\n \n\n \ As the Cartesian product is associative, the Cartesian product of a collect\ ion \n of digraphs gr1, gr2, ... is computed in the ob\ vious \n fashion. \n \n\n gr := ChainDigraph\ (4);\n\ngap> gr2 := CycleDigraph(3);\ \n\ngap> gr3 := DigraphCartesianProdu\ ct(gr, gr2);\n\ngap> IsIsomorphi\ cDigraph(gr3, \n> Digraph([[2, 5], [3, 6], [4, 7], [8], \n> [6, 9], [\ 7, 10], [8, 11], [12],\n> [10, 1], [11, 2], [12, 3], [4]]));\ntrue\n]\ ]>\n \n\n\n \n \n \n A digraph.\n \n In the first form, if gr1, gr2, etc. are digraphs, \ then\n DigraphDirectProduct returns a digraph isomorphic to their di\ rect\n product.\n \n\n In the second form, if list is a no\ n-empty list of digraphs, then\n DigraphDirectProduct returns a digr\ aph isomorphic to the direct\n product of the digraphs contained in the lis\ t.\n \n\n Mathematically, the direct product of two digraphs G, H is a\n digraph with vertex set Cartesian(DigraphVertices(G),\ \n DigraphVertices(H)) such that there is an edge from [u, u']\n\ to [v, v'] iff there is an edge from u to v in\n <\ C>G and an edge from u' to v' in H. \n \n\n \ Due to technical reasons, the digraph D returned by\n DigraphDi\ rectProduct has vertex set\n [1 .. DigraphNrVertices(G)*DigraphNrVe\ rtices(H)] instead, and the\n exact method of encoding pairs of vertice\ s into integers is implementation\n specific. The original vertex pair can\ be somewhat regained by using\n . In addition\n are preserved: if \ vertex pair\n [u,u'] was encoded as i then the vertex label o\ f i\n will be the pair of vertex labels of u and u' i.\ e.\n DigraphVertexLabel(D,i) = [DigraphVertexLabel(G,u),\n DigraphVer\ texLabel(H,u')].\n \n\n As the direct product is associative, th\ e direct product of a collection \n of digraphs gr1, gr2, \ ... is computed in the obvious \n fashion. \n \n\n gr := ChainDigraph(4);\n\n\ gap> gr2 := CycleDigraph(3);\n\ngap> \ gr3 := DigraphDirectProduct(gr, gr2);\n\ngap> IsIsomorphicDigraph(gr3, \n> Digraph([[6], [7], [8], [], \n> \ [10], [11], [12], [],\n> [2], [3], [4], []]));\ntrue\n]]>\n \n\n\n \n \n A list of transf\ ormations.\n \n If digraph is a Cartesian pr\ oduct digraph, \n digraph = DigraphCartesianProduct(gr_1, gr_2, ... ), then\n DigraphCartesianProductProjections returns a list proj<\ /C>\n such that proj[i] is the projection onto the i-th\n \ coordinate of the product.\n \n\n A projection is an idempotent endo\ morphism of digraph. If\n gr1, gr2, ... are all loopless digr\ aphs, then the induced \n subdigraph of digraph on the image of p\ roj[i] is \n isomorphic to gr_i.\n \n\n Currently this\ attribute is only set upon creating an immutable \n digraph via Digraph\ CartesianProduct and there is no way \n of calculating it for other dig\ raphs.\n \n\n For more information see\n \n\n D := DigraphCartesianProduct(ChainDigraph(3), CycleDigraph(4),\n> Digraph(\ [[2], [2]]));;\ngap> HasDigraphCartesianProductProjections(D);\ntrue\ngap> pro\ j := DigraphCartesianProductProjections(D);; Length(proj);\n3\ngap> IsIdempote\ nt(proj[2]);\ntrue\ngap> RankOfTransformation(proj[3]);\n2\ngap> S := ImageSet\ OfTransformation(proj[2]);;\ngap> IsIsomorphicDigraph(CycleDigraph(4), Induced\ Subdigraph(D, S));\ntrue\n]]>\n \n\n\n \ \n \n A list of transformations.\n \n If\ digraph is a direct product digraph, \n digraph = DigraphDirectP\ roduct(gr_1, gr_2, ... ), then\n DigraphDirectProductProjections\ returns a list proj\n such that proj[i] is the projection on\ to the i-th\n coordinate of the product.\n \n\n A projecti\ on is an idempotent endomorphism of digraph. If\n gr1, gr2, ... are all loopless digraphs, then the image \n of digraph under \ proj[i] is isomorphic to gr_i.\n \n\n Currently this attr\ ibute is only set upon creating an immutable\n digraph via DigraphDirect\ Product and there is no way of \n calculating it for other digraphs.\n \ \n\n For more information, see\n \n\n D := Di\ graphDirectProduct(ChainDigraph(3), CycleDigraph(4),\n> Digraph([[2], [2]]));;\ \ngap> HasDigraphDirectProductProjections(D);\ntrue\ngap> proj := DigraphDirec\ tProductProjections(D);; Length(proj);\n3\ngap> IsIdempotent(proj[2]);\ntrue\n\ gap> RankOfTransformation(proj[3]);\n2\ngap> P := DigraphRemoveAllMultipleEdge\ s(\n> ReducedDigraph(OnDigraphs(D, proj[2])));; \ngap> IsIsomorphicDigraph(Cyc\ leDigraph(4), P);\ntrue\n]]>\n \n\n\n \ \n \n \n A digraph.\n \n Given a digraph digraph, the operation returns the digraph\n \ obtained by associating a vertex with each edge of digraph, and\n \ creating an edge from a vertex v to a vertex u if and only\n \ if the terminal vertex of the edge associated with v is the start\n \ vertex of the edge associated with u. \n\n Note that the returned\ digraph is always a new immutable digraph, and the\n argument digraph is never modified.\n LineDigraph(CompleteDigrap\ h(3));\n\ngap> LineDigraph(ChainD\ igraph(3));\n\n]]>\n \n\n\n \n \n \n A digraph.\n \n Given a symmetri\ c digraph digraph, the operation returns the\n symmetric digraph obt\ ained by associating a vertex with each edge of\n digraph, ignoring \ directions and multiplicites, and adding an edge\n between two vertices if \ and only if the corresponding edges have a vertex in\n common. \n\n N\ ote that the returned digraph is always a new immutable digraph, and the\n \ argument digraph is never modified.\n LineU\ ndirectedDigraph(CompleteDigraph(3));\n\ngap> LineUndirectedDigraph(DigraphSymmetricClosure(ChainDigraph(3)));\n\ \n]]>\n \ \n\n\n \n \n A digraph.\n \n Let digraph<\ /A> be a digraph with vertex set V. This function\n returns the doub\ le digraph of digraph. The vertex set of the double\n digraph is the\ orginal vertex set together with a duplicate. The edges are\n [u_1, v_2\ ] and [u_2, v_1] if and only if [u, v] is an\n edge in digraph, together with the original edges and their\n duplicates. \ \n\n If digraph is mutable, then digraph is modified in-place\ . If\n digraph is immutable, then a new immutable digraph constructe\ d as\n described above is returned.\n gamma := \ Digraph([[2], [3], [1]]);\n\ngap> \ DoubleDigraph(gamma);\n\n]]>\n \n\n\n \n \n A digraph.\n \n Let digraph be a digraph with vertex set V. Thi\ s function\n returns the bipartite double digraph of digraph. The ve\ rtex set of\n the double digraph is the original vertex set together with a\ duplicate. The\n edges are [u_1, v_2] and [u_2, v_1] if and \ only if [u,\n v] is an edge in digraph. The resulting graph\ is bipartite,\n since the orignal edges are not included in the resulting \ digraph. \n\n If digraph is mutable, then digraph is modif\ ied in-place. If\n digraph is immutable, then a new immutable digrap\ h constructed as\n described above is returned.\n gamma := Digraph([[2], [3], [1]]);\n\ngap> BipartiteDoubleDigraph(gamma);\n\n]]>\n \n\n\n \ \n \n \n A digraph.\n \n For a digraph digraph this operation returns a new digrap\ h constructed\n from digraph, such that a loop is added for every ve\ rtex which did not\n have a loop in digraph. \n\n If digrap\ h is immutable, then a new immutable digraph is returned.\n Otherwise, \ the loops are added to the loopless vertices of the mutable digraph\n di\ graph in-place.\n D := EmptyDigraph(13);\n\ngap> D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n\ [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ], [ 8 ], [ 9 ], \n [ 10 ], [\ 11 ], [ 12 ], [ 13 ] ]\ngap> D := Digraph([[1, 2, 3], [1, 3], [1]]);\n\ngap> D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n[ [\ 1, 2, 3 ], [ 1, 3, 2 ], [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, \ 3);\n\ngap> DigraphAddAllLoops(D);\n\ \ngap> D;\n\n]]>\n \n\n\n \n \n \n A digraph.\ \n \n The first argument is a digraph, the second argument is\ a non-negative\n integer or a list of positive integers. This operation re\ turns a digraph on\n the same set of vertices as digraph, with two v\ ertices being adjacent\n if and only if the distance between them in dig\ raph equals i\n or is a number in list. See . \n\n If digraph is mutable, then digraph is modified in-plac\ e. If\n digraph is immutable, then a new immutable digraph construct\ ed as\n described above is returned.\n digraph \ := DigraphFromSparse6String(\n> \":]n?AL`BC_DEbEF`GIaGHdIJeGKcKL_@McDHfILaBJfH\ MjKM\");\n\ngap> DistanceDigraph\ (digraph, 1);\n\ngap> DistanceDi\ graph(digraph, [1, 2]);\n\n]]><\ /Example>\n \n\n\n \n \n A digraph.\n \n Given a symmetric loopless digraph with no multiple edges \ digraph,\n the k-closure of digraph is defined to \ be the unique\n smallest symmetric loopless digraph C with no multip\ le edges on the\n vertices of digraph that contains all the edges of\ digraph\n and satsifies the property that the sum of the degrees of\ every two\n non-adjacenct vertices in C is less than k. See \ , , , and . \n\ \n The operation DigraphClosure returns the k-closure of\n \ digraph.\n D := CompleteDigraph(6);\n\ngap> D := DigraphRemoveEdges(D, [[1, \ 2], [2, 1]]);\n\ngap> closure := \ DigraphClosure(D, 6);\n\ngap> IsC\ ompleteDigraph(closure);\ntrue\n]]>\n \n\ \n\n \n \n \ \n A digraph\ .\n \n If digraph is a symmetric digraph, th\ en\n DigraphMycielskian returns the Mycielskian of digraph.\n \n The Mycielskian of a symmetric digraph is a larger symmetric dig\ raph\n constructed from it, which has a larger chromatic number. For furthe\ r\n information, see https://en.wikipedia.org/wiki/Mycielskian.<\ P/>\n\n If digraph is immutable, then a new immutable digraph is ret\ urned.\n Otherwise, the mutable digraph digraph is changed in-place \ into\n its Mycielskian. \n\n D := CycleDigra\ ph(2);\n\ngap> ChromaticNumber(D);\n2\ \ngap> D := DigraphMycielskian(D);\n\ngap> ChromaticNumber(D);\n3\ngap> D := DigraphMycielskian(D);\n\ngap> ChromaticNumber(D);\n4\ngap> D := C\ ompleteBipartiteDigraph(IsMutable, 2, 3);\n\ngap> DigraphMycielskian(D);\n\ngap> D;\n\n]]>\n \ \n\n\n \n\n Random d\ igraphs\n \n \n A digraph.\n \n If the o\ ptional first argument filt is present, then this should\n specify t\ he category or representation the digraph being created will\n belong to. F\ or example, if filt is ,\n then the \ digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optional \ first argument filt is not present, then is used by default.\n\n If n is a positive integer, \ then this function returns a random\n digraph with n vertices and wi\ thout multiple edges. The result\n may or may not have loops. \n\n If\ the optional second argument p is a float with value\n 0 \\leq <\ /M> p \\leq 1, then an edge will exist between each\n pair\ of vertices with probability approximately p.\n If p is not \ specified, then a random probability will be assumed\n (chosen with uniform\ probability).\n RandomDigraph(1000);\n\ngap> RandomDigraph(10000, 0.023);\n\ngap> RandomDigraph(IsMutab\ leDigraph, 1000, 1 / 2);\n\n\ ]]>\n \n\n\n \n \n A digraph.\n \n If n is a positive integer, then this function returns\ a random\n digraph with n vertices. If the optional second argument\ \n m is a positive integer, then the digraph will have m\n \ edges. If m is not specified, then the number of edges will be\n ch\ osen randomly (with uniform probability) from the range\n [1 .. {\ n \\choose 2}].\n \n\n The method used by this function c\ hooses each edge from the set of\n all possible edges with uniform probabil\ ity. No effort is made to avoid\n creating multiple edges, so it is possibl\ e (but not guaranteed) that the\n result will have multiple edges. The resu\ lt may or may not have loops. \n RandomMultiDigraph\ (1000);\n\ngap> Rando\ mMultiDigraph(1000, 950);\n\n]]>\n \n\n\n \n \n A digraph.\ \n \n If the optional first argument filt is present, \ then this should\n specify the category or representation the digraph being\ created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if fi\ lt is , then the digraph will be im\ mutable. \n If the optional first argument filt is not present, then\ is used by default. \n\n If <\ A>n is a non-negative integer, this function returns a random\n tournam\ ent with n vertices. See . \n <\ ![CDATA[\ngap> RandomTournament(10);\n\ \ngap> RandomTournament(IsMutableDigraph, 10);\n\n]]>\n \n\n\n \n \n A digraph.\n \n If the optional first argument filt is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if \ filt is , then the digraph will \ be immutable. \n If the optional first argument filt is not present,\ then is used by default. \n\n \ If n is a positive integer, this function return a random lattice\n \ with m vertices, where it is guaranteed that m is between\n \ n and 2 * n. See . \n\ RandomLattice(10);\n\ngap> RandomLattice(IsMutableDigraph, 10);\n\n]]>\n \n\n\ \n \n\n Standard examples\n \n \n A digraph\ .\n \n If n is a positive integer, this func\ tion returns a chain\n with n vertices and n - 1 edges\ .\n Specifically,\n for each vertex i (with i < n), there is a directed\n edge with source i and range i + 1.\ \n\n If the optional first argument filt is present, then this \ should\n specify the category or representation the digraph being created w\ ill\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt is \ , then the digraph will be immutable. \ \n If the optional first argument filt is not present, then is used by default. \n\n The of a chain represents a\n total or\ der.\n ChainDigraph(42);\n\ngap> ChainDigraph(IsMutableDigraph, 10);\n\n]]>\n \n\n\ \n \n \n A digraph.\n \n If n is a non-negativ\ e integer, this function returns the complete\n digraph with n verti\ ces. See . \n\n If the optional first \ argument filt is present, then this should\n specify the category or\ representation the digraph being created will\n belong to. For example, if\ filt is ,\n then the digraph being \ created will be mutable, if filt is , then the digraph will be immutable. \n If the optional first argument\ filt is not present, then is\ used by default.\n CompleteDigraph(20);\n\ngap> CompleteDigraph(IsMutableDigraph, \ 10);\n\n]]>\n \n\n\n \n \n A digraph.\n \ \n A complete bipartite digraph is a digraph whose vertices\n can be par\ titioned into two non-empty vertex sets, such there exists\n a unique edge \ with source i and range j if and only if\n i and j<\ /C> lie in different vertex sets. \n\n If m and n are pos\ itive integers, this function\n returns the complete bipartite digraph with\ vertex sets of sizes\n m (containing the vertices [1 .. m]) \ and\n n (containing the vertices [m + 1 .. m + n]).\n \ \n\n If the optional first argument filt is present, then this shoul\ d\n specify the category or representation the digraph being created will\n\ belong to. For example, if filt is \ ,\n then the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n \ If the optional first argument filt is not present, then is used by default.\n\n CompleteBipartiteDigraph(2, 3);\n\ngap> CompleteBipartiteDigraph(IsMutableDigraph, 3\ , 2);\n\n]]>\n \n\n\n \n \n A digraph.\n \ \n For a list orders of n positive integers,\ \n this function returns the digraph containing n independent sets\ \n of vertices of orders [l[1] .. l[n]]. Moreover,\n\ each vertex is adjacent to every other not contained in the\n same \ independent set.\n \n\n If the optional first argument filt is present, then this should\n specify the category or representation \ the digraph being created will\n belong to. For example, if filt i\ s ,\n then the digraph being created will\ be mutable, if filt is , then\ the digraph will be immutable. \n If the optional first argument filt\ is not present, then is used \ by default.\n CompleteMultipartiteDigraph([5, 4,\ 2]);\n\ng\ ap> CompleteMultipartiteDigraph(IsMutableDigraph, [5, 4, 2]);\n\n]]>\n \n\ \n\n \n \n A digraph.\n \n If n is a positive int\ eger, this function returns a cycle\n digraph with n vertices\ and n edges. Specifically,\n for each vertex i (with i < n), there is a directed\n edge with source i and range\ i + 1. In addition, there is\n an edge with source n and ran\ ge 1. \n\n If the optional first argument filt is present\ , then this should\n specify the category or representation the digraph bei\ ng created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if \ filt is , then the digraph will be \ immutable. \n If the optional first argument filt is not present, th\ en is used by default.\n CycleDigraph(1);\n\ngap>\ CycleDigraph(123);\n\ngap> CycleDi\ graph(IsMutableDigraph, 10);\n\n]]\ >\n \n\n\n \n \n \n A digraph.\n \n If n i\ s a non-negative integer, this function returns the\n empty or nu\ ll digraph with n vertices. An empty\n digraph is one with no ed\ ges.\n \n \n If the optional first argument filt is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if \ filt is , then the digraph will \ be immutable. \n If the optional first argument filt is not present,\ then is used by default. \n\n \ NullDigraph is a synonym for EmptyDigraph.\n\n EmptyDigraph(20);\n\ngap\ > NullDigraph(10);\n\ngap> EmptyDigr\ aph(IsMutableDigraph, 10);\n]]>\n \n\n\n \n \n A digraph.\n \n If n and k are non-negative integers, then this oper\ ation\n returns a symmetric digraph which corresponds to the undirected \ Johnson\n graph J(n, k). \n\n If the optional first argum\ ent filt is present, then this should\n specify the category or repr\ esentation the digraph being created will\n belong to. For example, if f\ ilt is ,\n then the digraph being creat\ ed will be mutable, if filt is ,\ then the digraph will be immutable. \n If the optional first argument f\ ilt is not present, then is used\ by default. \n\n The Johnson graph J(n, k) has vertices g\ iven by all the\n k-subsets of the range [1 .. n], and\ two vertices are\n connected by an edge iff their intersection has size \ k - 1.\n\n gr := JohnsonDigraph(3, 1\ );\n\ngap> OutNeighbours\ (gr);\n[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ]\ngap> gr := JohnsonDigraph(4, 2);\n\ngap> OutNeighbours(gr);\ \n[ [ 2, 3, 4, 5 ], [ 1, 3, 4, 6 ], [ 1, 2, 5, 6 ], [ 1, 2, 5, 6 ], \n [ 1, 3\ , 4, 6 ], [ 2, 3, 4, 5 ] ]\ngap> JohnsonDigraph(1, 0);\n\ngap> JohnsonDigraph(IsMutableDigraph, 1, 0);\n\n]]>\n \n\n\n \ \n \n A \ digraph.\n \n\n From https://en.wikipedia.org/\ wiki/Petersen_graph: \n\n The Petersen graph is an undirected g\ raph with 10 vertices and 15 edges.\n It is a small graph that serves as a \ useful example and counterexample for\n many problems in graph theory. The \ Petersen graph is named after Julius\n Petersen, who in 1898 constructed it\ to be the smallest bridgeless cubic\n graph with no three-edge-coloring. \n \n If the optional first argument filt is present, then \ this should\n specify the category or representation the digraph being crea\ ted will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt is , then the digraph will be immutab\ le. \n If the optional first argument filt is not present, then is used by default. \n\n See also \ .\n\n Chromati\ cNumber(PetersenGraph());\n3\ngap> PetersenGraph(IsMutableDigraph);\n\n]]>\n \n\n\n \n \n A digraph.\n \n If n is a positive integer and k is a non-negative\n integer less than\ n / 2, then this operation returns\n the generalised Pete\ rsen graph GPG(n, k). \n\n From\n https://en.\ wikipedia.org/wiki/Generalized_Petersen_graph:\n \n\n The g\ eneralized Petersen graphs are a family of cubic\n graphs formed by connect\ ing the vertices of a regular polygon to the\n corresponding vertices of a \ star polygon. They include the Petersen graph\n and generalize one of the w\ ays of constructing the Petersen graph. The\n generalized Petersen graph fa\ mily was introduced in 1950 by H. S. M.\n Coxeter and was given its name in\ 1969 by Mark Watkins.\n \n\n If the optional first argument \ filt is present, then this should\n specify the category or representat\ ion the digraph being created will\n belong to. For example, if filt\ is ,\n then the digraph being created will\ be mutable, if filt is , then t\ he digraph will be immutable. \n If the optional first argument filt\ is not present, then is used by def\ ault. \n\n See also .\n\n GeneralisedPetersenGraph(7, 2);\n\ngap> GeneralisedPetersenGraph(40, 1);\n\ngap> D := GeneralisedPetersenGraph(5, \ 2);\n\ngap> IsIsomorph\ icDigraph(D, PetersenGraph());\ntrue\ngap> GeneralisedPetersenGraph(IsMutableD\ igraph, 9, 4);\n\n]]>\n \ \n\n\n \n\n\n \n Attributes and operations\n Vertices and edges\n \n \ \n A list of intege\ rs.\n \n Returns the vertices of the digraph dig\ raph. \n\n Note that the vertices of a digraph are always a range o\ f\n positive integers from 1 to the number of vertices of the graph.\ \n gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ \ngap> DigraphVertices(gr);\n[ 1 .\ . 3 ]\ngap> gr := Digraph([1, 2, 3, 4, 5, 7],\n> [1, 2, 2, 4,\ 4],\n> [2, 7, 5, 3, 7]);\n\ngap> DigraphVertices(gr);\n[ 1 .. 6 ]\ngap> DigraphVertices(Random\ Digraph(100));\n[ 1 .. 100 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphVertices(D);\n[ 1 .. 3 ]\ \n]]>\n \n\n\n \n \n An integer.\ \n \n Returns the number of vertices of the digraph digrap\ h. \n\n gr := Digraph([\"a\", \"b\", \"c\"\ ],\n> [\"a\", \"b\", \"b\"],\n> [\"b\", \"c\ \", \"a\"]);\n\ngap> DigraphNrVert\ ices(gr);\n3\ngap> gr := Digraph([1, 2, 3, 4, 5, 7],\n> [1, 2\ , 2, 4, 4],\n> [2, 7, 5, 3, 7]);\n\ngap> DigraphNrVertices(gr);\n6\ngap> DigraphNrVertices(Rand\ omDigraph(100));\n100\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphNrVertices(D);\n3\n]]>\n \n\n\n \n \n A list of lists.\n \n DigraphEdges returns a list of edges of the digraph digraph\ ,\n where each edge is a pair of elements of of\n the form [source,range].\n \n\n The entries of \ DigraphEdges(digraph) are in one-to-one\n corresponenc\ e with the edges of digraph. Hence\n DigraphEdges(digraph\ ) is duplicate-free if and only if\n digraph contains no \ multiple edges. \n\n The entries of DigraphEdges are guaranteed \ to be sorted by their\n first component (i.e. by the source of each edge), \ but they are not\n necessarily then sorted by the second component.\n gr := DigraphFromDiSparse6String(\".DaXbOe?EAM@G~\");\n\ \ngap> edges := ShallowCopy(\ DigraphEdges(gr));; Sort(edges);\ngap> edges;\n[ [ 1, 1 ], [ 1, 3 ], [ 1, 3 ],\ [ 1, 4 ], [ 1, 5 ], [ 2, 1 ], \n [ 2, 2 ], [ 2, 3 ], [ 2, 5 ], [ 3, 2 ], [ 3\ , 4 ], [ 3, 5 ], \n [ 4, 2 ], [ 4, 4 ], [ 4, 5 ], [ 5, 1 ] ]\ngap> D := Cycle\ Digraph(IsMutableDigraph, 3);\n\ngap\ > DigraphEdges(D);\n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\n]]>\n \n\n\n \n \n An integer.\n \n This f\ unction returns the number of edges of the digraph digraph.\n gr := Digraph([\n> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [\ 2, 4, 5], [1]]);;\ngap> DigraphNrEdges(gr);\n15\ngap> gr := Digraph([\"a\", \"\ b\", \"c\"],\n> [\"a\", \"b\", \"b\"],\n> [\ \"b\", \"a\", \"a\"]);\n\ngap\ > DigraphNrEdges(gr);\n3\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphNrEdges(D);\n3\n]]>\n \n\n\n \n \n A list of vertices.\n \n This function returns a list of the sinks of the digraph\n d\ igraph.\n A sink of a digraph is a vertex with out-degree zero.\n Se\ e .\n gr := Digra\ ph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);\n\ngap> DigraphSinks(gr);\n[ 3, 5 ]\ngap> D := CycleDigraph(\ IsMutableDigraph, 3);\n\ngap> Digrap\ hSinks(D);\n[ ]\n]]>\n \n\n\n \n \n A list o\ f vertices.\n \n This function returns an immutable\ list of the sources of the digraph\n digraph.\n A source of a di\ graph is a vertex with in-degree zero.\n See .\n gr := Digraph([[3, 5, 2, 2], [3], [], [5, 2,\ 5, 3], []]);\n\ngap> Digraph\ Sources(gr);\n[ 1, 4 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSources(D);\n[ ]\n]]>\n \n\n\n \n \n A list of positive integers, \ or fail.\n \n If digraph is a digraph\ whose only directed cycles are loops, then\n DigraphTopologicalSort\ returns the vertices of digraph ordered\n so that every edge's sour\ ce appears no earlier in the list than its range.\n If the digraph digra\ ph contains directed cycles of length greater\n than 1, then thi\ s operation returns fail.\n \n\n See section for the definition\n of a directed cycle, \ and the definition of a loop.\n\n \n\n The method used for this attr\ ibute has complexity O(m+n) where\n m is the number of edges \ (counting multiple edges as one) and\n n is the number of vertices i\ n the digraph. \n D := Digraph([\n> [2, 3], []\ , [4, 6], [5], [], [7, 8, 9], [], [], []]);\n\ngap> DigraphTopologicalSort(D);\n[ 2, 5, 4, 7, 8, 9, 6, 3, 1 ]\ng\ ap> D := Digraph(IsMutableDigraph, [[2, 3], [3], [4], []]);\n\ngap> DigraphTopologicalSort(D);\n[ 4, 3, 2, 1 ]\n]]\ >\n \n\n\n \n \n \n \n If digraph is\ a digraph, then the first operation returns the label\n of the vertex i\ . The second operation can be used to set the label\n of the vertex \ i in digraph to the arbitrary &GAP; object\n obj. \n\ \n The label of a vertex can be changed an arbitrary number of times. If no\ \n label has been set for the vertex i, then the default value is\n \ i. \n\n If digraph is a digraph created from a record \ with a component\n vertices, then the labels of the vertices are set\ to the value of\n this component. \n\n Induced subdigraphs, and some\ other operations which create new digraphs from\n old ones, inherit their \ labels from their parents.\n\n D := DigraphFromDig\ raph6String(\"&DHUEe_\");\n\ngap>\ DigraphVertexLabel(D, 3);\n3\ngap> D := Digraph([\"a\", \"b\", \"c\"], [], []\ );\n\ngap> DigraphVertexLabel(D, 2);\ \n\"b\"\ngap> SetDigraphVertexLabel(D, 2, \"d\");\ngap> DigraphVertexLabel(D, \ 2);\n\"d\"\ngap> D := InducedSubdigraph(D, [1, 2]);\n\ngap> DigraphVertexLabel(D, 2);\n\"d\"\ngap> D := Digraph(IsM\ utableDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> DigraphVertexLabel(D, 1);\n\"e\"\ngap> SetDigraphVertexLabel(D\ , 1, \"h\");\ngap> DigraphVertexLabel(D, 1);\n\"h\"\ngap> InducedSubdigraph(D,\ [1, 2]);\n\ngap> DigraphVertexLabel(D,\ 1);\n\"h\"\n]]>\n \n\n\n \n \n \n \n If digra\ ph is a digraph, then DigraphVertexLabels returns a\n copy of th\ e labels of the vertices in digraph.\n SetDigraphVertexLabels\ can be used to set the labels of the vertices\n in digraph to the l\ ist of\n arbitrary &GAP; objects list. \n\n The label of a ve\ rtex can be changed an arbitrary number of times. If no\n label has been se\ t for the vertex i, then the default value is\n i. \n\n \ If digraph is a digraph created from a record with a component\n <\ C>vertices, then the labels of the vertices are set to the value of\n t\ his component. \n\n Induced subdigraphs, and other operations which crea\ te new digraphs from\n old ones, inherit their labels from their parents.\n\ \n D := DigraphFromDigraph6String(\"&DHUEe_\");\n<\ immutable digraph with 5 vertices, 11 edges>\ngap> DigraphVertexLabels(D);\n[ \ 1 .. 5 ]\ngap> D := Digraph([\"a\", \"b\", \"c\"], [], []);\n\ngap> DigraphVertexLabels(D);\n[ \"a\", \"b\", \"c\" \ ]\ngap> SetDigraphVertexLabel(D, 2, \"d\");\ngap> DigraphVertexLabels(D);\n[ \ \"a\", \"d\", \"c\" ]\ngap> D := InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"a\", \"c\" ]\ng\ ap> D := Digraph(IsMutableDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> SetDigraphVertexLabels(D, [\"h\", \"i\", \ \"j\"]);\ngap> DigraphVertexLabels(D);\n[ \"h\", \"i\", \"j\" ]\ngap> InducedS\ ubdigraph(D, [1, 3]);\n\ngap> DigraphVe\ rtexLabels(D);\n[ \"h\", \"j\" ]\n]]>\n \n\n\n \n \n \n \ \n If digraph is a digraph without multiple edges, then\ the first\n operation returns the label of the edge from vertex i t\ o vertex\n j. The second operation can be used to set the label of \ the edge\n between vertex i and vertex j to the arbitrary &GA\ P; object\n obj. \n\n The label of an edge can be changed an \ arbitrary number of times. If no\n label has been set for the edge, then th\ e default value is 1. \n\n Induced subdigraphs, and some other o\ perations which create new digraphs from\n old ones, inherit their edge lab\ els from their parents.\n\n See also .\n\n\ D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphEdgeLabel(D, 3, 1);\n1\ \ngap> SetDigraphEdgeLabel(D, 2, 5, [42]);\ngap> DigraphEdgeLabel(D, 2, 5);\n[\ 42 ]\ngap> D := InducedSubdigraph(D, [2, 5]);\n\ngap> DigraphEdgeLabel(D, 1, 2);\n[ 42 ]\ngap> D := ChainDigrap\ h(IsMutableDigraph, 5);\n\ngap> Digr\ aphEdgeLabel(D, 2, 3);\n1\ngap> SetDigraphEdgeLabel(D, 4, 5, [1729]);\ngap> Di\ graphEdgeLabel(D, 4, 5);\n[ 1729 ]\ngap> InducedSubdigraph(D, [4, 5]);\n\ngap> DigraphEdgeLabel(D, 1, 2);\n[ 1729 ]\ \n]]>\n \n\n\n \n \n \n \n \n If digraph is a d\ igraph without multiple edges, then\n DigraphEdgeLabels returns a co\ py of the labels of the edges\n in digraph as a list of lists lab\ els such that\n labels[i][j] is the label on the edge from verte\ x i\n to vertex OutNeighbours(digraph)[i][j].\n\n SetDi\ graphEdgeLabels can be used to set the labels of the edges in\n digr\ aph without multiple edges to the list labels of lists of\n arbi\ trary &GAP; objects such that list[i][j] is the label on the edge\n \ from vertex i to the vertex OutNeighbours(digraph>[i][j].\n\n \ Alternatively SetDigraphEdgeLabels can be called with binary function\ \n func that as its second argument that when passed two vertices i\n and j returns the label for the edge between vertex i and vertex\n j. \n\n The label of an edge can be changed an\ arbitrary number of times. If no\n label has been set for an edge, then th\ e default value is 1. \n\n Induced subdigraphs, and some other o\ perations which create new digraphs\n from old ones, inherit their labels f\ rom their parents.\n\n D := DigraphFromDigraph6Str\ ing(\"&DHUEe_\");\n\ngap> Digraph\ EdgeLabels(D);\n[ [ 1 ], [ 1, 1, 1 ], [ 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ]\ngap> \ SetDigraphEdgeLabel(D, 2, 1, \"d\");\ngap> DigraphEdgeLabels(D);\n[ [ 1 ], [ \ \"d\", 1, 1 ], [ 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ]\ngap> D := InducedSubdigraph(\ D, [1, 2, 3]);\n\ngap> DigraphEdge\ Labels(D);\n[ [ 1 ], [ \"d\", 1 ], [ 1 ] ]\ngap> OutNeighbours(D);\n[ [ 3 ], [\ 1, 3 ], [ 1 ] ]\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\ \n\ngap> DigraphEdgeLabels(D);\n[ [\ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ]\ngap> SetDigraphEdgeLa\ bel(D, 2, 4, \"a\");\ngap> DigraphEdgeLabels(D);\n[ [ 1, 1, 1 ], [ 1, \"a\", 1\ ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ]\ngap> InducedSubdigraph(D, [1, 2, 3, 4]);\n\ \ngap> DigraphEdgeLabels(D);\n[ [ 1,\ 1 ], [ 1, \"a\" ], [ 1, 1 ], [ 1, 1 ] ]\ngap> OutNeighbors(D);\n[ [ 3, 4 ], [\ 3, 4 ], [ 1, 2 ], [ 1, 2 ] ]\n]]>\n \n\ \n\n \n \n A list of edges.\n \n DigraphInEdg\ es returns the list of all edges of digraph\n which have vert\ ex as their range.\n D := Digraph([[2, 2], [3,\ 3], [4, 4], [1, 1]]);\n\ngap\ > DigraphInEdges(D, 2);\n[ [ 1, 2 ], [ 1, 2 ] ]\n]]>\n \n\n\n \n \n A list of edges.\n \n \ DigraphOutEdges returns the list of all edges of digraph\n \ which have vertex as their source.\n D := \ Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n\ngap> DigraphOutEdges(D, 2);\n[ [ 2, 3 ], [ 2, 3 ] ]\n]]>\n \n\n\n \n \n\ \n true or false.\n \ \n In the first form, this function returns true if an\ d only if the list\n list specifies an edge in the digraph digrap\ h. Specifically,\n this operation returns true if list i\ s a pair of positive\n integers where list[1] is the source a\ nd list[2]\n is the range of an edge in digraph, and <\ K>false otherwise. \n\n The second form simply returns true \ if [u, v] is\n an edge in digraph, and false\ otherwise.\n\n D := Digraph([[2, 2], [6], [], [3]\ , [], [1]]);\n\ngap> IsDigrap\ hEdge(D, [1, 1]);\nfalse\ngap> IsDigraphEdge(D, [1, 2]);\ntrue\ngap> IsDigraph\ Edge(D, [1, 8]);\nfalse\n]]>\n \n\n\n \ \n \n \n \n \n true or false.\n \n If digraph is a digraph and list is a list of pairs of\n\ vertices of digraph, then IsMatching returns true if\ \n list is a matching of digraph. The operation\n IsMa\ ximalMatching returns true if list is a maximal \n matchi\ ng, IsMaximumMatching returns true if list is a\n maxi\ mum matching and IsPerfectMatching returns true if \n list\ is a perfect, matching of digraph, respectively. \n Otherwise,\ each of these operations return false.\n \n\n A matching<\ /E> M of a digraph digraph is a subset of the\n edges of d\ igraph, i.e. DigraphEdges(digraph), such\n that no pair o\ f distinct edges in M are incident to the same vertex\n of digrap\ h. Note that this definition allows a matching to contain\n loops. Se\ e . The matching M is\n maximal<\ /E> if it is contained in no larger matching of the digraph,\n is maximu\ m if it has the greatest cardinality among all matchings and\n is pe\ rfect if every vertex of the digraph is incident to an edge in\n the ma\ tching. Every maximum or perfect matching is maximal. Note, however,\n tha\ t not every perfect matching of digraphs with loops is maximum.\n\n D := Digraph([[1, 2], [1, 2], [2, 3, 4], [3, 5], [1]]);\n\ngap> IsMatching(D, [[2, 1], [3, 2]\ ]);\nfalse\ngap> edges := [[3, 2]];;\ngap> IsMatching(D, edges);\ntrue\ngap> I\ sMaximalMatching(D, edges);\nfalse\ngap> edges := [[2, 1], [3, 4]];;\ngap> IsM\ aximalMatching(D, edges);\ntrue\ngap> IsPerfectMatching(D, edges);\nfalse\ngap\ > edges := [[1, 2], [3, 3], [4, 5]];;\ngap> IsPerfectMatching(D, edges);\ntrue\ \ngap> IsMaximumMatching(D, edges);\nfalse\ngap> edges := [[1, 1], [2, 2], [3,\ 3], [4, 5]];;\ngap> IsMaximumMatching(D, edges);\ntrue\n]]>\n \n\n\n \n \n A list of pairs of vertices.\ \n \n This function returns a maximal matching of the digraph\ digraph.\n \n\n For the definition of a maximal matching, se\ e . \n\n D := Dig\ raphFromDiSparse6String(\".IeAoXCJU@|SHAe?d\");\n\ngap> M := DigraphMaximalMatching(D);; IsMaximalMatching(D, \ M);\ntrue\ngap> D := RandomDigraph(100);;\ngap> IsMaximalMatching(D, DigraphMa\ ximalMatching(D));\ntrue\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph,\ 9, 2);\n\ngap> IsMaximalMatching(\ D, DigraphMaximalMatching(D));\ntrue\n]]>\n \n\n\n \n \n A list of pairs of vertices.\n \n \ This function returns a maximum matching of the digraph digraph.\n \ \n\n For the definition of a maximum matching, see .\n If digraph is bipartite (see ), then \n the algorithm used has complexity O(m*sqrt(n))<\ /C>. Otherwise for general \n graphs the complexity is O(m*n*log(n))\ . Here n is the number of vertices \n and m is the number of \ edges.\n\n D := DigraphFromDigraph6String(\"&I@EA_\ A?AdDp[_c??OO\");\n\ngap> M := D\ igraphMaximumMatching(D);; IsMaximalMatching(D, M);\ntrue\ngap> Length(M);\n5\ \ngap> D := Digraph([[5, 6, 7, 8], [6, 7, 8], [7, 8], [8], \n> \ [], [], [], []]);;\ngap> M := DigraphMaximumMatching(D);\n[ [ 1, 5 ], [ 2, 6\ ], [ 3, 7 ], [ 4, 8 ] ]\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph,\ 9, 2);\n\ngap> M := DigraphMaximu\ mMatching(D);;\ngap> IsMaximalMatching(D, M);\ntrue\ngap> Length(M);\n9\n]]>\n \n\n\n \n\n Neighbours and degree\n \n \n \n A square matrix of non-negative integers.\n \ \n This function returns the adjacency matrix mat of t\ he digraph\n digraph.\n The value of the matrix entry mat[i][j\ ] is the number of edges\n in digraph with source i and r\ ange j. If digraph\n has no vertices, then the empty list is \ returned. \n\n The function AdjacencyMatrix returns an immutable\ list of\n lists, whereas the function AdjacencyMatrixMutableCopy re\ turns a copy\n of AdjacencyMatrix that is a mutable list of mutable \ lists. \n\n gr := Digraph([\n> [2, 2, 2], [1, \ 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, 3, 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],\ \n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4, 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\n<\ immutable multidigraph with 10 vertices, 44 edges>\ngap> mat := AdjacencyMatri\ x(gr);;\ngap> Display(mat);\n[ [ 0, 3, 0, 0, 0, 0, 0, 0, 0, 0 ],\n \ [ 1, 0, 1, 0, 0, 1, 0, 1, 1, 1 ],\n [ 0, 0, 0, 1, 0, 1, 0, \ 1, 0, 0 ],\n [ 1, 1, 1, 0, 0, 0, 0, 0, 1, 0 ],\n [ 0, 0, 2, \ 0, 0, 0, 0, 0, 0, 0 ],\n [ 0, 0, 1, 0, 1, 1, 0, 0, 0, 1 ],\n \ [ 1, 1, 0, 0, 0, 0, 1, 0, 0, 0 ],\n [ 1, 1, 1, 0, 1, 1, 0, \ 0, 0, 2 ],\n [ 1, 0, 1, 1, 1, 0, 0, 1, 0, 1 ],\n [ 0, 1, 1, \ 1, 0, 1, 1, 0, 0, 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ \ngap> Display(AdjacencyMatrix(D));\ \n[ [ 0, 1, 0 ],\n [ 0, 0, 1 ],\n [ 1, 0, 0 ] ]\n]]>\n <\ /Description>\n\n\n \n \n A polynomial with integer coeffici\ ents.\n \n This function returns the characteristic\ polynomial of the digraph\n digraph. That is it returns the charac\ teristic polynomial\n of the adjacency matrix of the digraph digraph\n D := Digraph([\n> [2, 2, 2], [1, 3, 6, 8, 9, \ 10], [4, 6, 8],\n> [1, 2, 3, 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],\n> [1, 2, 3\ , 10, 5, 6, 10], [1, 3, 4, 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\n\ngap> CharacteristicPolynomial(D);\nx_1\ ^10-3*x_1^9-7*x_1^8-x_1^7+14*x_1^6+x_1^5-26*x_1^4+51*x_1^3-10*x_1^2\\\n+18*x_1\ -30\ngap> D := CompleteDigraph(5);\n\ngap> CharacteristicPolynomial(D);\nx_1^5-10*x_1^3-20*x_1^2-15*x_1-4\ngap> \ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> CharacteristicPolynomial(D);\nx_1^3-1\n]]>\n \n\n\n \n \n \n A square matrix of booleans.\n \n If digraph is a digraph with a positive number of vertices\n \ n, then BooleanAdjacencyMatrix(digraph)\n ret\ urns the boolean adjacency matrix mat of digraph. The\n valu\ e of the matrix entry mat[j][i] is true if and only if\n ther\ e exists an edge in digraph with source j and range\n i. If digraph has no vertices, then the empty list is\n returned. <\ P/>\n\n Note that the boolean adjacency matrix loses information about mult\ iple\n edges. \n\n The attribute BooleanAdjacencyMatrix retu\ rns an immutable list of\n immutable lists, whereas the function\n Bo\ oleanAdjacencyMatrixMutableCopy returns a copy of the\n BooleanAdjac\ encyMatrix that is a mutable list of mutable lists. \n gr := Digraph([[3, 4], [2, 3], [1, 2, 4], [4]]);\n\ngap> PrintArray(BooleanAdjacencyMatrix(gr));\n[\ [ false, false, true, true ],\n [ false, true, true, false ],\n \ [ true, true, false, true ],\n [ false, false, false, true ] ]\n\ gap> gr := CycleDigraph(4);;\ngap> PrintArray(BooleanAdjacencyMatrix(gr));\n[ \ [ false, true, false, false ],\n [ false, false, true, false ],\n \ [ false, false, false, true ],\n [ true, false, false, false ] ]\ng\ ap> BooleanAdjacencyMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsM\ utableDigraph, 3);\n\ngap> PrintArra\ y(BooleanAdjacencyMatrix(D));\n[ [ false, true, false ],\n [ false, fal\ se, true ],\n [ true, false, false ] ]\n]]>\n \ \n\n\n \n \n A function.\n \n If \ digraph is a digraph, then DigraphAdjacencyFunction returns\n \ a function which takes two integer parameters x, y and returns\n true if there exists an edge from vertex x to vertex y\n \ in digraph and false if not.\n\n di\ graph := Digraph([[1, 2], [3], []]);\n\ngap> foo := DigraphAdjacencyFunction(digraph);\nfunction( u, v ) ... end\ \ngap> foo(1, 1);\ntrue\ngap> foo(1, 2);\ntrue\ngap> foo(1, 3);\nfalse\ngap> f\ oo(3, 1);\nfalse\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"a\"]);\n\ngap> foo := DigraphAdjacencyFunctio\ n(gr);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(3, 2);\nfals\ e\ngap> foo(3, 1);\nfalse\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> foo := DigraphAdjacencyFunction(D)\ ;\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(2, 1);\nfalse\n]]\ >\n \n\n\n \n \n \n A list of positive integers.\n \n \ DigraphRange and DigraphSource return the range and source of\ \n the digraph digraph. More precisely, position i in\n DigraphRange(digraph) is the range of the ith edge of\n \ digraph.\n\n gr := Digraph([\n> [2, 1, 3, 5\ ], [1, 3, 4], [2, 3], [2], [1, 2, 3, 4]]);\n\ngap> DigraphRange(gr);\n[ 2, 1, 3, 5, 1, 3, 4, 2, 3, 2, 1, 2, 3, \ 4 ]\ngap> DigraphSource(gr);\n[ 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5 ]\nga\ p> DigraphEdges(gr);\n[ [ 1, 2 ], [ 1, 1 ], [ 1, 3 ], [ 1, 5 ], [ 2, 1 ], [ 2,\ 3 ], \n [ 2, 4 ], [ 3, 2 ], [ 3, 3 ], [ 4, 2 ], [ 5, 1 ], [ 5, 2 ], \n [ 5,\ 3 ], [ 5, 4 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphRange(D);\n[ 2, 3, 1 ]\ngap> Digraph\ Source(D);\n[ 1, 2, 3 ]\n]]>\n \n\n\n \ \n \n \n \n \ \n The adjacencies of a digraph.\n \n Thi\ s function returns the list out of out-neighbours of each vertex\n o\ f the digraph digraph.\n More specifically, a vertex j appear\ s in out[i] each time\n there exists an edge with source i an\ d range j in\n digraph. \n\n The function OutNeighb\ ours returns an immutable list of\n lists, whereas the function OutN\ eighboursMutableCopy returns a copy\n of OutNeighbours which is \ a mutable list of mutable lists. \n\n gr := Di\ graph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \"b\"],\n> \ [\"b\", \"a\", \"c\"]);\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ 1, 3 ], [ ] ]\ngap> gr := Digraph(\ [[1, 2, 3], [2, 1], [3]]);\n\ngap>\ OutNeighbours(gr);\n[ [ 1, 2, 3 ], [ 2, 1 ], [ 3 ] ]\ngap> gr := DigraphByAdj\ acencyMatrix([\n> [1, 2, 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n\ngap> OutNeighbours(gr);\n[ [ 1, 2, 2, 3 ],\ [ 1, 2 ], [ 3 ] ]\ngap> OutNeighboursMutableCopy(gr);\n[ [ 1, 2, 2, 3 ], [ 1,\ 2 ], [ 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\n]]\ >\n \n\n\n \n \n \n \n \n A list of lists of \ vertices.\n \n This function returns the list in\ n of in-neighbours of each vertex\n of the digraph digraph.\n \ More specifically, a vertex j appears in inn[i] each time\n \ there exists an edge with source j and range i in\n digrap\ h. \n\n The function InNeighbours returns an immutable list \ of\n lists, whereas the function InNeighboursMutableCopy returns a c\ opy\n of InNeighbours which is a mutable list of mutable lists. \ \n\n Note that each entry of inn is sorted into ascending order. \n\n gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"c\"])\ ;\n\ngap> InNeighbours(gr);\n[ [ 2\ ], [ 1 ], [ 2 ] ]\ngap> gr := Digraph([[1, 2, 3], [2, 1], [3]]);\n\ngap> InNeighbours(gr);\n[ [ 1, 2 ], [ 1, 2 \ ], [ 1, 3 ] ]\ngap> gr := DigraphByAdjacencyMatrix([\n> [1, 2, 1],\n> [1, 1,\ 0],\n> [0, 0, 1]]);\n\ngap>\ InNeighbours(gr);\n[ [ 1, 2 ], [ 1, 1, 2 ], [ 1, 3 ] ]\ngap> InNeighboursMuta\ bleCopy(gr);\n[ [ 1, 2 ], [ 1, 1, 2 ], [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMu\ tableDigraph, 3);\n\ngap> InNeighbou\ rs(D);\n[ [ 3 ], [ 1 ], [ 2 ] ]\n]]>\n \n\n\n \n \n \n \n A list of non-negative integers.\n \n\n Given a digraph digraph with n vertices, the fu\ nction\n OutDegrees returns an immutable list out of length <\ M>n, such that\n for a vertex i in digraph, the value of \ out[i] is the\n out-degree of vertex i.\n See . \n\n The function OutDegreeSequence retur\ ns the same list,\n after it has been sorted into non-increasing order. \n\n The function OutDegreeSet returns the same list, sorted into\n\ increasing order with duplicate entries removed. \n\n D := Digraph([[1, 3, 2, 2], [], [2, 1], []]);\n\ngap> OutDegrees(D);\n[ 4, 0, 2, 0 ]\ngap> OutD\ egreeSequence(D);\n[ 4, 2, 0, 0 ]\ngap> OutDegreeSet(D);\n[ 0, 2, 4 ]\ngap> D \ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutDegrees(D);\n[ 1, 1, 1 ]\n]]>\n \n\n\n \n \n <\ Attr Name=\"InDegreeSequence\" Arg=\"digraph\"/>\n \n A list of non-negative integers.\n <\ Description>\n\n Given a digraph digraph with n vertices, the\ function\n InDegrees returns an immutable list inn of length\ n, such that\n for a vertex i in digraph, the value o\ f inn[i] is\n the in-degree of vertex i.\n See . \n\n The function InDegreeSequence retur\ ns the same list,\n after it has been sorted into non-increasing order. \n\n The function InDegreeSet returns the same list, sorted into\n \ increasing order with duplicate entries removed. \n\n D := Digraph([[1, 3, 2, 2, 4], [], [2, 1, 4], []]);\n\ngap> InDegrees(D);\n[ 2, 3, 1, 2 ]\ngap> \ InDegreeSequence(D);\n[ 3, 2, 2, 1 ]\ngap> InDegreeSet(D);\n[ 1, 2, 3 ]\ngap> \ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> InDegrees(D);\n[ 1, 1, 1 ]\n]]>\n \n\n\n \n \n The non-negative integer.\n \ \n This operation returns the out-degree of the vertex vertex in the\ \n digraph digraph.\n The out-degree of vertex is the numb\ er of edges in digraph\n whose source is vertex.\n \n \ D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2],\ [1, 1, 2, 2, 1, 2, 2]]);\n\ \ngap> OutDegreeOfVertex(D, 1);\n3\ngap> OutDegreeOfVertex(D, 2);\n2\ngap> Out\ DegreeOfVertex(D, 3);\n4\ngap> OutDegreeOfVertex(D, 4);\n7\n]]>\n <\ /Description>\n\n\n \n \n \n A list of vertices.\n \n\n This operation returns the list out of vertices of the digr\ aph\n digraph.\n A vertex i appears in the list out\ each time there exists an\n edge with source vertex and range i\ in digraph; in\n particular, this means that out may con\ tain duplicates. \n D := Digraph([\n> [2, 2, 3]\ , [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]]);\n\ngap> OutNeighboursOfVertex(D, 1);\n[ 2, 2, 3 ]\ngap> O\ utNeighboursOfVertex(D, 3);\n[ 2, 2, 3 ]\n]]>\n \n\n\n \n \n A non-negative integer.\n \n\ This operation returns the in-degree of the vertex vertex in the\n \ digraph digraph.\n The in-degree of vertex is the number o\ f edges in digraph\n whose range is vertex.\n D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1, 2, 2, 1\ , 2, 2]]);\n\ngap> InDegreeO\ fVertex(D, 1);\n5\ngap> InDegreeOfVertex(D, 2);\n9\ngap> InDegreeOfVertex(D, 3\ );\n0\ngap> InDegreeOfVertex(D, 4);\n2\n]]>\n \n\n\n \n \n \n A list of postitive vertices.\n \n\n \ This operation returns the list inn of vertices of the digraph\n \ digraph.\n A vertex i appears in the list inn each time t\ here exists an\n edge with source i and range vertex in d\ igraph; in\n particular, this means that inn may contain duplica\ tes. \n\n D := Digraph([\n> [2, 2, 3], [1, 3, \ 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]]);\n\ngap> InNeighboursOfVertex(D, 1);\n[ 2, 4, 4, 4 ]\ngap> InNeighb\ oursOfVertex(D, 2);\n[ 1, 1, 3, 3, 4, 4, 4, 4 ]\n]]>\n \n\n\n \n \n A list of vertices.\n \n If digraph is a digraph, then DigraphLoops returns the list\n cons\ isting of the of digraph\n at which \ there is a loop. See . \n\n D := Digraph([[2], [3], []]);\n\ngap> DigraphHasLoops(D);\nfalse\ngap> DigraphLoops(D);\n[ ]\ngap>\ D := Digraph([[3, 5], [1], [2, 4, 3], [4], [2, 1]]);\n\ngap> DigraphLoops(D);\n[ 3, 4 ]\ngap> D := Digraph(IsMu\ tableDigraph, [[1], [1]]);\n\ngap> D\ igraphLoops(D);\n[ 1 ]\n]]>\n \n\n\n \ \n \n \n A positive integer or fail\n \ \n If the first argument is a partial order digraph then these operations return the meet, or\ \n the join, of the two input vertices. If the meet (or join) is does not\n\ exist then fail is returned. The meet (or join) is guaranteed to\n \ exist when the first argument satisfies\n (or\n ) - see the docum\ entation for these\n properties for the definition of the meet (or the join\ ).\n\n D := Digraph([[1], [1, 2], [1, 3], [1, 2, 3\ , 4]]);\n\ngap> PartialOrderDigrap\ hMeetOfVertices(D, 2, 3);\n4\ngap> PartialOrderDigraphJoinOfVertices(D, 2, 3);\ \n1\ngap> PartialOrderDigraphMeetOfVertices(D, 1, 2);\n2\ngap> PartialOrderDig\ raphJoinOfVertices(D, 3, 4);\n3\ngap> D := Digraph([[1], [2], [1, 2, 3], [1, 2\ , 4]]);\n\ngap> PartialOrderDigrap\ hMeetOfVertices(D, 3, 4);\nfail\ngap> PartialOrderDigraphJoinOfVertices(D, 3, \ 4);\nfail\ngap> PartialOrderDigraphMeetOfVertices(D, 1, 2);\nfail\ngap> Partia\ lOrderDigraphJoinOfVertices(D, 1, 2);\nfail\n]]>\n \n\ \n\n \n \n A square matrix of non-negative integers.\n \n This function returns the outdegree matrix mat of the digr\ aph\n digraph. The value of the ith diagonal matrix entry is \ the\n outdegree of the vertex i in digraph. All off-diagonal \ entries \n are 0.\n If digraph has no vertices, then the e\ mpty list is returned. \n\n See for more inf\ ormation.\n D := Digraph([[1, 2, 3], [4], [1, 3, 4\ ], []]);\n\ngap> PrintArray(Degree\ Matrix(D));\n[ [ 3, 0, 0, 0 ],\n [ 0, 1, 0, 0 ],\n [ 0, 0, 3, 0 \ ],\n [ 0, 0, 0, 0 ] ]\ngap> D := CycleDigraph(5);;\ngap> PrintArray(Degre\ eMatrix(D));\n[ [ 1, 0, 0, 0, 0 ],\n [ 0, 1, 0, 0, 0 ],\n [ 0, 0\ , 1, 0, 0 ],\n [ 0, 0, 0, 1, 0 ],\n [ 0, 0, 0, 0, 1 ] ]\ngap> D\ egreeMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMutableDigraph, \ 3);\n\ngap> Display(DegreeMatrix(D))\ ;\n[ [ 1, 0, 0 ],\n [ 0, 1, 0 ],\n [ 0, 0, 1 ] ]\n]]>\n \ \n\n\n \n \n A square matrix of integers.\n \ \n This function returns the outdegree Laplacian matrix mat\ of the\n digraph digraph. The outdegree Laplacian matrix is def\ ined as\n DegreeMatrix(digraph) - AdjacencyMatrix(digraph). If\n \ digraph has no vertices, then the empty list is returned. \n\n \ See and for more\ \n information.\n gr := Digraph([[1, 2, 3], [4]\ , [1, 3, 4], []]);\n\ngap> PrintAr\ ray(LaplacianMatrix(gr));\n[ [ 2, -1, -1, 0 ],\n [ 0, 1, 0, -1 ]\ ,\n [ -1, 0, 2, -1 ],\n [ 0, 0, 0, 0 ] ]\ngap> LaplacianMatrix\ (EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> Display(LaplacianMatrix(D));\n[ [ \ 1, -1, 0 ],\n [ 0, 1, -1 ],\n [ -1, 0, 1 ] ]\n]]>\n \ \n\n\n \n\n Reachabi\ lity and connectivity\n \n \n An integer or fail.\n \n This function returns the diameter of the digraph digraph<\ /A>.\n \n\n If a digraph digraph is strongly connected and ha\ s at least 1\n vertex, then the diameter is the maximum shortest dis\ tance between\n any pair of distinct vertices. Otherwise then the diameter \ of\n digraph is undefined, and this function returns the value\n \ fail. \n\n See . \n \ D := Digraph([[2], [3], [4, 5], [5], [1, 2, 3, 4, \ 5]]);\n\ngap> DigraphDiameter(D);\ \n3\ngap> D := ChainDigraph(2);\n\nga\ p> DigraphDiameter(D);\nfail\ngap> IsStronglyConnectedDigraph(D);\nfalse\ngap>\ D := GeneralisedPetersenGraph(IsMutableDigraph, 6, 2);\n\ngap> DigraphDiameter(D);\n4\n]]>\n \n\n\n \n \n \n \n An integer or fail\n \n I\ f there is a directed path in the digraph digraph between vertex\n <\ A>u and vertex v, then this operation returns the length of the\n \ shortest such directed path. If no such directed path exists, then this\n \ operation returns fail. See section for the definition of a directed path. \n\n If t\ he second form is used, then list should be a list of length two,\n \ containing two positive integers which correspond to the vertices u\n \ and v. \n\n Note that as usual, a vertex is considered to be a\ t distance 0 from\n itself . \n\n If the third form is used, then \ list1 and list2 are both lists\n of vertices. The shortest direc\ ted path between list1 and\n list2 is then the length of the \ shortest directed path which starts\n with a vertex in list1 and ter\ minates at a vertex in list2, if\n such directed path exists. If \ list1 and list2 have non-empty\n intersection, the operation ret\ urns 0. \n D := Digraph([[2], [3], [1, 4\ ], [1, 3], [5]]);\n\ngap> DigraphS\ hortestDistance(D, 1, 3);\n2\ngap> DigraphShortestDistance(D, [3, 3]);\n0\ngap\ > DigraphShortestDistance(D, 5, 2);\nfail\ngap> DigraphShortestDistance(D, [1,\ 2], [4, 5]);\n2\ngap> DigraphShortestDistance(D, [1, 3], [3, 5]);\n0\n]]>\n \n\n\n \n \n A square matrix.\n \n If digraph is a digraph with n vertice\ s, then this\n function returns an n \\times n matrix mat, wh\ ere each entry is\n either a non-negative integer, or fail. If n\ = 0, then an\n empty list is returned. \n\n If there is a direc\ ted path from vertex i to vertex j, then\n the value of ma\ t[i][j] is the length of the shortest such directed\n path. If no such \ directed path exists, then the value of mat[i][j] is\n fail. \ We use the convention that the distance from every vertex to\n itself is <\ C>0, i.e. mat[i][i] = 0 for all vertices i.\n \n\n \ The method used in this function is a version of the Floyd-Warshall\n algo\ rithm, and has complexity O(n^3).\n\n D := \ Digraph([[1, 2], [3], [1, 2], [4]]);\n\ngap> mat := DigraphShortestDistances(D);;\ngap> PrintArray(mat);\n[ [ \ 0, 1, 2, fail ],\n [ 2, 0, 1, fail ],\n [ 1, \ 1, 0, fail ],\n [ fail, fail, fail, 0 ] ]\ngap> D := CycleDigrap\ h(IsMutableDigraph, 3);\n\ngap> Digr\ aphShortestDistances(D);\n[ [ 0, 1, 2 ], [ 2, 0, 1 ], [ 1, 2, 0 ] ]\n]]>\n \n\n\n \n \n An integer, or\ infinity.\n \n If digraph is a digra\ ph and v is a vertex in digraph,\n then this operation return\ s the length of the longest directed walk in\n digraph which begins \ at vertex v. See section for the definitions of directed\n walk, directed cycle, and loop\ .\n \n\n \n \n If there exists a directed wa\ lk starting at vertex v\n which traverses a loop or a directed c\ ycle,\n then we consider there to be a walk of infinite length from \ v\n (realised by repeatedly traversing the loop/directed cycle),\n \ and so the result is infinity.\n To disallow walks using \ loops, try using\n : \n\n <\ C>DigraphLongestDistanceFromVertex(DigraphRemoveLoops(digraph,v)\ ).\n \n\n \n Otherwise, if all directed walk\ s\n starting at vertex v have finite length,\n then the l\ ength of the longest such walk is returned.\n \n \n\n \ Note that the result is 0 if and only if v is a sink of\n digraph. See .\n\n D\ := Digraph([[2], [3, 4], [], [5], [], [6]]);\n\ngap> DigraphLongestDistanceFromVertex(D, 1);\n3\ngap> DigraphLo\ ngestDistanceFromVertex(D, 3);\n0\ngap> 3 in DigraphSinks(D);\ntrue\ngap> Digr\ aphLongestDistanceFromVertex(D, 6);\ninfinity\n]]>\n \ \n\n\n \n \ \n \n A list of vertices\n \n This operation returns the list of all vertices i\ n digraph digraph\n such that the shortest distance to a vertex v\ ertex is\n distance or is in the list distances. \n\n\ digraph should be a digraph, vertex should be a positive\n \ integer, distance should be a non-negative integer, and\n dista\ nces should be a list of non-negative integers.\n\n D := Digraph([[2], [3], [1, 4], [1, 3]]);\n\ngap> DigraphDistanceSet(D, 2, [1, 2]);\n[ 3, 1, 4 ]\ngap> Di\ graphDistanceSet(D, 3, 1);\n[ 1, 4 ]\ngap> DigraphDistanceSet(D, 2, 0);\n[ 2 ]\ \n]]>\n \n\n\n \n \n An integer, or infinity\ .\n \n This attribute returns the girth \ of the digraph digraph.\n The girth of a digraph is the lengt\ h of its shortest simple circuit.\n See section for the definitions\n of simple circuit, directed cy\ cle, and loop.\n \n\n If digraph has no directed cycles, then\ this function will return\n infinity. If digraph contains a\ loop, then this function will\n return 1.\n \n\n In the w\ orst case, the method used in this function is a version of the\n Floyd-War\ shall algorithm, and has complexity O(n ^ 3), where\n n is the number of vertices in digraph. If the digraph has\n known \ automorphisms [see ], then the performance\n is\ likely to be better.\n \n\n For symmetric digraphs, see also . \n D := Digr\ aph([[1], [1]]);\n\ngap> DigraphGi\ rth(D);\n1\ngap> D := Digraph([[2, 3], [3], [4], []]);\n\ngap> DigraphGirth(D);\ninfinity\ngap> D := Digraph([[2\ , 3], [3], [4], [1]]);\n\ngap> Dig\ raphGirth(D);\n3\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 6, 2);\ \n\ngap> DigraphGirth(D);\n2\n]]><\ /Example>\n \n\n\n \n \n An integer, or infinity.\n \n This attribute returns the odd girth of the digraph digraph.\n The odd girth of a digraph is the\ length of its shortest simple\n circuit of odd length. See Section for the definitions of simple \ circuit, directed\n cycle, and loop.\n \n\n If digraph has\ no directed cycles of odd length, then this function\n will return infi\ nity, even if digraph has a directed cycle\n of even length. If\ digraph contains a loop, then this function\n will return 1.\ \n \n\n See also . \n D := Digraph([[2], [3, 1], [1]]);\n\ngap> DigraphOddGirth(D);\n3\ngap> D := CycleDigraph(4);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap\ > D := Digraph([[2], [3], [], [3], [4]]);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := CycleDigraph(IsMutabl\ eDigraph, 4);\n\ngap> DigraphDisjoin\ tUnion(D, CycleDigraph(5));\n\ngap> \ DigraphOddGirth(D);\n5\n]]>\n \n\n\n \ \n \n An integer or infinity.\n \n If dig\ raph is a symmetric digraph, then this function returns the\n girth of \ digraph when treated as an undirected graph (i.e. each pair\n of edg\ es [i, j] and [j, i] is treated as a single edge between\n i and j). \n\n The girth of an undirected graph is t\ he length of its shortest simple\n cycle, i.e. the shortest non-trivial pat\ h starting and ending at the same\n vertex and passing through no vertex or\ edge more than once. \n\n If digraph has no cycles, then this f\ unction will return\n infinity. \n D\ := Digraph([[2, 4], [1, 3], [2, 4], [1, 3]]);\n\ngap> DigraphUndirectedGirth(D);\n4\ngap> D := Digraph([[2], [1\ , 3], [2]]);\n\ngap> DigraphUndire\ ctedGirth(D);\ninfinity\ngap> D := Digraph([[1], [], [4], [3]]);\n\ngap> DigraphUndirectedGirth(D);\n1\ngap> D :\ = GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> DigraphUndirectedGirth(D);\n5\n]]>\n \n\n\n \n \n \n A record.\n \n This f\ unction returns the record wcc corresponding to the weakly\n connect\ ed components of the digraph digraph. Two vertices of\n digraph<\ /A> are in the same weakly connected component whenever they are\n equal, o\ r there exists a directed path (ignoring the orientation of edges)\n betwee\ n them. More formally, two vertices are in the same weakly connected\n com\ ponent of digraph if and only if they are in the same strongly\n con\ nected component (see )\n \ of the of digraph. \n\n \ The set of weakly connected components is a partition of\n the vertex set\ of digraph. \n\n The record wcc has 2 components: com\ ps and id.\n The component comps is a list of the weakly \ connected components\n of digraph (each of which is a list of verti\ ces).\n The component id is a list such that the vertex i is \ an\n element of the weakly connected component comps[id[i]]. \n\ \n The method used in this function has complexity O(m+n), where\n \ m is the number of edges and\n n is the number of vertices\ in the digraph. \n\n DigraphNrConnectedComponents(digraph) is simply a shortcut\n for Length(DigraphConnectedComponents(digra\ ph).comps),\n and is no more efficient.\n\n gr := Digraph([[2], [3, 1], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ [ 1, 2, 3 ] ], i\ d := [ 1, 1, 1 ] )\ngap> gr := Digraph([[1], [1, 2], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps :\ = [ [ 1, 2 ], [ 3 ] ], id := [ 1, 1, 2 ] )\ngap> DigraphNrConnectedComponents(\ gr);\n2\ngap> gr := EmptyDigraph(0);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ ], id := [ ] )\ngap\ > D := CycleDigraph(IsMutableDigraph, 2);\n\ngap> G := CycleDigraph(3);\n\ \ngap> DigraphDisjointUnion(D, G);\n\ \ngap> DigraphConnectedComponents(D);\nrec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] ]\ , id := [ 1, 1, 2, 2, 2 ] )\n]]>\n \n\n\ \n \n \n A list of vertices.\n \n If \ vertex is a vertex in the digraph digraph, then\n this operat\ ion returns the connected component of vertex\n in digraph.\n\ See for more information.\n \ D := Digraph([[3], [2], [1, 2], [4]]);\n\ngap> DigraphConnectedComponent(D, 3);\n[ 1, 2\ , 3 ]\ngap> DigraphConnectedComponent(D, 2);\n[ 1, 2, 3 ]\ngap> DigraphConnect\ edComponent(D, 4);\n[ 4 ]\n]]>\n \n\n\n \ \n \n \n A record.\n \n This function retur\ ns the record scc corresponding to the strongly\n connected componen\ ts of the digraph digraph. Two vertices of\n digraph are in \ the same strongly connected component whenever they\n are equal, or there i\ s a directed path from each vertex to the other. The\n set of strongly con\ nected components is a partition of the vertex set of\n digraph.\n \ \n\n The record scc has 2 components: comps and id.\n The component comps is a list of the strongly connected compon\ ents\n of digraph (each of which is a list of vertices).\n The co\ mponent id is a list such that the vertex i is an\n element o\ f the strongly connected component comps[id[i]]. \n\n The method\ used in this function is a non-recursive version of Gabow's\n Algorithm and has complexity O(m+n) where\n m is th\ e number of edges (counting multiple edges as one) and\n n is the nu\ mber of vertices in the digraph. \n\n DigraphNrStronglyConnectedComp\ onents(digraph) is simply a\n shortcut for\n Length(Digrap\ hStronglyConnectedComponents(digraph).comps),\n and is no more e\ fficient.\n\n gr := Digraph([[2], [3, 1], []]);\n<\ immutable digraph with 3 vertices, 3 edges>\ngap> DigraphStronglyConnectedComp\ onents(gr);\nrec( comps := [ [ 3 ], [ 1, 2 ] ], id := [ 2, 2, 1 ] )\ngap> Digr\ aphNrStronglyConnectedComponents(gr);\n2\ngap> D := DigraphDisjointUnion(Cycle\ Digraph(4), CycleDigraph(5));\n\ng\ ap> DigraphStronglyConnectedComponents(D);\nrec( comps := [ [ 1, 2, 3, 4 ], [ \ 5, 6, 7, 8, 9 ] ], \n id := [ 1, 1, 1, 1, 2, 2, 2, 2, 2 ] )\ngap> DigraphNrSt\ ronglyConnectedComponents(D);\n2\ngap> D := CycleDigraph(IsMutableDigraph, 2);\ \n\ngap> G := CycleDigraph(3);\n\ngap> DigraphDisjointUnion(D, G);\n\ngap> DigraphStronglyConnectedComponents\ (D);\nrec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] ], id := [ 1, 1, 2, 2, 2 ] )\n]]><\ /Example>\n \n\n\n \n \n A \ list of vertices.\n \n If vertex is a vertex\ in the digraph digraph, then\n this operation returns the strongly \ connected component of vertex\n in digraph.\n See for more information.\n D := Digraph([[3], [2], [1, 2], [3]]);\n\ngap> DigraphStronglyConnectedComponent(D, 3);\n[ 1, 3\ ]\ngap> DigraphStronglyConnectedComponent(D, 2);\n[ 2 ]\ngap> DigraphStrongly\ ConnectedComponent(D, 4);\n[ 4 ]\n]]>\n \n\n\n \n \ \n A pair of lists of vertices, or fail.\n \n If digraph is a bipartite digraph, i.e. if it satisfies , then DigraphBicomponents return\ s a\n pair of bicomponents of digraph. Otherwise,\n DigraphBic\ omponents returns fail. \n\n For a bipartite digraph, the ver\ tices can be partitioned into two non-empty\n sets such that the source and\ range of any edge are in distinct sets. The\n parts of this partition are \ called bicomponents of digraph.\n Equivalently, a pair of bic\ omponents of digraph consists of the\n color-classes of a 2-coloring\ of digraph. \n\n For a bipartite digraph with at least 3 vertic\ es, there is a unique pair of\n bicomponents of bipartite if and only if th\ e digraph is connected. See for more\ information.\n D := CycleDigraph(3);\n\ngap> DigraphBicomponents(D);\nfail\ngap> D := \ ChainDigraph(5);\n\ngap> DigraphBicom\ ponents(D);\n[ [ 1, 3, 5 ], [ 2, 4 ] ]\ngap> D := Digraph([[5], [1, 4], [5], [\ 5], []]);\n\ngap> DigraphBicompone\ nts(D);\n[ [ 1, 3, 4 ], [ 2, 5 ] ]\ngap> D := CompleteBipartiteDigraph(IsMutab\ leDigraph, 2, 3);\n\ngap> DigraphBi\ components(D);\n[ [ 1, 2 ], [ 3, 4, 5 ] ]\n]]>\n \n\n\n \n \n A list of vertices.\n \n A con\ nected digraph is biconnected if it is still connected (in the\n sen\ se of ) when any vertex is removed. \n If\ the digraph digraph is not biconnected but is connected, then any\n \ vertex v of digraph whose removal makes the resulting digraph\n\ disconnected is called an articulation point. \n\n Articul\ ationPoints returns a list of the articulation points of\n digraph, if any, and, in particular, returns the empty list if\n digraph \ is not connected. \n\n Multiple edges are ignored by this method. \ \n\n The method used in this operation has complexity O(m+n) where\n\ m is the number of edges and n is the number of vertices in\ \n the digraph. \n \n If D has a bridge (see ), then a node incident\n to the bridge is an articulation point\ if and only if it has degree at\n least 2. It follows that if D\ has a bridge and at least\n 3 nodes, then at least one of the n\ odes incident to the bridge is\n an articulation point. The converse does \ not hold, there are digraphs\n with articulation points, but no bridges.\n \ \n\n See also and \n .\n ArticulationPoints(Cyc\ leDigraph(5));\n[ ]\ngap> D := Digraph([[2, 7], [3, 5], [4], [2], [6], [1], [\ ]]);;\ngap> ArticulationPoints(D);\n[ 2, 1 ]\ngap> ArticulationPoints(ChainDig\ raph(5));\n[ 4, 3, 2 ]\ngap> ArticulationPoints(NullDigraph(5));\n[ ]\ngap> D\ := ChainDigraph(IsMutableDigraph, 4);\n\ngap> ArticulationPoints(D);\n[ 3, 2 ]\n]]>\n \n\ \n\n \n \n A (possibly empty) list of edges.\n \n A conn\ ected digraph is 2-edge-connected if it is still connected (in\n the\ sense of ) when any edge is\n removed. \ If the digraph D is not 2-edge-connected but is\n connected, then an\ y edge [u, v] of D whose removal\n makes the resulting digrap\ h disconnected is called a bridge. \n\n Bridges returns a \ list of the bridges of D, if any, and, in\n particular, returns the \ empty list if D is not connected. \n\n Multiple edges are ignore\ d by this method. \n\n The method used in this operation has complexity\ O(m+n) where\n m is the number of edges and n is the \ number of vertices in\n the digraph.\n \n\n If D has a bri\ dge, then a node incident to the bridge is an\n articulation point (see ) if and only if\n it has degree at least 2. It follows that if D has a\n bridge and at least 3 n\ odes, then at least one of the nodes\n incident to the bridge is an articul\ ation point. The converse does not\n hold, there are digraphs with articula\ tion points, but no bridges.\n \n\n See also and \n .\n D := Digraph([[2, 5], [1, 3, 4, 5], [2, 4], [2, 3], [1, 2]]);\n\ngap> Bridges(D);\n[ ]\ngap> D := Dig\ raph([[2], [3], [4], [2]]);\n\ngap\ > Bridges(D);\n[ [ 1, 2 ] ]\ngap> Bridges(ChainDigraph(2));\n[ [ 1, 2 ] ]\ngap\ > ArticulationPoints(ChainDigraph(2));\n[ ]\n]]>\n \ \n\n\n \n \n \n A digrap\ h or fail.\n \n A strong orientation \ of a connected symmetric digraph D (if\n it exists) is a strongly co\ nnected subdigraph C of D such\n that for every edge [u, v\ ] of D either [u, v] or\n [v, u] is an edge of \ C but not both. Robbin's Theorem states\n that a digraph admits a stro\ ng orientation if and only if it is\n bridgeless (see ). \n \n\n This operation returns a strong orientation\ of the digraph D if \n D is symmetric and D admits a \ strong orientation. If D\n is symmetric but does not admit a strong \ orientation, then fail is\n returned. If D is not symmetric, \ then an error is given. \n \n\n If D is immutable, StrongO\ rientation(D) returns an\n immutable digraph, and if D is\ mutable, then\n StrongOrientation(D) returns a mutable digra\ ph. \n\n The method used in this operation has complexity O(m+n)\ where\n m is the number of edges and n is the number of vert\ ices in\n the digraph.\n StrongOrientation(DigraphS\ ymmetricClosure(CycleDigraph(5))) \n> = CycleDigraph(5);\ntrue\ngap> D := Digr\ aphSymmetricClosure(Digraph(\n> [[2, 7], [3, 5], [4], [2], [6], [1], []]));;\n\ gap> IsBridgelessDigraph(D);\nfalse\ngap> StrongOrientation(D);\nfail\ngap> St\ rongOrientation(NullDigraph(0));\n\ng\ ap> StrongOrientation(DigraphDisjointUnion(CompleteDigraph(3), \n> \ CompleteDigraph(3)));\nfail\n]]>\n <\ /Description>\n\n\n \n \n An integer.\n \n T\ his function returns the period of the digraph digraph. \n\n If a\ digraph digraph has at least one directed cycle, then the period\n \ is the greatest positive integer which divides the lengths of all directed\n \ cycles of digraph. If digraph has no directed cycles, then\n \ this function returns 0. See section for the definition of a directed cycle. \n\n \ A digraph with a period of 1 is said to be aperiodic. See\n \ . \n D := Di\ graph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> DigraphPeriod(D);\n6\ngap> D := Digraph([[2], [3, 5], [4]\ , [5], [1, 2]]);\n\ngap> DigraphPe\ riod(D);\n1\ngap> D := ChainDigraph(2);\n\ngap> DigraphPeriod(D);\n0\ngap> IsAcyclicDigraph(D);\ntrue\ngap> D := Ge\ neralisedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> DigraphPeriod(D);\n1\n]]>\n \n<\ /ManSection>\n\n \n \n A matrix.\n \n If digraph is a digraph with n vertices, then\n this \ operation returns an n \\times n matrix mat containing\n the \ output of a generalised version of the Floyd-Warshall algorithm,\n applied \ to digraph. \n\n The operation DigraphFloydWarshall is cu\ stomised by the arguments\n func, nopath, and edge.\n \ The arguments nopath and edge can be arbitrary &GAP; objects.\ \n The argument func must be a function which accepts 4 arguments:\n\ the matrix mat, followed by 3 postive integers. The function\n <\ A>func is where the work to calculate the desired outcome must be\n per\ formed. \n\n This method initialises the matrix mat by setting e\ ntry\n mat[i][j] to equal edge if there is an edge with sourc\ e\n i and range j, and by setting entry mat[i][j] to e\ qual\n nopath otherwise.\n The final part of DigraphFloydWarsh\ all then calls the function\n func inside three nested for loops\ , over the vertices of\n digraph: \n\n \n\n The matrix mat is then returned as the re\ sult. An example of using\n DigraphFloydWarshall to calculate the sh\ ortest (non-zero) distances\n between the vertices of a digraph is shown be\ low: \n\n D := DigraphFromDigraph6String(\"&EA\ HQeDB\");\n\ngap> func := functio\ n(mat, i, j, k)\n> if mat[i][k] <> -1 and mat[k][j] <> -1 then\n> if (ma\ t[i][j] = -1) or (mat[i][j] > mat[i][k] + mat[k][j]) then\n> mat[i][j] :\ = mat[i][k] + mat[k][j];\n> fi;\n> fi;\n> end;\nfunction( mat, i, j, k )\ ... end\ngap> shortest_distances := DigraphFloydWarshall(D, func, -1, 1);;\ng\ ap> Display(shortest_distances);\n[ [ 3, -1, -1, 2, 1, 2 ],\n [ 4\ , 2, 1, 3, 2, 1 ],\n [ 3, 1, 2, 2, 1, 2 ],\n [ 1, -\ 1, -1, 1, 1, 2 ],\n [ 2, -1, -1, 1, 2, 1 ],\n [ 3, -1, \ -1, 2, 1, 1 ] ]\n]]>\n \n\n\n <\ ManSection>\n \n \ true or false.\n \n This operation re\ turns true if there exists a non-trivial directed\n walk from vertex\ u to vertex v in the digraph digraph,\n and false<\ /K> if there does not exist such a directed walk. See section\n for the definition of a\n non-trivia\ l directed walk.\n \n\n The method for IsReachable has worst \ case complexity of O(m +\n n) where m is the number of edge\ s and n the number of\n vertices in digraph.\n\n D := Digraph([[2], [3], [2, 3]]);\n\ngap> IsReachable(D, 1, 3);\ntrue\ngap> IsReachable(D, 2, 1);\nf\ alse\ngap> IsReachable(D, 3, 3);\ntrue\ngap> IsReachable(D, 1, 1);\nfalse\n]]>\ \n \n\n\n \n \n A pair of lists, or fai\ l.\n \n If there exists a non-trivial directed \ path (or a non-trivial cycle, in the\n case that u = v\ ) from vertex u to vertex\n v in the digraph digraph, \ then this operation returns such a\n directed path (or directed cycle). Oth\ erwise, this operation returns\n fail. See Section for the\n definition of a directed path and a\ directed cycle.\n \n\n A directed path (or directed cycle) of non-z\ ero length n-1,\n (v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n),\n \ is represented by a pair of lists [v,a] as follows:\n\n \n \ \n v is the list [v_1, v_2, ..., v_n].\n \ \n \n a is the list of positive integers [a\ _1, a_2, ..., a_{n-1}]\n where for each each i < n, a_\ i is the position of\n v_{i+1} in\n OutNeighboursO\ fVertex(digraph,v_i)\n corresponding to\ the edge e_i. This is can be useful if the\n position of a ver\ tex in a list of out-neighours is significant, for\n example in orbit d\ igraphs.\n \n \n\n The method for DigraphPath \ has worst case complexity of O(m +\n n) where m is the numb\ er of edges and n the number of\n vertices in digraph.\n\n D := Digraph([[2], [3], [2, 3]]);\n\ngap> DigraphPath(D, 1, 3);\n[ [ 1, 2, 3 ], [ 1, 1 ] ]\ \ngap> DigraphPath(D, 2, 1);\nfail\ngap> DigraphPath(D, 3, 3);\n[ [ 3, 3 ], [ \ 2 ] ]\ngap> DigraphPath(D, 1, 1);\nfail\n]]>\n \n\n\n \n \n A pair of lists, or fail.\n \n Returns the shortest directed path in the digraph digraph from th\ e vertex\n u to the vertex v, if such a path exists. If u =\n v, then the shortest non-trivial cycle is returned, a\ gain, if it\n exists. Otherwise, this operation returns fail. See S\ ection\n for the definition o\ f a directed\n path and a directed cycle.\n \n\n See for details on the output.\n\n The method for DigraphSho\ rtestPath has worst case complexity of\n O(m + n) where m\ is the number of edges and n the\n number of vertices in digraph\ .\n\n D := Digraph([[1, 2], [3], [2, 4], [1], [2, \ 4]]);\n\ngap> DigraphShortestPath(\ D, 5, 1);\n[ [ 5, 4, 1 ], [ 2, 1 ] ]\ngap> DigraphShortestPath(D, 3, 3);\n[ [ \ 3, 2, 3 ], [ 1, 1 ] ]\ngap> DigraphShortestPath(D, 5, 5);\nfail\ngap> DigraphS\ hortestPath(D, 1, 1);\n[ [ 1, 1 ], [ 1 ] ]\n]]>\n \n<\ /ManSection>\n\n \n \n An iterator.\n \n If di\ graph is a digraph or a list of adjacencies which defines a\n digraph -\ see - then this operation returns an\n itera\ tor of the non-trivial directed paths (or directed cycles, in the case\n th\ at u = v) in digraph from the vertex u\n \ to the vertex v.\n \n\n See for \ more information about the repesentation\n of a directed path or directed c\ ycle which is used, and see \ for more information about iterators.\n See Section for the definition of\n a directed path and a direct\ ed cycle.\n \n\n D := Digraph([[1, 4, 4, 2], [3\ , 5], [2, 3], [1, 2], [4]]);\n\ngap> iter := IteratorOfPaths(D, 1, 4);\n\ngap> NextIterator(iter\ );\n[ [ 1, 4 ], [ 2 ] ]\ngap> NextIterator(iter);\n[ [ 1, 4 ], [ 3 ] ]\ngap> N\ extIterator(iter);\n[ [ 1, 2, 5, 4 ], [ 4, 2, 1 ] ]\ngap> IsDoneIterator(iter)\ ;\ntrue\ngap> iter := IteratorOfPaths(D, 4, 3);\n\ngap> NextIterator\ (iter);\n[ [ 4, 1, 2, 3 ], [ 1, 4, 1 ] ]\n]]>\n \n\n\n \n \n A list of lists of vertices.\n \n If digraph is a digraph, then DigraphAllSimpleCircuits\ \n returns a list of the simple circuits in digraph. \n\n\ See section for the defin\ ition\n of a simple circuit, and related notions. Note that a loop is a sim\ ple\n circuit. \n\n For a digraph without multiple edges, a simple c\ ircuit is uniquely\n determined by its subsequence of vertices. However thi\ s is not the case for\n a multidigraph. The attribute DigraphAllSimpleC\ ircuits ignores\n multiple edges, and identifies a simple circuit using\ only its subsequence\n of vertices. For example, although the simple circu\ its (v, e, v) and\n (v, e', v) (for distinct edges e a\ nd e') are\n mathematically distinct, DigraphAllSimpleCircuits considers them to\n be the same. \n\n With this approach, a direct\ ed circuit of length n can be determined\n by a list of its first n vertices. Thus a simple circuit (v_1,\n e_1, v_2, e_2, ..., e_{\ n-1}, v_n, e_{n+1}, v_1) can be represented as\n the list [v_1, \\ld\ ots, v_n], or any cyclic permutation thereof. For\n each simple circui\ t of digraph,\n DigraphAllSimpleCircuits(digraph) incl\ udes precisely one such\n list to represent the circuit. \n\n D := Digraph([[], [3], [2, 4], [5, 4], [4]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 4 ],\ [ 4, 5 ], [ 2, 3 ] ]\ngap> D := ChainDigraph(10);;\ngap> DigraphAllSimpleCirc\ uits(D);\n[ ]\ngap> D := Digraph([[3], [1], [1]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1, 3 ] ]\ngap> D :\ = Digraph([[1, 1]]);\n\ngap> Di\ graphAllSimpleCircuits(D);\n[ [ 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph\ , 3);\n\ngap> DigraphAllSimpleCircui\ ts(D);\n[ [ 1, 2, 3 ] ]\n]]>\n \n\n\n \ \n \ \n A list of vertices, or fail.\n \n \ If digraph is a digraph, then DigraphLongestSimpleCircuit\n \ returns the longest simple circuit in digraph. See section\n \ for the definition of simpl\ e\n circuit, and the definition of length for a simple circuit. \n\n \ This attribute computes\n DigraphAllSimpleCircuits(digraph\ ) to find all the\n simple circuits of digraph, and returns one \ of maximal length. A\n simple circuit is represented as a list of vertices\ , in the same way as\n described in . \n\n If digraph has no simple circuits, then this attribute re\ turns\n fail. If digraph has multiple simple circuits of max\ imal\n length, then this attribute returns one of them. \n\n D := Digraph([[], [3], [2, 4], [5, 4], [4]]);;\ngap> DigraphL\ ongestSimpleCircuit(D);\n[ 4, 5 ]\ngap> D := ChainDigraph(10);;\ngap> DigraphL\ ongestSimpleCircuit(D);\nfail\ngap> D := Digraph([[3], [1], [1, 4], [1, 1]]);;\ \ngap> DigraphLongestSimpleCircuit(D);\n[ 1, 3, 4 ]\ngap> D := GeneralisedPete\ rsenGraph(IsMutableDigraph, 4, 1);\n\ngap> DigraphLongestSimpleCircuit(D);\n[ 1, 2, 3, 4, 8, 7, 6, 5 ]\n]]>\n \n\n\n \n \n A list.\n \n This operation returns a list list such that list[i]\ is the\n list of vertices whose minimum distance from the vertex vertex\ in\n digraph is i - 1. Vertex vertex is\n assu\ med to be at distance 0 from itself.\n D :=\ CompleteDigraph(4);;\ngap> DigraphLayers(D, 1);\n[ [ 1 ], [ 2, 3, 4 ] ]\n]]><\ /Example>\n \n\n\n \n \n A non-negative integer, or\ fail.\n \n If digraph is a symmetric\ digraph without multiple edges - see\n \ and - then\n this attribute returns the dege\ neracy of digraph. \n\n The degeneracy of a digraph is the least\ integer k such\n that every induced of digraph contains a ve\ rtex whose number of\n neighbours (excluding itself) is at most k. N\ ote that this means\n that loops are ignored. \n\n If digraph \ is not symmetric or has multiple edges then this\n attribute returns fai\ l.\n D := DigraphSymmetricClosure(ChainDigraph\ (5));;\ngap> DigraphDegeneracy(D);\n1\ngap> D := CompleteDigraph(5);;\ngap> Di\ graphDegeneracy(D);\n4\ngap> D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], \ [2], []]);\n\ngap> DigraphDegener\ acy(D);\n1\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 10, 3);\n\ngap> DigraphDegeneracy(D);\n3\n]]>\n \n\n\n \n \n A list of integers, \ or fail.\n \n If digraph is a digraph\ for which\n DigraphDegeneracy(digraph) is a non-negat\ ive integer\n k - see - then this \ attribute\n returns a degeneracy ordering of the vertices of the vertices o\ f\n digraph. \n\n A degeneracy ordering of digraph is a\ list ordering of the\n vertices of digraph ordered such that\ for any\n position i of the list, the vertex ordering[i] has\ at most\n k neighbours in later position of the list. \n\n If\ DigraphDegeneracy(digraph) returns fail,\n the\ n this attribute returns fail.\n D := Digra\ phSymmetricClosure(ChainDigraph(5));;\ngap> DigraphDegeneracyOrdering(D);\n[ 5\ , 4, 3, 2, 1 ]\ngap> D := CompleteDigraph(5);;\ngap> DigraphDegeneracyOrdering\ (D);\n[ 5, 4, 3, 2, 1 ]\ngap> D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4],\ [2], []]);\n\ngap> DigraphDegene\ racyOrdering(D);\n[ 1, 6, 5, 2, 4, 3 ]\ngap> D := GeneralisedPetersenGraph(IsM\ utableDigraph, 3, 1);\n\ngap> Digra\ phDegeneracyOrdering(D);\n[ 6, 5, 4, 1, 3, 2 ]\n]]>\n \n\n\n \n \n A list or fail.\n \n \ Returns a Hamiltonian path if one exists, fail if not. \n\n A \ Hamiltonian path of a digraph with n vertices is directed\n cycl\ e of length n. If digraph is a digraph that contains a\n Hami\ ltonian path, then this function returns one, described in the form\n used \ by . Note if digraph has\n 0\ or 1 vertices, then HamiltonianPath returns [] or\n [1], respectively. \n\n The method used in this attribute has the same wo\ rst case complexity as\n .\n\n D := Digraph([[]]);\n\ngap> HamiltonianPath(D);\n[ 1 ]\ngap> D := Digraph([[2], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2 ]\ngap> D\ := Digraph([[3], [], [2]]);\n\nga\ p> HamiltonianPath(D);\nfail\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2, 3 ]\ngap>\ D := GeneralisedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> HamiltonianPath(D);\nfail\n]]>\n \n\n\n \n \n An integer.\n \n R\ eturns the number of spanning trees of the symmetric digraph\n digraph.\n NrSpanningTrees will return an error if digraph is not \ a\n symmetric digraph. \n\n See \ and\n for more information. \n\n \ D := CompleteDigraph(5);\n\ngap> NrSpanningTrees(D);\n125\ngap> D := DigraphSymmetri\ cClosure(CycleDigraph(24));;\ngap> NrSpanningTrees(D);\n24\ngap> NrSpanningTre\ es(EmptyDigraph(0));\n0\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, \ 9, 2);\n\ngap> NrSpanningTrees(D);\ \n1134225\n]]>\n \n\n\n \ \n \n \n Two lists.\n \n If digraph is a digraph and source and target are\n\ vertices of digraph, then DigraphDijkstra calculates the\n le\ ngth of the shortest path from source to target and returns\n t\ wo lists. Each element of the first list is the distance of the\n correspondi\ ng element from source. If a vertex was not visited in\n the process \ of calculating the shortest distance to target or if\n there is no pat\ h connecting that vertex with source, then\n the corresponding distanc\ e is infinity. Each element of the\n second list gives the previous v\ ertex in the shortest path\n from source to the corresponding vertex. \ For\n source and for any vertices that remained unvisited this\n w\ ill be -1. \n\n If the optional second argument target is \ not present, then\n DigraphDijkstra returns the shortest path from <\ A>source to\n every vertex that is reachable from source.\n\n <\ Example> mat := [[0, 1, 1], [0, 0, 1], [0, 0, 0]];\n[ [ 0, 1, 1\ ], [ 0, 0, 1 ], [ 0, 0, 0 ] ]\ngap> D := DigraphByAdjacencyMatrix(mat);\n\ngap> DigraphDijkstra(D, 2, 3);\n[ [ \ infinity, 0, 1 ], [ -1, -1, 2 ] ]\ngap> DigraphDijkstra(D, 1, 3);\n[ [ 0, 1, 1\ ], [ -1, 1, 1 ] ]\ngap> DigraphDijkstra(D, 1, 2);\n[ [ 0, 1, 1 ], [ -1, 1, 1 \ ] ]\n]]>\n\n\n\n \n\n Cayley graphs of groups\n \n \n \n A group or semigroup.\n \n If digraph is an immutable Cayley graph of a group G and\n digraph belongs to the category , then\n GroupOfCayleyDigraph returns G.\n \n\n \ If digraph is a Cayley graph of a semigroup S and\n digra\ ph belongs to the category , then\n S\ emigroupOfCayleyDigraph returns S.\n \n\n See also .\n G := Dihedr\ alGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> digraph := CayleyD\ igraph(G);\n\ngap> GroupOfCayleyD\ igraph(digraph) = G;\ntrue\n]]>\n \n\n\n\ \n \ \n A list of generators.\n \n If digra\ ph is an immutable Cayley graph of a group or semigroup with\n respect \ to a set of generators gens and digraph belongs to\n the cate\ gory , then\n GeneratorsOfCayleyDigraph return the list of generators gens\n over which digraph is\ defined.\n \n\n See also \n \ or .\n G \ := DihedralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> digraph :\ = CayleyDigraph(G);\n\ngap> Gener\ atorsOfCayleyDigraph(digraph) = GeneratorsOfGroup(G);\ntrue\ngap> digraph := C\ ayleyDigraph(G, [()]);\n\ngap> Gen\ eratorsOfCayleyDigraph(digraph) = [()];\ntrue]]>\n \n\ \n\n \n\n Associated semigroups\n \n \n \n A semilattice of partial perms.\n \n The operation AsSemigroup requires that \ filt be equal to\n .\n If digraph is a\n or \n then AsSemigroup retu\ rns a semigroup of partial perms which is\n isomorphic to the semigroup w\ hose elements are the vertices of\n digraph with the binary operat\ ion\n .\n If dig\ raph\n satisfies but not\n \ then\n AsSemigroup r\ eturns a semigroup of partial perms which is\n isomorphic to the semigrou\ p whose elements are the vertices of\n digraph with the binary ope\ ration\n .\n \n\ \n The operation AsMonoid behaves similarly to AsSemigroup\ \n except that filt may also be equal to\n ,\n digraph must satisfy , and the\n output satisfies . \n\n The output of both of these operations i\ s guaranteed to be of minimal\n degree (see ).\n Furthermore the of\n the output is guaranteed to be th\ e unique generating set of minimal size.\n di :=\ Digraph([[1], [1, 2], [1, 3], [1, 4], [1, 2, 3, 5]]);\n\ngap> S := AsSemigroup(IsPartialPermSemigroup, di);\n<\ partial perm semigroup of rank 3 with 4 generators>\ngap> ForAll(Elements(S), \ IsIdempotent);\ntrue\ngap> IsInverseSemigroup(S);\ntrue\ngap> Size(S);\n5\ngap\ > di := Digraph([[1], [1, 2], [1, 2, 3]]);\n\ngap> M := AsMonoid(IsPartialPermMonoid, di);\n\ngap> Size(M);\n3]]>\n \n \n\n \n \n A Clifford semigroup of partial perms. \ \n \n The operation AsSemigroup requires that f\ ilt be equal to\n .\n If Y is a or\n \ ,\n gps is a list of gr\ oups corresponding to each vertex, and\n homs is a list containing\ for each edge (i, j) in\n the transitive reduction of digraph<\ /A> a triple [i, j, hom]\n where hom is a group homomorphis\ m from gps[i] to\n gps[j], and the diagram of homomorphisms\ commutes, then\n AsSemigroup returns a semigroup of partial perms\ which is\n isomorphic to the strong semilattice of groups S[Y; gps; h\ oms].\n G1 := AlternatingGroup(4);;\ngap> G2\ := SymmetricGroup(2);;\ngap> G3 := SymmetricGroup(3);;\ngap> gr := Digraph([[\ 1, 3], [2, 3], [3]]);;\ngap> sgn := function(x)\n> if SignPerm(x) = 1 then\n> \ return ();\n> fi;\n> return (1, 2);\n> end;;\ngap> hom13 := GroupHomomorphismB\ yFunction(G1, G3, sgn);;\ngap> hom23 := GroupHomomorphismByFunction(G2, G3, sg\ n);;\ngap> T := AsSemigroup(IsPartialPermSemigroup,\n> gr,\n> [G1, G2, G3], [[\ 1, 3, hom13], [2, 3, hom23]]);;\ngap> Size(T);\n20\ngap> D := GreensDClasses(T\ );;\ngap> List(D, x -> Size(x));\n[ 6, 12, 2 ]\n]]>\n \n \n\n \n\n Planarity\n \n \n A list or fail.\n \n Kur\ atowskiPlanarSubdigraph returns the immutable list of\n lists of out-ne\ ighbours of a (not necessarily induced) subdigraph of the\n digraph digr\ aph that witnesses the fact that digraph is not\n planar, or \ fail if digraph is planar. In other words,\n KuratowskiPlana\ rSubdigraph returns the out-neighbours of a\n subdigraph of digraph<\ /A> that is homeomorphic to the complete graph\n with 5 vertices, or\ to the complete bipartite graph with vertex sets\n of sizes 3 and <\ C>3. \n \n The directions and multiplicities of any edges in \ digraph are\n ignored when considering whether or not digraph is\ planar. \n\n See also \n and .\n \n \n This method uses th\ e reference implementation in\n &edge-addition-planarity-suite; by John Boy\ er of the algorithms described\n in .\n\n D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11\ ], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiPlanarSubdigraph(D);\nfail\ngap> D := \ Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1,\ 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5\ , 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsPlanarDigraph(D);\nfalse\ngap> KuratowskiPlanarSubdigraph(D);\ \n[ [ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 ], [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], \ [ ], [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1,\ 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\ \n\ngap> KuratowskiPlanarSubdigrap\ h(D);\nfail\ngap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, \ 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7\ , 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]\ );\n\ngap> IsPlanarDigraph(D);\nfa\ lse\ngap> KuratowskiPlanarSubdigraph(D);\n[ [ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 \ ], [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ], [ ] ]\n]]>\n \n\n\n \n \n A list or fail.\ \n \n KuratowskiOuterPlanarSubdigraph returns the immu\ table list of\n immutable lists of out-neighbours of a (not necessarily ind\ uced)\n subdigraph of the digraph digraph that witnesses the fact\n \ that digraph is not outer planar, or fail if\n digraph<\ /A> is outer planar. In other words,\n KuratowskiOuterPlanarSubdigraph<\ /C> returns the out-neighbours of a\n subdigraph of digraph that is \ homeomorphic to the complete graph\n with 4 vertices, or to the comp\ lete bipartite graph with vertex sets\n of sizes 2 and 3. \n \n The directions and multiplicities of any edges in digraph \ are\n ignored when considering whether or not digraph is outer plana\ r. \n \n\n See also \n ,\n \ , and \n .\n \n\n This method uses the reference implemen\ tation in\n &edge-addition-planarity-suite; by John Boyer of the algorithms\ described\n in .\n\n D := Digr\ aph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4, 7], [6, 8], [\ 2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [\ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := D\ igraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, \ 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5,\ 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsOuterPlanarDigraph(D);\nfalse\ngap> KuratowskiOuterPlanarSubdi\ graph(D);\n[ [ ], [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ],\ \n [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, \ 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\ \n\ngap> KuratowskiOuterPlanarSubd\ igraph(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \ \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[2, 4, 7,\ 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, \ 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5\ , 6, 7, 8], [3, 5]]);\n\ngap> IsOu\ terPlanarDigraph(D);\nfalse\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ ],\ [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ], \n [ ] ]]]>\n \n\n\n \n \n A list or fail.\ \n \n If digraph is a planar digraph, then \n Pl\ anarEmbedding returns the immutable list of lists of\n out-neighbours o\ f a subdigraph of digraph such that each vertex's\n neighbours are g\ iven in clockwise order. If digraph is not planar,\n then fail is returned. \n \n The directions and multiplicities of any edges \ in digraph are\n ignored by PlanarEmbedding.\n \n\n \ See also \n .\n \n\n This method \ uses the reference implementation in\n &edge-addition-planarity-suite; by J\ ohn Boyer of the algorithms described\n in .\n\n D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1\ , 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10\ , 8, 9 ], [ 4 ], [ 6 ], [ 11, 7 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] \ ]\ngap> D := Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, \ 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, \ 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> PlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMut\ ableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [\ 6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 \ ], [ 11, 7 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(\ IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8\ , 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10\ ], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> PlanarEmbedding(D);\nfail\n]]>\n \n\n\n \n \n A list or fail.\n \n If digraph is an outer planar digraph, then\n OuterPlanar\ Embedding returns the immutable list of lists\n of out-neighbours of a \ subdigraph of digraph such that each\n vertex's neighbours are given\ in clockwise order. If digraph is not\n outer planar, then fail<\ /K> is returned. \n \n The directions and multiplicities of any edge\ s in digraph are\n ignored by OuterPlanarEmbedding.\n \ \n\n See also .\n \n\n This m\ ethod uses the reference implementation in\n &edge-addition-planarity-suite\ ; by John Boyer of the algorithms described\n in .\n\n<\ Example> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \ \n> [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap\ > D := Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, \ 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [\ 3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbeddin\ g(CompleteBipartiteDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\ngap> D\ := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, \ 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Dig\ raph(IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2,\ 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, \ 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarE\ mbedding(CompleteBipartiteDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\ \n]]>\n \n\n\n \n \n \n \n A list or fail.\n \n These attributes return the immutable list of lists of\n out\ -neighbours of a subdigraph of the digraph digraph which is\n homeom\ orphic to one of the following:\n the complete bipartite graph with vertex \ sets of sizes 2 and\n 3; the complete bipartite graph with ve\ rtex sets of sizes 3\n and 3; or the complete graph with 4\ vertices. If\n digraph has no such subdigraphs, then fail is returned. \n \n\n See also and \ for more details. \n\n This \ method uses the reference implementation in\n &edge-addition-planarity-suit\ e; by John Boyer of the algorithms described\n in .\n\n\ D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6],\ [1, 7, 11], \n> [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ \ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [\ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9\ , 8, 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ \ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 11], \n> [4,\ 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphH\ omeomorphicToK23(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 \ ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\ \nfail\ngap> SubdigraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> Subdigra\ phHomeomorphicToK33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK\ 33(CompleteBipartiteDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], \ [ ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\ \ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, \ 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3\ , 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11\ ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8\ , 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\nga\ p> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], \ [1, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> S\ ubdigraphHomeomorphicToK23(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ \ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphi\ cToK33(D);\nfail\ngap> SubdigraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap\ > SubdigraphHomeomorphicToK33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeo\ morphicToK33(CompleteBipartiteDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6\ , 5, 4 ], [ ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK4(CompleteDigraph(\ 3));\nfail\n]]>\n \n\n\n \n\n\ \n \n Properties of digraphs\n\n E\ dge properties\n \n \n true or false.\n \n Returns true if the digraph digraph has loops, and\n\ false if it does not. A loop is an edge with equal source and range\ .\n \n \n If the argument digraph is mutable, then the r\ eturn value of\n this property is recomputed every time it is called. \n\n<\ Example> D := Digraph([[1, 2], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 2 ] ]\ \ngap> DigraphHasLoops(D);\ntrue\ngap> D := Digraph([[2, 3], [1], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [\ 1, 3 ], [ 2, 1 ], [ 3, 2 ] ]\ngap> DigraphHasLoops(D);\nfalse\ngap> D := Comp\ leteDigraph(IsMutableDigraph, 4);\n\ \ngap> DigraphHasLoops(D);\nfalse]]>\n \n\n\n \n \n \n t\ rue or false.\n \n This property is t\ rue if the digraph digraph\n is antisymmetric, and false \ if it is not.\n \n\n A digraph is antisymmetric if whenever t\ here is an edge with source\n u and range v, and an edge with\ source v and range\n u, then the vertices u and v<\ /C> are equal.\n \n \n If the argument digraph is mutabl\ e, then the return value of\n this property is recomputed every time it is \ called. \n\n gr1 := Digraph([[2], [1, 3], [2, 3]])\ ;\n\ngap> IsAntisymmetricDigraph(g\ r1);\nfalse\ngap> DigraphEdges(gr1){[1, 2]};\n[ [ 1, 2 ], [ 2, 1 ] ]\ngap> gr2\ := Digraph([[1, 2], [3, 3], [1]]);\n\ngap> IsAntisymmetricDigraph(gr2);\ntrue\ngap> DigraphEdges(gr2);\n[ \ [ 1, 1 ], [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ]\n]]>\n \n\n\n \n \n true or false.\n \n This property is true if the digraph digraph is bipar\ tite, and\n false if it is not. A digraph is bipartite if and only if\ the\n vertices of digraph can be partitioned into two non-empty sets \ such\n that the source and range of any edge of digraph lie in distinc\ t sets.\n Equivalently, a digraph is bipartite if and only if it is 2-colorab\ le; see\n . \n\n See also .\n \n \n If\ the argument digraph is mutable, then the return value of\n this pr\ operty is recomputed every time it is called. \n D\ := ChainDigraph(4);\n\ngap> IsBipart\ iteDigraph(D);\ntrue\ngap> D := CycleDigraph(3);\n\ngap> IsBipartiteDigraph(D);\nfalse\ngap> D := CompleteBipartite\ Digraph(IsMutableDigraph, 5, 4);\n\ \ngap> IsBipartiteDigraph(D);\ntrue]]>\n \n\n\n \n \n true or false.\n \n\ Returns true if the digraph digraph is a complete bipartite\ \n digraph, and false if it is not. \n\n A digraph is a co\ mplete bipartite digraph if it is bipartite, see\n , and there exists a unique edge with\n source i and r\ ange j if and only if i and j lie\n in different bicom\ ponents of digraph, see . \n\n Equivalently, a bipartite digraph with bicomponents of size m \ and\n n is complete precisely when it has 2mn edges, none of \ which\n are multiple edges. \n\n See also .\n \n \n If the argument digraph is mutab\ le, then the return value of\n this property is recomputed every time it is\ called. \n\n D := CycleDigraph(2);\n\ngap> IsCompleteBipartiteDigraph(D);\ntrue\ngap> \ D := CycleDigraph(4);\n\ngap> IsBipar\ titeDigraph(D);\ntrue\ngap> IsCompleteBipartiteDigraph(D);\nfalse\ngap> D := C\ ompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteBipartiteDigraph(D);\ntrue]]>\n \n\n\n \n \n true or false.\n <\ Description>\n Returns true if the digraph digraph is complet\ e, and\n false if it is not. \n\n A digraph is complete if it has no loops, and for all\n distinct vertices i and <\ C>j,\n there is exactly one edge with source i and range j.\n\n Equivalently, a digraph with n vertices is complete precisely\ when\n it has n(n - 1) edges, no loops, and no multiple edges.\n \ \n \n If the argument digraph is mutable, then the return \ value of\n this property is recomputed every time it is called. \n\n D := Digraph([[2, 3], [1, 3], [1, 2]]);\n\ngap> IsCompleteDigraph(D);\ntrue\ngap> D := Dig\ raph([[2, 2], [1]]);\n\ngap> \ IsCompleteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDigr\ aph, 5, 4);\n\ngap> IsCompleteDigra\ ph(D);\nfalse]]>\n \n\n\n \n \n true or false.\n \n This property r\ eturns true if digraph is a complete\n multipartite digraph, \ and false if not. \n\n A digraph is a complete multipartite d\ igraph if and only if\n its vertices can be partitioned into at least t\ wo maximal independent sets,\n where every possible edge between these inde\ pendent sets occurs in the\n digraph exactly once.\n \n \n I\ f the argument digraph is mutable, then the return value of\n this p\ roperty is recomputed every time it is called. \n\n D := CompleteMultipartiteDigraph([2, 4, 6]);\n\ngap> IsCompleteMultipartiteDigraph(D);\ \ntrue\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteMultipartiteDigraph(D);\ntr\ ue]]>\n \n\n\n \n \n \n true or false.\n \n Returns true if the digraph digraph is empty, and\n \ false if it is not. A digraph is empty if it has no\n edges.<\ P/>\n\n IsNullDigraph is a synonym for IsEmptyDigraph.\n <\ P/>\n \n If the argument digraph is mutable, then the return va\ lue of\n this property is recomputed every time it is called. \n\n D := Digraph([[], []]);\n\ngap> IsEmptyDigraph(D);\ntrue\ngap> IsNullDigraph(D);\ntrue\ngap> D \ := Digraph([[], [1]]);\n\ngap> IsEm\ ptyDigraph(D);\nfalse\ngap> IsNullDigraph(D);\nfalse]]>\n \n\n\n \n \n true or false.\n \n A digraph is an equivalence digraph if and only if the digraph\ satisifies\n all of ,\n and .\n A partial \ order digraph corresponds to an equivalence relation.\n \n \ \n If the argument digraph is mutable, then the return value of\n \ this property is recomputed every time it is called. \n\n D := Digraph([[1, 3], [2], [1, 3]]);\n\ngap> IsEquivalenceDigraph(D);\ntrue\n]]>\n \n\n\n \n \n true or false.\n \n This property is true if the digraph digraph is\n f\ unctional. \n\n A digraph is functional if every vertex is the s\ ource of a\n unique edge.\n \n \n If the argument digraph\ is mutable, then the return value of\n this property is recomputed eve\ ry time it is called. \n gr1 := Digraph([[3], [2],\ [2], [1], [6], [5]]);\n\ngap> IsF\ unctionalDigraph(gr1);\ntrue\ngap> gr2 := Digraph([[1, 2], [1]]);\n\ngap> IsFunctionalDigraph(gr2);\nfalse\ngap>\ gr3 := Digraph(3, [1, 2, 3], [2, 3, 1]);\n\ngap> IsFunctionalDigraph(gr3);\ntrue\n]]>\n \n\n\n \n \n true or false.\n \ \n A multidigraph is one that has at least two\n edges with equal\ source and range. \n\n If the argument digraph is mutable, then \ the return value of\n this property is recomputed every time it is called. \ \n\n D := Digraph([\"a\", \"b\", \"c\"], [\"a\", \ \"b\", \"b\"], [\"b\", \"c\", \"a\"]);\n\ngap> IsMultiDigraph(D);\nfalse\ngap> D := DigraphFromDigraph6String(\"\ &Bug\");\n\ngap> IsDuplicateFree(D\ igraphEdges(D));\ntrue\ngap> IsMultiDigraph(D);\nfalse\ngap> D := Digraph([[1,\ 2, 3, 2], [2, 1], [3]]);\n\n\ gap> IsDuplicateFree(DigraphEdges(D));\nfalse\ngap> IsMultiDigraph(D);\ntrue\n\ gap> D := DigraphMutableCopy(D); \n\ngap> IsMultiDigraph(D);\ntrue]]>\n \n\n\n \n \ \n true or false.\n \n This\ property is true if the digraph digraph is\n reflexive, and \ false if it is not.\n A digraph is reflexive if it has a loop\ at every vertex. \n \n \n If the argument digraph i\ s mutable, then the return value of\n this property is recomputed every tim\ e it is called. \n D := Digraph([[1, 2], [2]]);\n<\ immutable digraph with 2 vertices, 3 edges>\ngap> IsReflexiveDigraph(D);\ntrue\ \ngap> D := Digraph([[3, 1], [4, 2], [3], [2, 1]]);\n\ngap> IsReflexiveDigraph(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n <\ Description>\n This property is true if the digraph digraph\n\ is symmetric, and false if it is not. \n\n A symmetric dig\ raph is one where for each non-loop edge, having\n source u and \ range v, there is a corresponding edge with\n source v and ra\ nge u. If there are n edges with\n source u and range\ v, then there must be precisely n\n edges with source v and range u. In other words, a symmetric\n digraph has a symmetr\ ic adjacency matrix .\n \n \n If\ the argument digraph is mutable, then the return value of\n this pr\ operty is recomputed every time it is called. \n g\ r1 := Digraph([[2], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr1);\ntrue\ngap> adj1 := AdjacencyMatrix(gr1);\ ;\ngap> Display(adj1);\n[ [ 0, 1, 0 ],\n [ 1, 0, 1 ],\n [ 0, 1, 1 ]\ ]\ngap> adj1 = TransposedMat(adj1);\ntrue\ngap> gr1 = DigraphReverse(gr1);\nt\ rue\ngap> gr2 := Digraph([[2, 3], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr2);\nfalse\ngap> adj2 := Adjace\ ncyMatrix(gr2);;\ngap> Display(adj2);\n[ [ 0, 1, 1 ],\n [ 1, 0, 1 ],\n \ [ 0, 1, 1 ] ]\ngap> adj2 = TransposedMat(adj2);\nfalse\n]]>\n <\ /Description>\n\n\n \n \n true or false.\n \n This property is true if the digraph digraph is \ a tournament,\n and false if it is not. \n\n A tournament is \ an orientation of a complete (undirected) graph.\n Specifically, a tourname\ nt is a digraph which has a unique directed edge\n (of some orientation) be\ tween any pair of distinct vertices, and no loops.\n \n \n If t\ he argument digraph is mutable, then the return value of\n this prop\ erty is recomputed every time it is called. \n D :\ = Digraph([[2, 3, 4], [3, 4], [4], []]);\n\ngap> IsTournament(D);\ntrue\ngap> D := Digraph([[2], [1], [3]]);\n\ngap> IsTournament(D);\nfalse\ngap>\ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> IsTournament(D);\ntrue\ngap> DigraphRemoveEdge(D, 1, 2);\n\ngap> IsTournament(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n This property is true if the digraph di\ graph\n is transitive, and false if it is not.\n\n A digraph \ is transitive if whenever [ i, j ] and\n [ j, k ] are \ edges of the digraph, then [ i, k ] is also an\n edge of the digraph\ . \n\n Let n be the number of vertices of an arbitrary digraph, \ and let\n m be the number of edges.\n For general digraphs, t\ he methods used for this property use a version\n of the Floyd-Warshall alg\ orithm, and have complexity O(n^3).\n\n However for digraphs which a\ re topologically sortable\n [], then \ methods with\n complexity O(m + n + m \\cdot n) will be used when ap\ propriate.\n \n \n If the argument digraph is mutable, t\ hen the return value of\n this property is recomputed every time it is call\ ed. \n \n D := Digraph([[1, 2], [3], [3]]);\ \n\ngap> IsTransitiveDigraph(D);\n\ false\ngap> gr2 := Digraph([[1, 2, 3], [3], [3]]);\n\ngap> IsTransitiveDigraph(gr2);\ntrue\ngap> gr2 = DigraphTr\ ansitiveClosure(D);\ntrue\ngap> gr3 := Digraph([[1, 2, 2, 3], [3, 3], [3]]);\n\ \ngap> IsTransitiveDigraph(gr\ 3);\ntrue\n]]>\n \n\n\n \ \n \n \n true or false.\n \n A digraph is a preorder digraph if and only if the\ digraph satisifies both\n and .\n A preorder digraph (or quasiorder digraph) digraph corresponds to\n the preorder relation \\leq defined by\ x \\leq y if and only\n if [x, y] is an edge of digraph.\n \n \n If the argument digraph is mutable, then the\ return value of\n this property is recomputed every time it is called. \n\ \n D := Digraph([[1], [2, 3], [2, 3]]);\n\ngap> IsPreorderDigraph(D);\ntrue\ngap> D := D\ igraph([[1 .. 4], [1 .. 4], [1 .. 4], [1 .. 4]]);\n\ngap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[2], \ [3], [4], [5], [1]]);\n\ngap> IsPr\ eorderDigraph(D);\nfalse\ngap> D := Digraph([[1], [1, 2], [2, 3]]);\n\ngap> IsQuasiorderDigraph(D);\nfalse\n]]><\ /Example>\n \n\n\n \n \n true or fals\ e.\n \n A digraph is a partial order digraph if\ and only if the digraph satisifies\n all of ,\n and .\n A partial order digraph corresponds\n to the parti\ al order relation \\leq defined by x \\leq y if and\n only if\ [x, y] is an edge of digraph.\n \n \n If the arg\ ument digraph is mutable, then the return value of\n this property i\ s recomputed every time it is called. \n\n D := Digrap\ h([[1, 3], [2, 3], [3]]);\n\ngap> \ IsPartialOrderDigraph(D);\ntrue\ngap> D := CycleDigraph(5);\n\ngap> IsPartialOrderDigraph(D);\nfalse\ngap> D := Dig\ raph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4]]);\n\ngap> IsPartialOrderDigraph(D);\ntrue\n]]>\n \n\n\n \n \n \n \n true or false.\n \n IsMeetSemilat\ ticeDigraph returns true if the digraph\n digraph is a me\ et semilattice; IsJoinSemilatticeDigraph\n returns true if th\ e digraph digraph is a join semilattice;\n and IsLatticeDigraph<\ /C> returns true if the digraph\n digraph is both a meet and \ a join semilattice.\n \n\n For a partial order digraph the\n corresponding partial order is the relation \ \\leq, defined by\n x \\leq y if and only if [x, y] is\ an edge.\n A digraph is a meet semilattice if it is a partial order\ and every\n pair of vertices has a greatest lower bound (meet) with respec\ t to the\n aforementioned relation. A join semilattice is a partial \ order where\n every pair of vertices has a least upper bound (join) with re\ spect to\n the relation.\n \n \n If the argument digraph<\ /A> is mutable, then the return value of\n this property is recomputed ever\ y time it is called. \n\n D := Digraph([[1, 3], [2\ , 3], [3]]);\n\ngap> IsMeetSemilat\ ticeDigraph(D);\nfalse\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLattic\ eDigraph(D);\nfalse\ngap> D := Digraph([[1], [2], [1 .. 3]]);\n\ngap> IsJoinSemilatticeDigraph(D);\nfalse\ngap> \ IsMeetSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ngap> D :\ = Digraph([[1 .. 4], [2, 4], [3, 4], [4]]);\n\ngap> IsMeetSemilatticeDigraph(D);\ntrue\ngap> IsJoinSemilatticeDi\ graph(D);\ntrue\ngap> IsLatticeDigraph(D);\ntrue\ngap> D := Digraph([[1, 1, 1]\ , [1, 1, 2, 2],\n> [1, 3, 3], [1, 2, 3, 3, 4]]);\n\ngap> IsMeetSemilatticeDigraph(D);\nt\ rue\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\ntrue\ \n]]>\n \n\n\n \n\n Regularity\n \n \n true or false.\n\ \n This property is true if there is an integer n<\ /C> such that\n for every vertex v of digraph digraph there a\ re exactly\n n edges terminating in v.\n\n See also and\n .\n \ \n \n If the argument digraph is mutable, then the return v\ alue of\n this property is recomputed every time it is called. \n IsInRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsInReg\ ularDigraph(ChainDigraph(4));\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n \ This property is true if there is an integer n such that\n fo\ r every vertex v of digraph digraph there are exactly\n n<\ /C> edges starting at v.\n \n\n See also and\n .\n \n \n \ If the argument digraph is mutable, then the return value of\n th\ is property is recomputed every time it is called. \n IsOutRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsOutRegularDigraph(C\ hainDigraph(4));\nfalse\n]]>\n \n\n\n \ \n \n true or false.\n \n This property is\ true if there is an integer n such that\n for every vertex <\ C>v of digraph digraph there are exactly\n n edges starti\ ng and terminating at v. In other words,\n the property is true if digraph is both in-regular and\n and out-regular.\n\n See a\ lso and\n .\n \n \n If the argument digraph is mutable, then \ the return value of\n this property is recomputed every time it is called. \ \n IsRegularDigraph(CompleteDigraph(4));\ntrue\nga\ p> IsRegularDigraph(ChainDigraph(4));\nfalse\n]]>\n \ \n\n\n \n \n true or false.\n \n If digraph is a connected symmetric graph, this property \ returns\n true if for any two vertices u and v of d\ igraph\n and any two integers i and j between 0 an\ d the\n diameter of digraph, the number of vertices at distance i\ \n from u and distance j from v depends only on\n \ i, j, and the distance between vertices u and\n \ v. \n\n Alternatively, a distance regular graph is a graph for which\ there exist\n integers b_i, c_i, and i such that for \ any two\n vertices u, v in digraph which are distance \ i\n apart, there are exactly b_i neighbors of v which \ are at\n distance i - 1 away from u, and c_i neighbors\ of\n v which are at distance i + 1 away from u. This\ \n definition is used to check whether digraph is distance regular.<\ P/>\n\n In the case where digraph is not symmetric or not connected,\ the\n property is false.\n \n \n If the argument \ digraph is mutable, then the return value of\n this property is recompu\ ted every time it is called. \n\n D := DigraphSymm\ etricClosure(ChainDigraph(5));;\ngap> IsDistanceRegularDigraph(D);\nfalse\ngap\ > D := Digraph([[2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]]);\n\ngap> IsDistanceRegularDigraph(D);\ntrue\n]]>\n \n\n\n \n\n Connectivity and cycles\n \n \n true or false.\n \n This property is true if the digraph digrap\ h is acyclic, and\n false if it is not. A digraph is acyclic<\ /E> if every directed\n cycle on the digraph is trivial. See Section for the definition of a dire\ cted cycle, and of a trivial\n directed cycle.\n\n The method used i\ n this operation has complexity O(m+n) where\n m is the numbe\ r of edges (counting multiple edges as one) and\n n is the number of\ vertices in the digraph. \n\n If the argument digraph is mutabl\ e, then the return value of\n this property is recomputed every time it is \ called. \n\n Petersen := Graph(SymmetricGroup(5), \ [[1, 2]], OnSets,\n> function(x, y)\n> return IsEmpty(Intersection(x, y));\n\ > end);;\ngap> D := Digraph(Petersen);\n\ngap> IsAcyclicDigraph(D);\nfalse\ngap> D := DigraphFromDiSparse6Stri\ ng(\n> \".b_OGCIDBaPGkULEbQHCeRIdrHcuZMfRyDAbPhTi|zF\");\n\ngap> IsAcyclicDigraph(D);\ntrue\ngap> IsAcyclicDig\ raph(ChainDigraph(10));\ntrue\ngap> D := CompleteDigraph(IsMutableDigraph, 4);\ \n\ngap> IsAcyclicDigraph(D);\nfals\ e\ngap> IsAcyclicDigraph(CycleDigraph(10));\nfalse]]>\n \n\n\n \n \n true or false.\n \n IsChainDigraph returns true if the digraph digraph\ is\n isomorphic to the chain digraph with the same number of vertices as\n\ digraph, and false if it is not; see . \n\n A digraph is a chain if and only if it is a di\ rected tree, in which\n every vertex has out degree at most one; see \n and .\n \n \ \n If the argument digraph is mutable, then the return value of\n \ this property is recomputed every time it is called. \n\n D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsChainDigraph(D);\nfalse\ngap> D := ChainDigraph(5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> D :=\ DigraphReverse(D);\n\ngap> IsChai\ nDigraph(D);\ntrue\ngap> D := ChainDigraph(IsMutableDigraph, 5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> DigraphRev\ erse(D);\n\ngap> IsChainDigraph(D);\ \ntrue]]>\n \n\n\n \n \n true or <\ K>false.\n \n This property is true if t\ he digraph digraph\n is weakly connected and false if it is n\ ot. A digraph\n digraph is weakly connected if it is possibl\ e to travel\n from any vertex to any other vertex by traversing edges in ei\ ther\n direction (possibly against the orientation of some of them). \n\ \n The method used in this function has complexity O(m) if the\n \ digraph's attribute is set, otherwise it has\n \ complexity O(m+n) (where\n m is the number of edges and\n \ n is the number of vertices of the digraph).\n \n \n I\ f the argument digraph is mutable, then the return value of\n this p\ roperty is recomputed every time it is called. \n\n D := Digraph([[2], [3], []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> D :=\ Digraph([[1, 3], [4], [3], []]);;\ngap> IsConnectedDigraph(D);\nfalse\ngap> D\ := Digraph(IsMutableDigraph, [[2], [3], []]);;\ngap> IsConnectedDigraph(D);\n\ true\ngap> D := Digraph(IsMutableDigraph, [[1, 3], [4], [3], []]);;\ngap> IsCo\ nnectedDigraph(D);\nfalse]]>\n \n\n\n \ \n \n true or false.\n \n A connected \ digraph is biconnected if it is still connected (in the\n sense of <\ Ref Prop=\"IsConnectedDigraph\"/>) when any vertex is removed. \n If D\ has at least 3 vertices, then IsBiconnectedDigraph\n implies \ ;\n see or for a more\n detailed explanation.\n \ \n\n IsBiconnectedDigraph returns true if the digraph\n digraph is biconnected, and false if it is not. In\n particular\ , IsBiconnectedDigraph returns false if\n digraph is n\ ot connected. \n \n If the argument digraph is mutable, the\ n the return value of\n this property is recomputed every time it is called\ . \n\n Multiple edges are ignored by this method. \n\n The metho\ d used in this operation has complexity O(m+n) where\n m is t\ he number of edges and n is the number of vertices in\n the digraph.\ \n \n\n See also , , and \n .\n \n \n IsBiconnectedDigraph(Digraph([[1, 3], [2, 3], [3]]));\n\ false\ngap> IsBiconnectedDigraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[\ 1, 1], [1, 1, 2], [3], [3, 3, 4, 4]]);;\ngap> IsBiconnectedDigraph(D);\nfalse\ \ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBiconnectedDigraph(D);\ntrue]]>\ \n \n\n\n \n \n true or false.\n\ \n A connected digraph is bridgeless if it is still c\ onnected (in the\n sense of ) when any ed\ ge is removed. \n If D has at least 3 vertices, then implies IsBridgelessDigraph;\n see or for a more\n detail\ ed explanation.\n \n\n IsBridgelessDigraph returns true if the digraph\n D is bridgeless, and false if it is not. I\ n\n particular, IsBridgelessDigraph returns false if\n \ D is not connected. \n \n If the argument D is mutable,\ then the return value of\n this property is recomputed every time it is ca\ lled. \n\n Multiple edges are ignored by this method. \n\n The m\ ethod used in this operation has complexity O(m+n) where\n m \ is the number of edges and n is the number of vertices in\n the digr\ aph.\n \n\n See also , , and \n .\n \n IsBridgelessDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ \ngap> IsBridgelessDigraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1],\ [1, 1, 2], [3], [3, 3, 4, 4]]);;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> D\ := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBridgelessDigraph(D);\ntrue\ngap> D := Digraph([[\ 2, 5], [1, 3, 4, 5], [2, 4], [2, 3], [1, 2]]);\n\ngap> IsBridgelessDigraph(D);\ntrue\ngap> IsBiconnectedDigraph\ (D);\nfalse\ngap> D := Digraph([[2], [3], [4], [2]]);\n\ngap> IsBridgelessDigraph(D);\nfalse\ngap> IsBiconnected\ Digraph(D);\nfalse\ngap> IsBridgelessDigraph(ChainDigraph(2));\nfalse\ngap> Is\ BiconnectedDigraph(ChainDigraph(2));\ntrue\n]]>\n \n<\ /ManSection>\n\n \n \n true or false.\n \n This property is true if the digraph digraph is str\ ongly\n connected and false if it is not. \n\n A digraph d\ igraph is strongly connected if there is a directed\n path from \ every vertex to every other vertex. See Section for the definition of a directed\n path. \ \n\n The method used in this operation is based on Gabow's Algorithm and has complexity O(m+n), where m is\n \ the number of edges (counting multiple edges as one) and n is the\n \ number of vertices in the digraph.\n \n \n If the argument <\ A>digraph is mutable, then the return value of\n this property is recom\ puted every time it is called. \n\n D := CycleDigr\ aph(250000);\n\ngap> IsStronglyC\ onnectedDigraph(D);\ntrue\ngap> D := DigraphRemoveEdges(D, [[250000, 1]]);\n\ngap> IsStronglyConnected\ Digraph(D);\nfalse\ngap> D := CycleDigraph(IsMutableDigraph, 250000);\n\ngap> IsStronglyConnectedDigraph\ (D);\ntrue\ngap> DigraphRemoveEdge(D, [250000, 1]);\n\ngap> IsStronglyConnectedDigraph(D);\nfalse\n]]>\n \n\n\n \n \n true or false\ .\n \n This property is true if the digraph \ digraph\n is aperiodic, i.e. if its \ is equal to 1.\n Otherwise, the property is false.\n \n \ \n If the argument digraph is mutable, then the return value of\n \ this property is recomputed every time it is called. \n D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> IsAperiodicDigraph(D);\nfalse\ngap> D := Di\ graph([[2], [3, 5], [4], [5], [1, 2]]);\n\ngap> IsAperiodicDigraph(D);\ntrue\ngap> D := Digraph(IsMutableDigraph\ , [[2], [3, 5], [4], [5], [1, 2]]);\n\ngap> IsAperiodicDigraph(D);\ntrue]]>\n \n\n\n \n \n \ true or false.\n \n Returns\ true if the digraph digraph is a directed tree, and\n fal\ se if it is not. \n\n A directed tree is an acyclic digraph \ with precisely 1 source,\n such that no two vertices share an out-neighbour\ .\n Note the empty digraph is not considered a directed\n tree as it has\ no source. \n\n See also .\n \n \ \n If the argument digraph is mutable, then the return value of\ \n this property is recomputed every time it is called. \n\n D := Digraph([[], [2]]);\n\ngap> IsDirectedTree(D);\nfalse\ngap> D := Digraph([[3], [3], []]);\n\ngap> IsDirectedTree(D);\nfalse\ngap\ > D := Digraph([[2], [3], []]);\n\ \ngap> IsDirectedTree(D);\ntrue\ngap> D := Digraph([[2, 3], [6], [4, 5], [], [\ ], []]);\n\ngap> IsDirectedTree(D)\ ;\ntrue\n]]>\n \n\n\n \n \ \n \n true or false.\ \n \n The property IsUndirectedTree returns true if the digraph\n digraph is an undirected tree, and the property\n\ IsUndirectedForest returns true if digraph is an\n \ undirected forest; otherwise, these properties return false. \n\n \ An undirected tree is a symmetric digraph without loops, in which fo\ r\n any pair of distinct vertices u and v, there is exactly o\ ne\n directed path from u to v. See and , and see\n Section for the definition of\n direc\ ted path. This definition implies that an undirected tree has\n no multiple\ edges. \n\n An undirected forest is a digraph, each of whose co\ nnected components\n is an undirected tree. In other words, an undirected f\ orest is isomorphic to\n a disjoint union of undirected trees. See and . In particular, every\n undi\ rected tree is an undirected forest. \n\n Please note that the digraph \ with zero vertices is considered to be neither\n an undirected tree nor an \ undirected forest.\n \n \n If the argument digraph is mu\ table, then the return value of\n this property is recomputed every time it\ is called. \n\n D := Digraph([[3], [3], [1, 2]]);\ \n\ngap> IsUndirectedTree(D);\ntru\ e\ngap> IsSymmetricDigraph(D) and not DigraphHasLoops(D);\ntrue\ngap> D := Dig\ raph([[3], [5], [1, 4], [3], [2]]);\n\ngap> IsConnectedDigraph(D);\nfalse\ngap> IsUndirectedTree(D);\nfalse\ngap\ > IsUndirectedForest(D);\ntrue\ngap> D := Digraph([[1, 2], [1], [2]]);\n\ngap> IsUndirectedTree(D) or IsUndirect\ edForest(D);\nfalse\ngap> IsSymmetricDigraph(D) or not DigraphHasLoops(D);\nfa\ lse]]>\n \n\n\n \n \n true or fa\ lse.\n \n This property returns true if the dig\ raph digraph is Eulerian.\n \n\n A connected digraph is calle\ d Eulerian if there exists a directed\n circuit on the digraph which\ includes every edge exactly once. See\n Section for the definition of\n a directed circuit. Note th\ at the empty digraph with at most one vertex is\n considered to be Eulerian\ .\n \n \n If the argument digraph is mutable, then the r\ eturn value of\n this property is recomputed every time it is called. \n\n \ D := Digraph([[]]);\n\ngap> IsEulerianDigraph(D);\ntrue\ngap> D := Digraph([[2], []]);\n<\ immutable digraph with 2 vertices, 1 edge>\ngap> IsEulerianDigraph(D);\nfalse\ \ngap> D := Digraph([[3], [], [2]]);\n\ngap> IsEulerianDigraph(D);\nfalse\ngap> D := Digraph([[2], [3], [1]]);\n\ \ngap> IsEulerianDigraph(D);\ntrue\ \n]]>\n \n\n\n \n \n true or \ false.\n \n If digraph is Hamiltonian, t\ hen this property returns\n true, and false if it is not. \n\n A digraph with n vertices is Hamiltonian if it has a\n \ directed cycle of length n. See Section for the definition of a directed cycle.\n Note t\ he empty digraphs on 0 and 1 vertices are considered to be\n Hamiltonian.\n\n The method used in this operation has the worst case complexity as\n\ .\n \n \n If the argumen\ t digraph is mutable, then the return value of\n this property is re\ computed every time it is called. \n\n g := Digrap\ h([[]]);\n\ngap> IsHamiltonianDigraph(g\ );\ntrue\ngap> g := Digraph([[2], [1]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\ngap> g := Digraph([[3], [], [2]\ ]);\n\ngap> IsHamiltonianDigraph(g\ );\nfalse\ngap> g := Digraph([[2], [3], [1]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\n]]>\n \n\n\n \n \n true or false.\n \n IsCycleDigraph returns true if the digraph digraph<\ /A> is\n isomorphic to the cycle digraph with the same number of vertices a\ s\n digraph, and false if it is not; see . \n\n A digraph is a cycle if and only if it is s\ trongly connected and has\n the same number of edges as vertices. \n \n \n If the argument digraph is mutable, then the return valu\ e of\n this property is recomputed every time it is called. \n\n D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsCycleDigraph(D);\nfalse\ngap> D := CycleDigraph(5);\ \n\ngap> IsCycleDigraph(D);\ntrue\nga\ p> D := OnDigraphs(D, (1, 2, 3));\n\ngap> D = CycleDigraph(5);\nfalse\ngap> IsCycleDigraph(D);\ntrue]]>\n \n\n\n \n \n Pl\ anarity\n \n \n true or false.\n \ \n A planar digraph is a digraph that can be embedded in the plane i\ n\n such a way that its edges do not intersect. A digraph is planar if and \ only\n if it does not have a subdigraph that is homeomorphic to either the\ \n complete graph on 5 vertices or the complete bipartite graph with\ \n vertex sets of sizes 3 and 3.\n \n \n IsPl\ anarDigraph returns true if the digraph digraph is\n plan\ ar and false if it is not. The directions and multiplicities of\n an\ y edges in digraph are ignored by IsPlanarDigraph.\n \n\ \n See also .\n \n\n This met\ hod uses the reference implementation in\n &edge-addition-planarity-suite; \ by John Boyer of the algorithms described\n in .\n\n \ IsPlanarDigraph(CompleteDigraph(4));\ntrue\ngap> IsP\ lanarDigraph(CompleteDigraph(5));\nfalse\ngap> IsPlanarDigraph(CompleteBiparti\ teDigraph(2, 3));\ntrue\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(3, 3));\ \nfalse\ngap> IsPlanarDigraph(CompleteDigraph(IsMutableDigraph, 4));\ntrue\nga\ p> IsPlanarDigraph(CompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsPlana\ rDigraph(CompleteBipartiteDigraph(IsMutableDigraph, 2, 3));\ntrue\ngap> IsPlan\ arDigraph(CompleteBipartiteDigraph(IsMutableDigraph, 3, 3));\nfalse\n]]>\n \n\n\n \n \n true or false.<\ /Returns>\n \n An outer planar digraph is a digraph th\ at can be embedded in the\n plane in such a way that its edges do not inter\ sect, and all vertices\n belong to the unbounded face of the embedding. A \ digraph is outer planar\n if and only if it does not have a subdigraph that\ is homeomorphic to either\n the complete graph on 4 vertices or the\ complete bipartite graph\n with vertex sets of sizes 2 and 3\ .\n \n \n IsOuterPlanarDigraph returns true if the \ digraph\n digraph is outer planar and false if it is not. The\ \n directions and multiplicities of any edges in digraph are ignored\ by\n IsPlanarDigraph. \n\n See also .\n\n This method uses the reference implementation in\n &edge-a\ ddition-planarity-suite; by John Boyer of the algorithms described\n in .\n\n IsOuterPlanarDigraph(CompleteDi\ graph(4));\nfalse\ngap> IsOuterPlanarDigraph(CompleteDigraph(5));\nfalse\ngap>\ IsOuterPlanarDigraph(CompleteBipartiteDigraph(2, 3));\nfalse\ngap> IsOuterPla\ narDigraph(CompleteBipartiteDigraph(3, 3));\nfalse\ngap> IsOuterPlanarDigraph(\ CycleDigraph(10));\ntrue\ngap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableD\ igraph, 4));\nfalse\ngap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableDigrap\ h, 5));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(IsMutableDi\ graph,\n> 2, 3));\nfalse\ngap\ > IsOuterPlanarDigraph(CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 3, 3));\nfalse\ngap> IsOuterPlanarDigr\ aph(CycleDigraph(IsMutableDigraph, 10));\ntrue\n\n]]>\n \n\n\n \n\n Homomorphisms and tr\ ansformations\n \n \n true or false.\n \n This property returns true if digraph is a core,\n a\ nd false if it is not.\n\n A digraph D is a core i\ f and only if it has no proper\n subdigraphs A such that there exist\ s a homomorphism from D\n to A. In other words, a digraph \ D is a core if and only if\n every endomorphism on D is an autom\ orphism on D.\n \n \n If the argument digraph is \ mutable, then the return value of\n this property is recomputed every time \ it is called. \n\n D := CompleteDigraph(6);\n\ngap> IsDigraphCore(D);\ntrue\ngap> D \ := DigraphSymmetricClosure(CycleDigraph(6));\n\ngap> DigraphHomomorphism(D, CompleteDigraph(2));\nTra\ nsformation( [ 1, 2, 1, 2, 1, 2 ] )\ngap> IsDigraphCore(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n If digraph is a digraph without multiple edg\ es, then\n IsEdgeTransitive returns true if digraph\n \ is edge transitive, and false otherwise. A digraph is\n edge t\ ransitive if its automorphism group acts\n transitively on its edges (v\ ia the action\n ).\n \n \ \n If the argument digraph is mutable, then the return value of\n \ this property is recomputed every time it is called. \n\n IsEdgeTransitive(CompleteDigraph(2));\ntrue\ngap> IsEdgeTransitive(C\ hainDigraph(3));\nfalse\ngap> IsEdgeTransitive(Digraph([[2], [3, 3, 3], []]));\ \nError, the argument must be a digraph with no multiple edges,\n]]>\n \n\n\n \n \n true or false.\n \n If digraph is a digraph, then IsVertex\ Transitive returns\n true if digraph is vertex transitive\ , and false\n otherwise. A digraph is vertex transitive if it\ s automorphism group\n acts transitively on its vertices.\n \n \ \n If the argument digraph is mutable, then the return value of\n \ this property is recomputed every time it is called. \n\n IsVertexTransitive(CompleteDigraph(2));\ntrue\ngap> IsVertexTransitiv\ e(ChainDigraph(3));\nfalse\n]]>\n \n\n\n\ \n\n \n\n \n Homomorphisms\n \n\n Act\ ing on digraphs\n \n \n \n A digraph.\n \n If digraph is a digraph, and the second argument perm is a\n permutation<\ /E> of the vertices of digraph, then this operation\n returns a digr\ aph constructed by relabelling the vertices of\n digraph according t\ o perm. Note that for an automorphism\n f of a digraph, we h\ ave OnDigraphs(digraph, f) =\n digraph.\n \n\n \ If the second argument is a transformation trans of the\n \ vertices of digraph, then this operation returns a digraph\n constru\ cted by transforming the source and range of each edge according to\n tr\ ans. Thus a vertex which does not appear in the image of\n trans\ will be isolated in the returned digraph, and the returned\n digraph may c\ ontain multiple edges, even if digraph does not.\n If trans i\ s mathematically a permutation, then the result coincides\n with OnDigra\ phs(digraph, AsPermutation(trans)).\n \n\n The of digraph will not be retained\n i\ n the returned digraph. \n\n If digraph belongs to , then\n relabelling of the vertices is performed direct\ ly on digraph.\n If digraph belongs to , an immutable\n copy of digraph with the vertices relab\ elled is returned. \n\n D := Digraph([[3], [1, \ 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> new := OnDigraphs(D, (1, 2));\n\ngap> OutNeighbours(new);\n[ [ 2, 3, 5 ], [ 3 ], [ 2 ], [ 2, 1, 4 ],\ [ 1, 3, 5 ] ]\ngap> D := Digraph([[2], [], [2]]);\n\ngap> t := Transformation([1, 2, 1]);;\ngap> new := OnDigra\ phs(D, t);\n\ngap> OutNeighbo\ urs(new);\n[ [ 2, 2 ], [ ], [ ] ]\ngap> ForAll(DigraphEdges(D),\n> e -> IsD\ igraphEdge(new, [e[1] ^ t, e[2] ^ t]));\ntrue\n]]>\n \ \n\n\n \n \n \n A digraph.\n \n If digraph is a digraph, and pair is \ a pair consisting of a\n permutation of the vertices and a permutation of t\ he edges of\n digraph, then this operation returns a digraph\n co\ nstructed by relabelling the vertices and edges of digraph\n accordi\ ng to perm[1] and perm[2], respectively. \n\n In its seco\ nd form, OnMultiDigraphs returns a digraph with vertices\n and edges\ permuted by perm1 and perm2, respectively. \n\n Note tha\ t OnDigraphs(digraph,\n perm)=OnMultiDigraphs(digraph, [\ perm, ()]) where perm is\n a permutation of the vertices of d\ igraph. If you are only interested\n in the action of a permutation on \ the vertices of a digraph, then you can\n use OnDigraphs instead of \ OnMultiDigraphs.\n\n If digraph belongs to , then\n relabelling of the vertices is performed directly on\ digraph.\n If digraph belongs to , an immutable\n copy of digraph with the vertices relabelled\ is returned. \n\n D1 := Digraph([\n> [3, 6, 3]\ , [], [3], [9, 10], [9], [], [], [10, 4, 10], [], []]);\n\ngap> p := BlissCanonicalLabelling(D1);\n[ (1,\ 7)(3,6)(4,10)(5,9), () ]\ngap> D2 := OnMultiDigraphs(D1, p);\n\ngap> OutNeighbours(D2);\n[ [ ], [ ], [ \ ], [ ], [ ], [ 6 ], [ 6, 3, 6 ], [ 4, 10, 4 ], \n [ 5 ], [ 5, 4 ] ]]]>\n \n\n\n \n\n \n Isomorphisms and canoni\ cal labellings\n From version 0.11.0 of &Digraphs; it is possible\ to use either &bliss; or\n &nauty; (via &NautyTracesInterface;) to calcula\ te canonical labellings and\n automorphism groups of digraphs; see and for more details about &bliss; an\ d &nauty;,\n respectively. \n\n \n \n \n N\ othing.\n \n These functions can be used to specify\ whether &nauty; or &bliss; should be\n used by default by &Digraphs;. If &\ NautyTracesInterface; is not available,\n then these functions do nothing. \ Otherwise, by calling\n DigraphsUseNauty subsequent computations wil\ l default to using\n &nauty; rather than &bliss;, where possible.\n \ \n\n You can call these functions at any point in a &GAP; session, as many\ \n times as you like, it is guaranteed that existing digraphs remain\n v\ alid, and that comparison of existing digraphs and newly created digraphs\n \ via\n ,\n ,\n , and\n\ are also valid. \n\n It is also possible to compute the au\ tomorphism group of a specific digraph\n using both &nauty; and &bliss; usi\ ng \n and , respectively.\n \n\n\n \n \n A permutation group.\n \n If digraph is a digraph, then this attribute contains the \ group of\n automorphisms of digraph. An automorphism of d\ igraph\n is an isomorphism from digraph to itself. See for more information\ \n about isomorphisms of digraphs. \n\n If digraph is not a m\ ultidigraph then the automorphism group is\n returned as a group of permuta\ tions on the set of vertices of\n digraph. \n\n If digraph\ is a multidigraph then the automorphism group is returned\n as the dir\ ect product of a group of permutations on the set of vertices of\n digra\ ph with a group of permutations on the set of edges of\n digraph\ . These groups can be accessed using on the\n returned g\ roup. \n\n By default, the automorphism group is found using &bliss; by \ Tommi Junttila\n and Petteri Kaski. If &NautyTracesInterface; is available,\ then &nauty; by\n Brendan Mckay and Adolfo Piperno can be used instead; se\ e\n , \n <\ Ref Attr=\"NautyAutomorphismGroup\"/>, ,\n \ and .\n \n\n If the argument digr\ aph is mutable, then the return value of\n this attribute is recomputed\ every time it is called. \n\n johnson := DigraphF\ romGraph6String(\"E}lw\");\n\ngap\ > G := AutomorphismGroup(johnson);\nGroup([ (3,4), (2,3)(4,5), (1,2)(5,6) ])\n\ gap> cycle := CycleDigraph(9);\n\ngap\ > G := AutomorphismGroup(cycle);\nGroup([ (1,2,3,4,5,6,7,8,9) ])\ngap> IsCycli\ c(G) and Size(G) = 9;\ntrue]]>\n \n \ \n\n \n \n \n \n A p\ ermutation group.\n \n If digraph is a digra\ ph, then this attribute contains the group of\n automorphisms of digraph\ as calculated using &bliss; by Tommi Junttila\n and Petteri Kaski. \n\n The attribute and\n operation returns\n the value of either BlissAuto\ morphismGroup or . These grou\ ps are, of course, equal but\n their generating sets may differ. \n \ The attribute returns the value of\n BlissAutomorphismGr\ oup as it is not implemented for &nauty; The \n requirements for the op\ tional arguments vertex_colours and \n edge_colours are docum\ ented in .\n\n See also , an\ d .\n \n\n If the argument digrap\ h is mutable, then the return value of\n this attribute is recomputed e\ very time it is called. \n\n G := BlissAutomorphis\ mGroup(JohnsonDigraph(5, 2));;\ngap> IsSymmetricGroup(G);\ntrue\ngap> Size(G);\ \n120]]>\n \n \n\n \n \ \n A permutation group.\n \n If digraph \ is a digraph, then this attribute contains the group of\n automorphisms of \ digraph as calculated using &nauty; by\n Brendan Mckay and Adolfo Pi\ perno via &NautyTracesInterface;.\n\n \n\n The attribute and\n operation returns\n the value \ of either NautyAutomorphismGroup or .\n These groups are, of course, equal b\ ut their generating sets may differ. \n\n See also , and .\n \n\n If the arg\ ument digraph is mutable, then the return value of\n this attribute \ is recomputed every time it is called. \n\n NautyAutom\ orphismGroup(JohnsonDigraph(5, 2));\nGroup([ (3,4)(6,7)(8,9), (2,3)(5,6)(9,10)\ , (2,5)(3,6)(4,7),\n (1,2)(6,8)(7,9) ])]]>\n \n \n\n \n \n A permutation group.\n \n\n This operation comp\ utes the automorphism group of a vertex-coloured digraph.\n A vertex-colour\ ed digraph can be specified by its underlying digraph\n digraph and \ its colouring vert_colours. Let n be the\n number of vertices\ of digraph. The colouring vert_colours may\n have one of the\ following two forms:\n\n \n \n a list of n \ integers, where vert_colours[i] is the\n colour of vertex\ i, using the colours [1 .. m] for some\n m <= n; or\n \n \n a list of non-empty disjoint lists\ whose union is\n DigraphVertices(digraph), such that\n \ vert_colours[i] is the list of all vertices with colour\n \ i.\n \n \n\n The automorphism group<\ /E> of a coloured digraph digraph with\n colouring vert_colours is the group consisting of its automorphisms; an\n automorphism o\ f digraph is an isomorphism of coloured\n digraphs from digraph to itself. This group is equal to the\n subgroup of AutomorphismGrou\ p(digraph) consisting of those\n automorphisms that preserve the\ colouring specified by vert_colours. See\n , and see \n for more informati\ on about isomorphisms of coloured digraphs. \n\n If digraph is n\ ot a multidigraph then the automorphism group is\n returned as a group of p\ ermutations on the set of vertices of\n digraph. \n\n If d\ igraph is a multidigraph then the automorphism group is returned\n as t\ he direct product of a group of permutations on the set of vertices of\n digraph with a group of permutations on the set of edges of\n digra\ ph. These groups can be accessed using on the\n retu\ rned group. \n\n By default, the automorphism group is found using &blis\ s; by Tommi Junttila\n and Petteri Kaski. If &NautyTracesInterface; is avai\ lable, then &nauty; by\n Brendan Mckay and Adolfo Piperno can be used inste\ ad; see\n ,\n , ,\n and .\n\n cycle := CycleDigraph(9);\n\ngap> G := AutomorphismGroup(cycle);;\ngap> IsCyclic(G) and Size(\ G) = 9;\ntrue\ngap> colours := [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;\ngap> H := \ AutomorphismGroup(cycle, colours);;\ngap> Size(H);\n3\ngap> H = AutomorphismGr\ oup(cycle, [1, 2, 3, 1, 2, 3, 1, 2, 3]);\ntrue\ngap> H = SubgroupByProperty(G,\ p -> OnTuplesSets(colours, p) = colours);\ntrue\ngap> IsTrivial(AutomorphismG\ roup(cycle, [1, 1, 2, 2, 2, 2, 2, 2, 2]));\ntrue]]>\n \n\n\n \n \n A permutation group.\n \n\n This operation computes the automorphism group of a vertex- and/\ or\n edge-coloured digraph.\n A coloured digraph can be specified by its\ underlying digraph\n digraph and colourings vert_colours, edge_colours.\n Let n be the number of vertices of digraph. The colourings\n must have the following forms:\n\n \n \n vert_colours must be fail or a list of n in\ tegers,\n where vert_colours[i] is the colour of vertex <\ C>i,\n using the colours [1 .. m] for some m <= n;\ \n \n \n edge_colours must be fail\ or a list of n lists of\n integers of the same shape as OutN\ eighbours(digraph), where\n edge_colours[i][j] is the\ colour of the edge\n OutNeighbours(digraph)[i][j], using the co\ lours [1 .. k]\n for some k <= n;\n \n \ \n\n Giving vert_colours [edge_colours] as fail\ is\n equivalent to setting all vertices [edges] to be the same colour. \n\n Unlike , it \ is\n possible to obtain the automorphism group of an edge-coloured multidig\ raph\n (see ) when no two edges share the sa\ me\n source, range, and colour.\n\n The automorphism group of a v\ ertex/edge-coloured digraph\n digraph with colouring c is the\ group consisting of its\n vertex/edge-colour preserving automorphisms; an\ \n automorphism of digraph is an isomorphism of vertex/edge-c\ oloured\n digraphs from digraph to itself. This group is equal to t\ he\n subgroup of AutomorphismGroup(digraph) consisting of tho\ se\n automorphisms that preserve the colouring specified by colours.\ See\n , and see \ \n for more information about isomorphisms of coloured digraphs. \ \n\n If digraph is not a multidigraph then the automorphism grou\ p is\n returned as a group of permutations on the set of vertices of\n <\ A>digraph. \n\n If digraph is a multidigraph then the automo\ rphism group is returned\n as the direct product of a group of permutations\ on the set of vertices of\n digraph with a group of permutations on\ the set of edges of\n digraph. These groups can be accessed using <\ Ref Oper=\"Projection\"\n Label=\"for a domain and a positive integer\" B\ ookName=\"ref\"/> on the\n returned group. \n\n By default, the autom\ orphism group is found using &bliss; by Tommi Junttila\n and Petteri Kaski.\ If &NautyTracesInterface; is available, then &nauty; by\n Brendan Mckay an\ d Adolfo Piperno can be used instead; see\n ,\n , , and\n \ .\n\n cycle := Cy\ cleDigraph(12);\n\ngap> vert_colours\ := List([1 .. 12], x -> x mod 3 + 1);;\ngap> edge_colours := List([1 .. 12], \ x -> [x mod 2 + 1]);;\ngap> Size(AutomorphismGroup(cycle));\n12\ngap> Size(Aut\ omorphismGroup(cycle, vert_colours));\n4\ngap> Size(AutomorphismGroup(cycle, f\ ail, edge_colours));\n6\ngap> Size(AutomorphismGroup(cycle, vert_colours, edge\ _colours));\n2\ngap> IsTrivial(AutomorphismGroup(cycle,\n> vert_colours, List(\ [1 .. 12], x -> [x mod 4 + 1])));\ntrue\n]]>\n \n\n\n \n \n \n A permutation, or a list\ of two permutations.\n \n A function \\rho \ that maps a digraph to a digraph is a canonical\n representative map if the following two conditions hold for all\n digraphs G and \ H: \n\n \n \n \\rho(G) and G \ are isomorphic as digraphs; and\n \n \n \\rho\ (G)=\\rho(H) if and only if G and H are\n isomorphic \ as digraphs.\n \n \n\n A canonical labelling o\ f a digraph G (under \\rho) is an\n isomorphism of G o\ nto its canonical representative,\n \\rho(G). See \n for more information ab\ out isomorphisms of digraphs. \n\n BlissCanonicalLabelling retur\ ns a canonical labelling of the digraph\n digraph found using &bliss\ ; by Tommi Junttila and Petteri Kaski.\n NautyCanonicalLabelling ret\ urns a canonical labelling of the digraph\n digraph found using &nau\ ty; by Brendan McKay and Adolfo Piperno.\n Note that the canonical labellin\ gs returned by &bliss; and &nauty; are not\n usually the same (and may depe\ nd of the version used). \n\n BlissCanonicalLabelling can only be\ computed if digraph has\n no multiple edges; see . \n\n digraph1 := DigraphFromDiS\ parse6String(\".ImNS_AiB?qRN\");\n\ngap> BlissCanonicalLabelling(digraph1);\n(1,9,5,7)(3,6,4,10)\ngap> p := (1,\ 2, 7, 5)(3, 9)(6, 10, 8);;\ngap> digraph2 := OnDigraphs(digraph1, p);\n\ngap> digraph1 = digraph2;\nfalse\ngap\ > OnDigraphs(digraph1, BlissCanonicalLabelling(digraph1)) =\n> OnDigraphs(d\ igraph2, BlissCanonicalLabelling(digraph2));\ntrue]]>\n \n\n\n \n \n \n A permutation.\n \n\ A function \\rho that maps a coloured digraph to a coloured digraph\ is\n a canonical representative map if the following two conditions\ hold\n for all coloured digraphs G and H:\n\n \n \ \n \\rho(G) and G are isomorphic as coloured dig\ raphs; and\n \n \n \\rho(G)=\\rho(H) if a\ nd only if G and H are\n isomorphic as coloured digraphs.\ \n \n \n\n A canonical labelling of a coloured\ digraph G (under\n \\rho) is an isomorphism of G onto\ its canonical\n representative, \\rho(G). See \ for more information about\n isomorphisms of coloured digraphs. \n\n \ A coloured digraph can be specified by its underlying digraph digraph\ \n and its colouring colours. Let n be the number of vertice\ s of\n digraph. The colouring colours may have one of the fol\ lowing\n two forms: \n\n \n \n a list of \ n integers, where colours[i] is the\n colour of verte\ x i, using the colours [1 .. m] for some\n m <= n; or\n \n \n a list of non-empty disjoint list\ s whose union is\n DigraphVertices(digraph), such that\n \ colours[i] is the list of all vertices with colour\n \ i.\n \n \n\n If digraph and colou\ rs together form a coloured digraph,\n BlissCanonicalLabelling r\ eturns a canonical labelling of the digraph\n digraph found using &b\ liss; by Tommi Junttila and Petteri Kaski.\n Similarly, NautyCanonicalLa\ belling returns a canonical labelling of\n the digraph digraph f\ ound using &nauty; by Brendan McKay and Adolfo\n Piperno. Note that the ca\ nonical labellings returned by &bliss; and\n &nauty; are not usually the sa\ me (and may depend of the version used). \n\n BlissCanonicalLabelling\ can only be computed if digraph has\n no multiple edges; see . The canonical\n labelling of digraph \ is given as a permutation of its vertices.\n The canonical representative o\ f digraph can be created from\n digraph and its canonical lab\ elling p by using the\n operation :\n gap> OnDigraphs(digraph, p);\n\n\ The colouring of the canonical representative can easily be\n construct\ ed. A vertex v (in digraph) has colour i if\n and only\ if the vertex v ^ p (in the canonical representative) has\n colour \ i, where p is the permutation of the canonical\n labelling th\ at acts on the vertices of digraph. In particular, if\n colours has the first form that is described above, then the\n colouring of the \ canonical representative is given by:\n\n gap> List(DigraphVertices(di\ graph), i -> colours[i / p]);\n\n On the other hand, if colours has the second form above, then the\n canonical representative has colour\ ing:\n\n gap> OnTuplesSets(colours, p);\n\n \n\n digraph := DigraphFromDiSparse6String(\".ImNS_AiB?qRN\");\ \n\ngap> colours := [[1, 2, 8, 9,\ 10], [3, 4, 5, 6, 7]];;\ngap> p := BlissCanonicalLabelling(digraph, colours);\ \n(1,5,8,4,10,3,9)(6,7)\ngap> OnDigraphs(digraph, p);\n\ngap> OnTuplesSets(colours, p);\n[ [ 1, 2, 3, 4, 5 ], [\ 6, 7, 8, 9, 10 ] ]\ngap> colours := [1, 1, 1, 1, 2, 3, 1, 3, 2, 1];;\ngap> p \ := BlissCanonicalLabelling(digraph, colours);\n(1,6,9,7)(3,4,5,8,10)\ngap> OnD\ igraphs(digraph, p);\n\ngap> List\ (DigraphVertices(digraph), i -> colours[i / p]);\n[ 1, 1, 1, 1, 1, 1, 2, 2, 3,\ 3 ]]]>\n \n\n\n \n \n \n A digraph.\n \n The attribute BlissCanonicalLabelling returns the canonical\n \ representative found by applying . The digraph returned is canonical in the sense\n \ that\n\n \n \n BlissCanonicalDigraph(digra\ ph) and digraph are\n isomorphic as digraphs; and\n \ \n \n If gr is any digraph then BlissCanon\ icalDigraph(gr)\n and BlissCanonicalDigraph(digraph) \ are equal if and only\n if gr and digraph are isomorphic \ as digraphs.\n \n \n\n Analogously, the attribute \ NautyCanonicalLabelling returns the\n canonical representative found by\ applying .\n \n\n If the argument digraph is mutable, then the retur\ n value of\n this attribute is recomputed every time it is called. \n\n \ digraph := Digraph([[1], [2, 3], [3], [1, 2, 3]]);\n<\ immutable digraph with 4 vertices, 7 edges>\ngap> canon := BlissCanonicalDigra\ ph(digraph);\n\ngap> OutNeighbours\ (canon);\n[ [ 1 ], [ 2 ], [ 3, 2 ], [ 1, 3, 2 ] ]\n]]>\n \n\n\n \n \n A permutation group.\n \n If\ digraph is immutable and was created knowing a subgroup of\n its au\ tomorphism group, then this group is stored in the attribute\n DigraphGr\ oup. If digraph is mutable, or was not created\n knowing a subgr\ oup of its automorphism group, then DigraphGroup\n returns the entir\ e automorphism group of digraph. \n Note that if digraph i\ s mutable, then the automorphism group is\n recomputed every time this func\ tion is called.\n \n\n Note that certain other constructor operation\ s such as , , and , may not require a group a\ s one of the arguments,\n but use the standard constructor method using a g\ roup, and hence set the\n DigraphGroup attribute for the resulting d\ igraph.\n\n n := 4;;\ngap> adj := function(x, y)\n\ > return (((x - y) mod n) = 1) or (((x - y) mod n) = n - 1);\n> end;;\ \ngap> group := CyclicGroup(IsPermGroup, n);\nGroup([ (1,2,3,4) ])\ngap> D := \ Digraph(IsMutableDigraph, group, [1 .. n], \\^, adj);\n\ngap> HasDigraphGroup(D);\nfalse\ngap> DigraphGroup(D);\nG\ roup([ (2,4), (1,2)(3,4) ])\ngap> AutomorphismGroup(D);\nGroup([ (2,4), (1,2)(\ 3,4) ])\ngap> D := Digraph(group, [1 .. n], \\^, adj);\n\ngap> HasDigraphGroup(D);\ntrue\ngap> DigraphGroup(D);\ \nGroup([ (1,2,3,4) ])\ngap> D := DoubleDigraph(D);\n\ngap> HasDigraphGroup(D);\ntrue\ngap> DigraphGroup(D);\nG\ roup([ (1,2,3,4)(5,6,7,8), (1,5)(2,6)(3,7)(4,8) ])\ngap> AutomorphismGroup(D) \ =\n> Group([(6, 8), (5, 7), (4, 6), (3, 5), (2, 4),\n> (1, 2)(3, 4)(5, \ 6)(7, 8)]);\ntrue\ngap> D := Digraph([[2, 3], [], []]);\n\ngap> HasDigraphGroup(D);\nfalse\ngap> HasAutomorphism\ Group(D);\nfalse\ngap> DigraphGroup(D);\nGroup([ (2,3) ])\ngap> HasAutomorphis\ mGroup(D);\ntrue\ngap> group := DihedralGroup(8);\n\ngap> D := CayleyDigraph(group);\n\ngap> HasDigraphGroup(D);\ntrue\ngap> GeneratorsOfGroup(DigraphG\ roup(D));\n[ (1,2)(3,8)(4,6)(5,7), (1,3,4,7)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) ]]\ ]>\n \n\n\n \n \n \n An immutable list of l\ ists of integers.\n \n \n DigraphOrbits ret\ urns the orbits of the action of the\n on the \ set of vertices of digraph.\n\n G := Group(\ [(2, 3)(7, 8, 9), (1, 2, 3)(4, 5, 6)(8, 9)]);;\ngap> D := EdgeOrbitsDigraph(G,\ [1, 2]);\n\ngap> DigraphOrbits(D)\ ;\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]\ngap> D := DigraphMutableCopy(D);\ \n\ngap> DigraphOrbits(D);\n[ [ 1, 2\ , 3 ], [ 4, 5, 6, 7, 8, 9 ] ]\n]]>\n \n\ \n\n \n \n <\ Returns>\n An immutable list of integers.\n \n \n \ DigraphOrbitReps returns a list of orbit representatives\n of the\ action of the on the set of vertices of\n \ digraph.\n\n D := CayleyDigraph(AlternatingGro\ up(4));\n\ngap> DigraphOrbitReps\ (D);\n[ 1 ]\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphOrbitReps(D);\n[ 1 ]\ngap> D := DigraphFromDigraph6\ String(\"&IGO??S?`?_@?a?CK?O\");\n\ngap> DigraphOrbitReps(D);\n[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\ngap> Digraph\ OrbitReps(DigraphMutableCopy(D));\n[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\n]]>\n \n\n\n \n \n \n An immutable list of \ integers.\n \n \n DigraphSchreierVector ret\ urns the so-called Schreier vector\n of the action of the on the set of vertices of\n digraph. The Schreier\ vector is a list sch of integers with\n length DigraphNrVertices\ (digraph) where:\n \n sch[i] < 0:\n \n implies that i is an orbit representative an\ d\n DigraphOrbitReps(digraph)[-sch[i]] = i.\n \n\n sch[i] > 0:\n \n implies t\ hat i / gens[sch[i]]\n is one step closer to the root (or repres\ entative) of the tree, where\n gens is the generators of Digr\ aphGroup(digraph).\n \n \n\n n := 4;;\ngap> adj := function(x, y)\n> return (((x - y) mod \ n) = 1) or (((x - y) mod n) = n - 1);\n> end;;\ngap> group := CyclicGroup(I\ sPermGroup, n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph, gro\ up, [1 .. n], \\^, adj);\n\ngap> sch\ := DigraphSchreierVector(D);\n[ -1, 2, 2, 1 ]\ngap> D := CayleyDigraph(Altern\ atingGroup(4));\n\ngap> sch := D\ igraphSchreierVector(D);\n[ -1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1 ]\ngap> Digrap\ hOrbitReps(D);\n[ 1 ]\ngap> gens := GeneratorsOfGroup(DigraphGroup(D));\n[ (1,\ 5,7)(2,4,8)(3,6,9)(10,11,12), (1,2,3)(4,7,10)(5,9,11)(6,8,12) ]\ngap> 10 / gen\ s[sch[10]];\n7\ngap> 7 / gens[sch[7]];\n5\ngap> 5 / gens[sch[5]];\n1]]>\n \n\n\n \n \n \n A permutation group.\n \ \n \n DigraphStabilizer returns the stabiliz\ er of the vertex v\n under of the action of the on the set of\n vertices of digraph.\n\n D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> \ DigraphStabilizer(D, 2);\nGroup(())\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ \ngap> DigraphStabilizer(D, 2);\nGroup(())\n]]>\n \n<\ /ManSection>\n\n \n \n true or fa\ lse.\n \n This operation returns true if\ there exists an isomorphism from the\n digraph digraph1 to the digr\ aph digraph2. See for more information\n about isomorphisms of digraphs. \ \n\n By default, an isomorphism is found using the canonical labellings of \ the\n digraphs obtained from &bliss; by Tommi Junttila\n and Petteri Kas\ ki. If &NautyTracesInterface; is available, then &nauty; by\n Brendan Mckay\ and Adolfo Piperno can be used instead; see\n , and .\n\n di\ graph1 := CycleDigraph(4);\n\ngap> di\ graph2 := CycleDigraph(5);\n\ngap> Is\ IsomorphicDigraph(digraph1, digraph2);\nfalse\ngap> digraph2 := DigraphReverse\ (digraph1);\n\ngap> IsIsomorphicDi\ graph(digraph1, digraph2);\ntrue\ngap> digraph1 := Digraph([[3], [], []]);\n\ngap> digraph2 := Digraph([[], [], [\ 2]]);\n\ngap> IsIsomorphicDigraph(d\ igraph1, digraph2);\ntrue]]>\n \n\n\n \ \n \n true or false.\n \n This opera\ tion tests for isomorphism of coloured digraphs. A coloured\n digraph can \ be specified by its underlying digraph digraph1 and its\n colouring \ colours1. Let n be the number of vertices of\n digraph1. The colouring colours1 may have one of the following\n two forms\ :\n\n \n \n a list of n integers, where c\ olours[i] is the\n colour of vertex i, using the colo\ urs [1 .. m] for some\n m <= n; or\n \n \ \n a list of non-empty disjoint lists whose union is\n \ DigraphVertices(digraph), such that\n colours[i\ ] is the list of all vertices with colour\n i.\n \n \n\n If digraph1 and digraph2 are digraphs with\ out multiple edges,\n and colours1 and colours2 are colouring\ s of digraph1\n and digraph2, respectively, then this operati\ on returns true\n if there exists an isomorphism between these two c\ oloured digraphs. See\n for more information about isomorphisms\ of coloured digraphs. \n\n By default, an isomorphism is found using t\ he canonical labellings of the\n digraphs obtained from &bliss; by Tommi Ju\ nttila\n and Petteri Kaski. If &NautyTracesInterface; is available, then &n\ auty; by\n Brendan Mckay and Adolfo Piperno can be used instead; see\n <\ Ref Func=\"DigraphsUseBliss\"/>, and .\n\n \ digraph1 := ChainDigraph(4);\n\ngap> digraph2 := ChainDigraph(3);\n\ngap> IsIsomorphicDigraph(digraph1, digraph2,\n> [[1, 4], \ [2, 3]], [[1, 2], [3]]);\nfalse\ngap> digraph2 := DigraphReverse(digraph1);\n<\ immutable digraph with 4 vertices, 3 edges>\ngap> IsIsomorphicDigraph(digraph1\ , digraph2,\n> [1, 1, 1, 1], [1, 1, 1, 1]);\ntrue\ngap> IsIsomorphicDigraph(d\ igraph1, digraph2,\n> [1, 2, 2, 1], [1, 2, 2, 1]);\ntrue\ngap> IsIsomorphicDi\ graph(digraph1, digraph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfalse]]>\ \n \n\n\n \n \n \ A permutation, or a pair of permutations, or fail.\n \n This operation returns an isomorphism between the digraphs dig\ raph1\n and digraph2 if one exists, else this operation returns \ fail.\n \n\n An isomorphism from a digraph digraph1\ to a digraph\n digraph2 is a bijection p from the vertic\ es of\n digraph1 to the vertices of digraph2 with the followi\ ng\n property: for all vertices i and j of digraph1,\n\ [i, j] is an edge of digraph1 if and only if [i ^ p, j\n \ ^ p] is an edge of digraph2. \n\n If there exists such \ an isomorphism, then this operation returns one.\n The form of this isomorp\ hism is a permutation p of the vertices of\n digraph1 such th\ at \n\n OnDigraphs(digraph1, p) = digraph2.\n\n By def\ ault, an isomorphism is found using the canonical labellings of the\n digra\ phs obtained from &bliss; by Tommi Junttila\n and Petteri Kaski. If &NautyT\ racesInterface; is available, then &nauty; by\n Brendan Mckay and Adolfo Pi\ perno can be used instead; see\n , and .\n\n digraph1 := Cycl\ eDigraph(4);\n\ngap> digraph2 := Cycl\ eDigraph(5);\n\ngap> IsomorphismDigra\ phs(digraph1, digraph2);\nfail\ngap> digraph1 := CompleteBipartiteDigraph(10, \ 5);\n\ng\ ap> digraph2 := CompleteBipartiteDigraph(5, 10);\n\ngap> p := IsomorphismDigraphs(digr\ aph1, digraph2);\n(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)\ngap> OnDigraphs(d\ igraph1, p) = digraph2;\ntrue\n]]>\n \n\ \n\n \n \n A permutation, or fail.\n \n \ This operation searches for an isomorphism between coloured digraphs. A\n \ coloured digraph can be specified by its underlying digraph digraph1\n \ and its colouring colours1. Let n be the number of vertices\ \n of digraph1. The colouring colours1 may have one of the\n \ following two forms:\n\n \n \n a list of n integers, where colours[i] is the\n colour of vertex i, using the colours [1 .. m] for some\n m <= n; \ or\n \n \n a list of non-empty disjoint lists wh\ ose union is\n DigraphVertices(digraph), such that\n \ colours[i] is the list of all vertices with colour\n <\ C>i.\n \n \n\n An isomorphism between colo\ ured digraphs is an isomorphism between\n the underlying digraphs that pres\ erves the colourings. See for more information\n about isomorphisms of digraphs. More \ precisely, let f be an\n isomorphism of digraphs from the digraph digraph1 (with colouring\n colours1) to the digraph digraph2\ (with colouring\n colours2), and let p be the permutatio\ n of the vertices of\n digraph1 that corresponds to f. Then \ f preserves the\n colourings of digraph1 and digraph2 \ – and hence is an\n isomorphism of coloured digraphs – if colours\ 1[i] =\n colours2[i ^ p] for all vertices i in d\ igraph1.\n \n\n This operation returns such an isomorphism if on\ e exists, else it returns\n fail. \n\n By default, an isomorp\ hism is found using the canonical labellings of the\n digraphs obtained fro\ m &bliss; by Tommi Junttila\n and Petteri Kaski. If &NautyTracesInterface; \ is available, then &nauty; by\n Brendan Mckay and Adolfo Piperno can be use\ d instead; see\n , and .\n\n digraph1 := ChainDigraph(4);\n\ngap> digraph2 := ChainDigraph(3);\n\ngap> IsomorphismDigraphs(digraph1, di\ graph2,\n> [[1, 4], [2, 3]], [[1, 2], [3]]);\nfail\ngap> digraph2 := DigraphR\ everse(digraph1);\n\ngap> colours1\ := [1, 1, 1, 1];;\ngap> colours2 := [1, 1, 1, 1];;\ngap> p := IsomorphismDigr\ aphs(digraph1, digraph2, colours1, colours2);\n(1,4)(2,3)\ngap> OnDigraphs(dig\ raph1, p) = digraph2;\ntrue\ngap> List(DigraphVertices(digraph1), i -> colours\ 1[i ^ p]) = colours2;\ntrue\ngap> colours1 := [1, 1, 2, 2];;\ngap> colours2 :=\ [2, 2, 1, 1];;\ngap> p := IsomorphismDigraphs(digraph1, digraph2, colours1, c\ olours2);\n(1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\ngap> Li\ st(DigraphVertices(digraph1), i -> colours1[i ^ p]) = colours2;\ntrue\ngap> Is\ omorphismDigraphs(digraph1, digraph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfail]]\ >\n \n\n\n \n \n An immutable li\ st of lists.\n \n This function returns the list out of out-neighbours of each\n representative of the orbits of\ the action of on\n the vertex set of the digr\ aph digraph. \n\n More specifically, if reps is the list \ of orbit representatives,\n then a vertex j appears in out[i]\ each time\n there exists an edge with source reps[i] and range j\ in\n digraph.\n \n\n If \ is trivial, then \n is returned.\n\n D := Digraph([\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, \ 5], [1, 2, 3, 4]]);\n\ngap> Digra\ phGroup(D);\nGroup(())\ngap> RepresentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, \ 5 ], [ 3, 5 ], [ 2 ], [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := Digraph(IsMu\ tableDigraph, [\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3, 4]])\ ;\n\ngap> DigraphGroup(D);\nGroup((\ ))\ngap> RepresentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ], [ 2 ]\ , [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := DigraphFromDigraph6String(\"&GYH\ PQgWTIIPW\");\n\ngap> G := Digrap\ hGroup(D);;\ngap> GeneratorsOfGroup(G);\n[ (1,2)(3,4)(5,6)(7,8), (1,3,2,4)(5,7\ ,6,8), (1,5)(2,6)(3,8)(4,7) ]\ngap> Set(RepresentativeOutNeighbours(D), Set);\ \n[ [ 2, 3, 5 ] ]]]>\n \n\n\n \n \n \n \n \n true or false.\n \n IsDigraphIsomorphism returns true if the permutation or\ \n transformation x is an isomorphism from the digraph src to\ \n the digraph ran.\n \n\n IsDigraphAutomorphism re\ turns true if the permutation or\n transformation x is an aut\ omorphism of the digraph digraph.\n \n\n A permutation or tra\ nsformation x is an isomorphism from a\n digraph src t\ o a digraph ran if the following hold:\n \n \n \ x is a bijection from the vertices of src to those of\n \ ran;\n \n \n [u ^ x, v ^ x\ ] is an edge of\n ran if and only if [u, v] is an\ \n edge of src; and\n \n \n x fixes every i which is not a vertex of src.\n \n \ \n See also . \n \n If col1 and col2, or col, ar\ e given, then they must\n represent vertex colourings; see \n \n f\ or details of the permissible values for\n these arguments. The homomorphis\ m must then also have the property:\n \n \n \n \ col1[i] = col2[i ^ x] for all vertices i of src,\n fo\ r IsDigraphIsomorphism. \n \n col[i] = col[\ i ^ x] for all vertices i of digraph,\n for IsDigr\ aphAutomorphism. \n \n\n For some digraphs, it can be \ faster to use IsDigraphAutomorphism\n than to test membership in the\ automorphism group of digraph.\n\n src := \ Digraph([[1], [1, 2], [1, 3]]);\n\ \ngap> IsDigraphAutomorphism(src, (1, 2, 3));\nfalse\ngap> IsDigraphAutomorphi\ sm(src, (2, 3));\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2, 1, 1]);\nt\ rue\ngap> IsDigraphAutomorphism(src, (2, 3), [2, 2, 1]);\nfalse\ngap> IsDigrap\ hAutomorphism(src, (2, 3)(4, 5));\nfalse\ngap> IsDigraphAutomorphism(src, (1, \ 4));\nfalse\ngap> IsDigraphAutomorphism(src, ());\ntrue\ngap> ran := Digraph([\ [2, 1], [2], [2, 3]]);\n\ngap> IsD\ igraphIsomorphism(src, ran, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(ran, src\ , (1, 2));\ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ntrue\ngap> IsD\ igraphIsomorphism(src, Digraph([[3], [1, 3], [2]]), (1, 2, 3));\nfalse\ngap> I\ sDigraphIsomorphism(src, ran, (1, 2), [1, 2, 3], [2, 1, 3]);\ntrue\ngap> IsDig\ raphIsomorphism(src, ran, (1, 2), [1, 2, 2], [2, 1, 3]);\nfalse\n]]>\ \n \n\n\n \n \n \n \ true or false. \n \n The operatio\ n IsDigraphColouring verifies whether or not \n the list list describes a proper colouring of the digraph \n digraph. \n \ \n\n A list list describes a proper colouring of a dig\ raph\n digraph if list consists of positive integers, the l\ ength of\n list equals the number of vertices in digraph, a\ nd for any\n vertices u, v of digraph if u and v<\ /C> are adjacent,\n then list[u] >< list[v].\n\ \n\n A transformation t describes a proper colouring of \ a digraph\n digraph, if ImageListOfTransformation(t,\n \ DigraphNrVertices(digraph)) is a proper colouring of \n digraph. \n\n See also .\n \ \n D := JohnsonDigraph(5, 3);\n\ngap> IsDigraphColouring(D, [1, 2\ , 3, 3, 2, 1, 4, 5, 6, 7]);\ntrue\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, \ 1, 2, 5, 6, 7]);\nfalse\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6\ , -1]);\nfalse\ngap> IsDigraphColouring(D, [1, 2, 3]);\nfalse\ngap> IsDigraphC\ olouring(D, IdentityTransformation);\ntrue\n]]>\n \ \n \n\n \n\n Homomorphisms of digr\ aphs\n\n The following methods exist to find homomorphisms betwee\ n digraphs.\n If an argument to one of these methods is a digraph with mult\ iple edges,\n then the multiplicity of edges will be ignored in order to pe\ rform the\n calculation; the digraph will be treated as if it has no multip\ le edges.\n\n \n \n The argument user_param\ .\n \n This function finds homomorphisms from t\ he digraph D1 to the digraph\n D2 subject to the conditions i\ mposed by the other arguments as\n described below.\n\n If f \ and g are homomorphisms found by\n HomomorphismDigraphsFinder\ , then f cannot be obtained from\n g by right multiplying by \ an automorphism of D2 in\n aut_grp.\n\n \n hook\n \n This argument should be a function\ or fail. \n\n If hook is a function, then it must hav\ e two arguments\n user_param (see below) and a transformation t. The\n function hook(user_param, t) is call\ ed every time\n a new homomorphism t is found by\n Hom\ omorphismDigraphsFinder. If the function returns true,\n the\ n HomomorphismDigraphsFinder stops and does not find any\n furth\ er homomorphisms. This feature might be useful if you are\n searching f\ or a homomorphism that satisfies some condition that you\n cannot speci\ fy via the other arguments to\n HomomorphismDigraphsFinder.\n \ \n\n If hook is fail, then a default function is\ used which\n simply adds every new homomorphism found by\n H\ omomorphismDigraphsFinder to user_param, which must be a\n m\ utable list in this case.\n \n\n user_param\n \n If hook is a function, then user_param\ can be any &GAP;\n object. The object user_param is used as the\ first argument of\n the function hook. For example, user_par\ am might be a\n transformation semigroup, and hook(use\ r_param, t)\n might set user_param to be the closure of \ user_param and\n t. \n\n If the value of ho\ ok is fail, then the value of\n user_param must be a \ mutable list.\n \n\n max_results\n <\ Item>\n This argument should be a positive integer or infinity.\ \n HomomorphismDigraphsFinder will return after it has found\n \ max_results homomorphisms or the search is complete, whichever\n \ happens first.\n \n\n hint\n \ \n This argument should be a positive integer or fail. \n\n If hint is a positive integer, then only homorphisms of ra\ nk\n hint are found. \n\n If hint is fail, then no restriction is put on the rank\n of homomorphisms found.\n \ \n\n injective\n \n This\ argument should be 0, 1, or 2. If it is\n 2, then only embeddings are found, if it is 1, then only\n inje\ ctive homomorphisms are found, and if it is 0 there are no\n res\ trictions imposed by this argument.\n \n\n For backwards com\ patibility, injective can also be false\n or true \ which correspond to the values 0 and 1\n described in the\ previous paragraph, respectively. \n \n\n image\ \n \n This argument should be a subset of the vertice\ s of the graph D2.\n HomomorphismDigraphsFinder only find\ s homomorphisms from\n D1 to the subgraph of D2 induced b\ y the vertices\n image.\n \n\n partial\ _map\n \n This argument should be a partial map f\ rom D1 to D2,\n that is, a (not necessarily dense) list o\ f vertices of the digraph\n D2 of length no greater than the num\ ber vertices in the digraph\n D1. HomomorphismDigraphsFinder<\ /C> only finds homomorphisms\n extending partial_map (if any).\n\ \n \n colors1\n \n \ This should be a list representing possible colours of vertices in the\n \ digraph D1; see \n \n for details of t\ he permissible values for this argument.\n \n \n <\ A>colors2\n \n This should be a list representing\ possible colours of vertices in the\n digraph D2; see \n \ \n for details of the permissible values for this argum\ ent.\n \n order\n \n Th\ e optional argument order specifies the order the\n vertices in \ D1 appear in the search for homomorphisms. \n The value of this \ parameter can have a large impact\n on the runtime of the function. It \ seems in many cases to be a good\n idea for this to be the , i.e.\n vertices ordered from highest to lo\ west degree.\n \n \n The optional argument au\ t_grp should be a subgroup of the\n automorphism group of D2\ . This function returns unique\n representatives of the homomorphisms f\ ound up to right multiplication\n by aut_grp. If this argument i\ s not specific, it defaults to the\n full automorphism group of D2, which may be costly to calculate.\n \n \n\n D := ChainDigraph(10);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> HomomorphismDigraphsFinder(D, D, fail, [], i\ nfinity, 2, 0,\n> [3, 4], [], fail, fail);\n[ Transformation( [ 3, 4, 3, 4, 3,\ 4, 3, 4, 3, 4 ] ), \n Transformation( [ 4, 3, 4, 3, 4, 3, 4, 3, 4, 3 ] ) ]\n\ gap> D2 := CompleteDigraph(6);;\ngap> HomomorphismDigraphsFinder(D, D2, fail, \ [], 1, fail, 0,\n> [1 .. 6], [1, 2, 1], fail, fail);\n[ Transformation( [ 1, 2\ , 1, 3, 4, 5, 6, 1, 2, 1 ] ) ]\ngap> func := function(user_param, t)\n> Add(us\ er_param, t * user_param[1]);\n> end;;\ngap> HomomorphismDigraphsFinder(D, D2,\ func, [Transformation([2, 2])],\n> 3, fail, 0, [1 .. 6], [1, 2, 1], fail, fai\ l);\n[ Transformation( [ 2, 2 ] ), \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, \ 2, 2, 2 ] ), \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 3 ] ), \n Trans\ formation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 4 ] ) ]\ngap> HomomorphismDigraphsFind\ er(NullDigraph(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [1, 2, 3], \ fail, fail, fail, fail,\n> Group(()));\n[ IdentityTransformation, Transformati\ on( [ 1, 3, 3 ] ), \n Transformation( [ 2, 1 ] ), Transformation( [ 2, 3, 3 ]\ ), \n Transformation( [ 3, 1, 3 ] ), Transformation( [ 3, 2, 3 ] ) ]\ngap> H\ omomorphismDigraphsFinder(NullDigraph(2), NullDigraph(3), fail,\n> [], infinit\ y, fail, 1, [1, 2, 3], fail, fail, fail, fail,\n> Group((1, 2)));\n[ IdentityT\ ransformation, Transformation( [ 1, 3, 3 ] ), \n Transformation( [ 3, 1, 3 ] \ ) ]]]>\n \n\n\n \n \n A tran\ sformation, or fail.\n \n A homomorphism fro\ m digraph1 to digraph2 is a mapping from\n the vertex set of \ digraph1 to a subset of the vertices of\n digraph2, such that\ every pair of vertices [i,j] which has\n an edge i->j is map\ ped to a pair of vertices [a,b] which\n has an edge a->b. No\ te that non-adjacent vertices can still be\n mapped to adjacent vertices. <\ P/>\n\n DigraphHomomorphism returns a single homomorphism between\n \ digraph1 and digraph2 if it exists, otherwise it returns\n \ fail.\n gr1 := ChainDigraph(3);;\ngap> gr2 := \ Digraph([[3, 5], [2], [3, 1], [], [4]]);\n\ngap> DigraphHomomorphism(gr1, gr1);\nIdentityTransformation\ngap> ma\ p := DigraphHomomorphism(gr1, gr2);\nTransformation( [ 3, 1, 5, 4, 5 ] )\ngap>\ IsDigraphHomomorphism(gr1, gr2, map);\ntrue\n]]>\n \ \n\n\n \n \n \n A list of transformations.\n \n HomomorphismsDigraphsRepresentatives finds e\ very\n between digraph1 and\n \ digraph2, up to right multiplication by an element of the\n of digraph2.\n In\ other words, every homomorphism f between digraph1 and\n \ digraph2 can be written as the composition f = g * x,\n where g is one of the HomomorphismsDigraphsRepresentatives\n and x\ is an automorphism of digraph2.\n \n\n HomomorphismsD\ igraphs returns all homomorphisms between\n digraph1 and digr\ aph2.\n gr1 := ChainDigraph(3);;\ngap> gr2 := \ Digraph([[3, 5], [2], [3, 1], [], [4]]);\n\ngap> HomomorphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 3, 1 ]\ ), Transformation( [ 1, 3, 3 ] ), \n Transformation( [ 1, 5, 4, 4, 5 ] ), Tr\ ansformation( [ 2, 2, 2 ] ), \n Transformation( [ 3, 1, 3 ] ), Transformation\ ( [ 3, 1, 5, 4, 5 ] ), \n Transformation( [ 3, 3, 1 ] ), Transformation( [ 3,\ 3, 3 ] ) ]\ngap> HomomorphismsDigraphsRepresentatives(gr1, CompleteDigraph(3)\ );\n[ Transformation( [ 2, 1 ] ), Transformation( [ 2, 1, 2 ] ) ]\n]]>\n \n\n\n \n \n A transformation, or \ fail.\n \n DigraphMonomorphism return\ s a single injective between\ digraph1 and digraph2\n if one exists, otherwise it returns \ fail.\n gr1 := ChainDigraph(3);;\ngap> gr2 \ := Digraph([[3, 5], [2], [3, 1], [], [4]]);\n\ngap> DigraphMonomorphism(gr1, gr1);\nIdentityTransformation\ngap>\ DigraphMonomorphism(gr1, gr2);\nTransformation( [ 3, 1, 5, 4, 5 ] )\n]]>\n \n\n\n \n \n \n A list of \ transformations.\n \n These operations behave the s\ ame as and\n , except they only return\n injective ho\ momorphisms.\n gr1 := ChainDigraph(3);;\ngap> gr2 \ := Digraph([[3, 5], [2], [3, 1], [], [4]]);\n\ngap> MonomorphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 5, \ 4, 4, 5 ] ), \n Transformation( [ 3, 1, 5, 4, 5 ] ) ]\ngap> MonomorphismsDigr\ aphsRepresentatives(gr1, CompleteDigraph(3));\n[ Transformation( [ 2, 1 ] ) ]\ \n]]>\n \n\n\n \n \n A transfo\ rmation, or fail.\n \n DigraphEpimorphism\ returns a single surjective between digraph1 and digraph2\n if one exists, otherwise\ it returns fail.\n gr1 := DigraphReverse(C\ hainDigraph(4));\n\ngap> gr2 := Di\ graphRemoveEdge(CompleteDigraph(3), [1, 2]);\n\ngap> DigraphEpimorphism(gr2, gr1);\nfail\ngap> DigraphEpimorphis\ m(gr1, gr2);\nTransformation( [ 3, 1, 2, 3 ] )\n]]>\n \n\n\n \n \n \n A list of transformations.\n \n These operations behave the same as and\n , except they only return\n surjective homomorphisms.\n gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphSymmetricClosure(CycleDigraph(\ 3));\n\ngap> Epimorphism\ sDigraphsRepresentatives(gr1, gr2);\n[ Transformation( [ 3, 1, 2, 1 ] ), Trans\ formation( [ 3, 1, 2, 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ) ]\ngap> Epim\ orphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 2, 1, 3 ] ), Transformatio\ n( [ 1, 2, 3, 1 ] ), \n Transformation( [ 1, 2, 3, 2 ] ), Transformation( [ 1\ , 3, 1, 2 ] ), \n Transformation( [ 1, 3, 2, 1 ] ), Transformation( [ 1, 3, 2\ , 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ), Transformation( [ 2, 1, 3, 1 ] \ ), \n Transformation( [ 2, 1, 3, 2 ] ), Transformation( [ 2, 3, 1, 2 ] ), \n \ Transformation( [ 2, 3, 1, 3 ] ), Transformation( [ 2, 3, 2, 1 ] ), \n Trans\ formation( [ 3, 1, 2, 1 ] ), Transformation( [ 3, 1, 2, 3 ] ), \n Transformat\ ion( [ 3, 1, 3, 2 ] ), Transformation( [ 3, 2, 1, 2 ] ), \n Transformation( [\ 3, 2, 1, 3 ] ), Transformation( [ 3, 2, 3, 1 ] ) ]\n]]>\n \n\n\n \n \n A transformation, or fail.\n \n An embedding of a digraph digraph1 into an\ other digraph\n digraph2 is a fr\ om digraph1\n to digraph2 which has the additional property t\ hat a pair of\n vertices [i, j] which have no edge i -> j in \ digraph1 are\n\n mapped to a pair of vertices [a, b] which ha\ ve no edge a->b in\n digraph2. \n\n In other words, an \ embedding t is an isomorphism from\n digraph1 to the of digraph2\n on the image of t. \n\n DigraphEmbedding returns a single embedding if one exists, ot\ herwise\n it returns fail.\n gr := Chain\ Digraph(3);\n\ngap> DigraphEmbedding(\ gr, CompleteDigraph(4));\nfail\ngap> DigraphEmbedding(gr, Digraph([[3], [1, 4]\ , [1], [3]]));\nTransformation( [ 2, 4, 3, 4 ] )\n]]>\n \n\n\n \n \n \n A list of transformations.\n \n T\ hese operations behave the same as and\n\ , except they only ret\ urn\n embeddings of D1 into D2. \n\n See also .\n D1 := NullDigraph(2);\ \n\ngap> D2 := CycleDigraph(5);\n\ngap> EmbeddingsDigraphsRepresentatives(\ D1, D2);\n[ Transformation( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 ] ) ]\ \ngap> EmbeddingsDigraphs(D1, D2);\n[ Transformation( [ 1, 3, 3 ] ), Transform\ ation( [ 1, 4, 3, 4 ] ), \n Transformation( [ 2, 4, 4, 5, 1 ] ), \n Transfor\ mation( [ 2, 5, 4, 5, 1 ] ), \n Transformation( [ 3, 1, 5, 1, 2 ] ), \n Tran\ sformation( [ 3, 5, 5, 1, 2 ] ), \n Transformation( [ 4, 1, 1, 2, 3 ] ), \n \ Transformation( [ 4, 2, 1, 2, 3 ] ), \n Transformation( [ 5, 2, 2, 3, 4 ] ), \ \n Transformation( [ 5, 3, 2, 3, 4 ] ) ]\n]]>\n \n\n\n \n \n \n \n \n \n \n \n \n true\ or false.\n \n IsDigraphHomomorphism\ returns true if the permutation\n or transformation x is\ a homomorphism from the digraph\n src to the digraph ran.\n \ \n\n IsDigraphEpimorphism returns true if the permutat\ ion\n or transformation x is a surjective homomorphism from the digr\ aph\n src to the digraph ran.\n \n\n IsDigraphMo\ nomorphism returns true if the permutation\n or transformation <\ A>x is an injective homomorphism from the digraph\n src to the d\ igraph ran.\n \n\n IsDigraphEndomorphism returns tr\ ue if the permutation\n or transformation x is an endomorphism o\ f the digraph\n digraph.\n \n\n A permutation or transform\ ation x is a homomorphism from a\n digraph src to a di\ graph ran if the following hold:\n \n \n \ [u ^ x, v ^ x] is an edge of\n ran whenever [u, v] is an\n edge of src; and \n \n \ x fixes every i which is not a vertex of src.\n \ \n \n See also . \n\n\n If col1 and col2, or col, are given, t\ hen they must\n represent vertex colourings; see \n \n for details\ of the permissible values for these argument. The\n homomorphism must then\ also have the property:\n \n \n \n col[i] = co\ l[i ^ x] for all vertices i of digraph,\n in the case\ of IsDigraphEndomorphism.\n \n col1[i] = c\ ol2[i ^ x] for all vertices i of src,\n in the cases \ of the other operations.\n \n\n See also .\n \n src := Digraph([\ [1], [1, 2], [1, 3]]);\n\ngap> ran\ := Digraph([[1], [1, 2]]);\n\ngap\ > IsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]));\ntrue\ngap> IsDi\ graphHomomorphism(src, ran, Transformation([2, 1, 2]));\nfalse\ngap> IsDigraph\ Homomorphism(src, ran, Transformation([3, 3, 3]));\nfalse\ngap> IsDigraphHomom\ orphism(src, src, Transformation([3, 3, 3]));\ntrue\ngap> IsDigraphHomomorphis\ m(src, ran, Transformation([1, 2, 2]),\n> [1, 2, 2], \ [1, 2]);\ntrue\ngap> IsDigraphHomomorphism(src, ran, Transformation([1, 2, 2])\ ,\n> [2, 1, 1], [1, 2]);\nfalse\ngap> IsDigraphEndomo\ rphism(src, Transformation([3, 3, 3]));\ntrue\ngap> IsDigraphEndomorphism(src,\ Transformation([3, 3, 3]), [1, 1, 1]);\ntrue\ngap> IsDigraphEndomorphism(src,\ Transformation([3, 3, 3]), [1, 1, 2]);\nfalse\ngap> IsDigraphEpimorphism(src,\ ran, Transformation([3, 3, 3]));\nfalse\ngap> IsDigraphMonomorphism(src, ran,\ Transformation([1, 2, 2]));\nfalse\ngap> IsDigraphEpimorphism(src, ran, Trans\ formation([1, 2, 2]));\ntrue\ngap> IsDigraphMonomorphism(ran, src, ());\ntrue]\ ]>\n \n\n\n \n \n \n true or false.\n \n IsDigraphEmbedding returns true if the \ permutation\n or transformation x is a embedding of the digraph\n \ src into the digraph ran, while respecting the colourings\n \ col1 and col2 if given.\n \n\n A permutation or transf\ ormation x is a embedding of a digraph\n src into a di\ graph ran if x is a monomorphism from\n src to ran<\ /A> and the inverse of x is a monomorphism from\n the subdigraph of \ ran induced by the image of x to\n src.\n\n See als\ o . \n\n s\ rc := Digraph([[1], [1, 2]]);\n\ng\ ap> ran := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, ());\ntrue\ngap> IsDigraphEmb\ edding(src, ran, ());\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, 1], [2,\ 1, 1]);\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, 1], [1, 2, 1]);\nfal\ se\ngap> ran := Digraph([[1, 2], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, IdentityTransformatio\ n);\ntrue\ngap> IsDigraphEmbedding(src, ran, IdentityTransformation);\nfalse]]\ >\n \n\n\n \n \n true or false. \n \n The op\ eration DigraphsRespectsColouring verifies whether or not \n the p\ ermutation or transformation x respects the vertex colourings\n col1 and col2 of the digraphs src and range.\n \ That is, DigraphsRespectsColouring returns true if and only if f\ or\n all vertices i of src, col1[i] = col2[i ^ x].\n\ \n\n src := Digraph([[1], [1, 2]]);\n<\ immutable digraph with 2 vertices, 3 edges>\ngap> ran := Digraph([[1], [1, 2],\ [1, 3]]);\n\ngap> DigraphsRespect\ sColouring(src, ran, (1, 2), [2, 1], [1, 2, 1]);\ntrue\ngap> DigraphsRespectsC\ olouring(src, ran, (1, 2), [2, 1], [2, 1, 1]);\nfalse\n]]>\n \n \n\n \n \n \n A list of transf\ ormations.\n \n An endomorphism of digraph i\ s a homomorphism\n from digraph \ back to itself.\n\n GeneratorsOfEndomorphismMonoid, called with a si\ ngle argument,\n returns a generating set for the monoid of all endomorphis\ ms of\n digraph. If digraph belongs to , then the value of\n GeneratorsOfEndomorphismMonoid\ will not be recomputed on future\n calls. \n\n If the colors<\ /A> argument is specified, then\n GeneratorsOfEndomorphismMonoid wil\ l return a generating\n set for the monoid of endomorphisms which respect t\ he given colouring.\n The colouring colors can be in one of two form\ s: \n\n \n \n A list of positive integers of si\ ze the number of vertices of\n digraph, where colors[i\ ] is the colour of vertex\n i.\n \n \ \n A list of lists, such that colors[i] is a list of all\ \n vertices with colour i.\n \n \n\n If\ the limit argument is specified, then it will return only the\n fir\ st limit homomorphisms, where limit must be a positive\n inte\ ger or infinity. \n\n gr := Digraph(Lis\ t([1 .. 3], x -> [1 .. 3]));;\ngap> GeneratorsOfEndomorphismMonoid(gr);\n[ Tra\ nsformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransforma\ tion, Transformation( [ 1, 2, 1 ] ), \n Transformation( [ 1, 2, 2 ] ), Transf\ ormation( [ 1, 1, 2 ] ), \n Transformation( [ 1, 1, 1 ] ) ]\ngap> GeneratorsO\ fEndomorphismMonoid(gr, 3);\n[ Transformation( [ 1, 3, 2 ] ), Transformation( \ [ 2, 1 ] ), \n IdentityTransformation ]\ngap> gr := CompleteDigraph(3);;\ngap\ > GeneratorsOfEndomorphismMonoid(gr);\n[ Transformation( [ 2, 3, 1 ] ), Transf\ ormation( [ 2, 1 ] ), \n IdentityTransformation ]\ngap> GeneratorsOfEndomorph\ ismMonoid(gr, [1, 2, 2]);\n[ Transformation( [ 1, 3, 2 ] ), IdentityTransforma\ tion ]\ngap> GeneratorsOfEndomorphismMonoid(gr, [[1], [2, 3]]);\n[ Transformat\ ion( [ 1, 3, 2 ] ), IdentityTransformation ]\n]]>\n \ \n\n\n \n \n A transformation, or fail.\n \n A prope\ r colouring of a digraph is a labelling of its vertices in\n such a way\ that adjacent vertices have different labels. A proper\n n-col\ ouring is a proper colouring that uses exactly n\n colours. Equ\ ivalently, a proper (n-)colouring of a digraph can be\n defined to b\ e a from a digraph onto the\n complete d\ igraph (with n vertices); see . \ Note that a digraph with loops () does n\ ot have a proper n-colouring for\n any value n.\n \n\n\ If digraph is a digraph and n is a non-negative integer, the\ n\n DigraphColouring(digraph, n) returns an epimorphis\ m\n from digraph onto the complete digraph with n vertices if\ one\n exists, else it returns fail. \n\n See also and \n \n\n Not\ e that a digraph with at least two vertices has a 2-colouring if and only\n \ if it is bipartite, see .\n\n <\ ![CDATA[\ngap> DigraphColouring(CompleteDigraph(5), 4);\nfail\ngap> DigraphCol\ ouring(ChainDigraph(10), 1);\nfail\ngap> D := ChainDigraph(10);;\ngap> t := Di\ graphColouring(D, 2);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\ngap\ > IsDigraphColouring(D, t); \ntrue\ngap> DigraphGreedyColouring(D);\nTransform\ ation( [ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\n]]>\n \n\n\n \n \n \n \n A transformation, or fail.\n \n A proper colouring of a digraph is a labelling of its vertices \ in\n such a way that adjacent vertices have different labels. Note that a d\ igraph \n with loops () does not have any pr\ oper \n colouring.\n \n \n If digraph is a digraph and \ order is a dense list consisting \n of all of the vertices of dig\ raph (in any order), then \n DigraphGreedyColouring\n uses a\ greedy algorithm with the specified order to obtain some proper\n colourin\ g of digraph, which may not use the minimal number of\n colours. \n \n If digraph is a digraph and func is a function whos\ e argument \n is a digraph, and that returns a dense list order, the\ n \n DigraphGreedyColouring(digraph, func) returns\n \ DigraphGreedyColouring(digraph, func(digraph)).\ \n \n\n If the optional second argument (either a list or a function\ ), is not \n specified, then is use\ d by default. \n \n \n See also\n .\n \n\n\ DigraphGreedyColouring(ChainDigraph(10));\nTransf\ ormation( [ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\ngap> DigraphGreedyColouring(Chai\ nDigraph(10), [1 .. 10]);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\ \n]]>\n \n\n\n \n \n A list of the \ vertices.\n \n DigraphWelshPowellOrder retur\ ns a list of all of the vertices of \n the digraph digraph ordered a\ ccording to the sum of the number of \n out- and in-neighbours, from highes\ t to lowest. \n \n\n DigraphWelshPowellOrde\ r(Digraph([[4], [9], [9], [], \n> [4, 6, 9\ ], [1], [], [], \n> [4, 5], [4, 5]]));\n[ \ 5, 9, 4, 1, 6, 10, 2, 3, 7, 8 ]\n]]>\n \n\n\n \n \n <\ Returns> A non-negative integer.\n \n A proper c\ olouring of a digraph is a labelling of its\n vertices in such a way th\ at adjacent vertices have different labels.\n Equivalently, a proper digrap\ h colouring can be defined to be a f\ rom a digraph onto a complete digraph. \n\n If digraph is a digr\ aph without loops (see , then Chromat\ icNumber returns the least\n non-negative integer n such that th\ ere is a proper colouring of\n digraph with n colours. In ot\ her words, for a digraph with at\n least one vertex, ChromaticNumber\ returns the least number n\n such that DigraphColouring(digra\ ph, n) does not return\n fail. See .\n\n ChromaticNumber(NullDigraph(10));\n1\ngap> ChromaticNumber(C\ ompleteDigraph(10));\n10\ngap> ChromaticNumber(CompleteBipartiteDigraph(5, 5))\ ;\n2\ngap> ChromaticNumber(Digraph([[], [3], [5], [2, 3], [4]]));\n3\ngap> Chr\ omaticNumber(NullDigraph(0));\n0\ngap> D := PetersenGraph(IsMutableDigraph);\n\ \ngap> ChromaticNumber(D);\n3\n]]>\ \n \n\n\n \n \n A list of positive integers.\ \n \n If D is a digraph, then Digraph\ Core returns a list of vertices\n corresponding to the core of\ D. In particular, the subdigraph\n of D induced by this li\ st is isomorphic to the core of D. \n\n The core of a digr\ aph D is the minimal subdigraph C of\n D which is a ho\ momorphic image of D. The core of a digraph \n is unique up to isomo\ rphism.\n D := DigraphSymmetricClosure(CycleDigr\ aph(8));\n\ngap> Digrap\ hCore(D);\n[ 1, 2 ]\ngap> D := PetersenGraph();\n\ngap> DigraphCore(D);\n[ 1 .. 10 ]\ngap> D := Digraph(IsMuta\ bleDigraph, [[3], [3], [4], [5], [2]]);\n\ngap> DigraphCore(D);\n[ 2, 3, 4, 5 ]\n]]>\n \ \n \n\n \n\n\n \n \n Finding cli\ ques and independent sets\n\n In &Digraphs;, a clique of a\ digraph is a set of mutually adjacent\n vertices of the digraph, and an independent set is a set of mutually\n non-adjacent vertices of the di\ graph. A maximal clique is a clique\n which is not properly containe\ d in another clique, and a maximal\n independent set is an indepen\ dent set which is not properly contained\n in another independent set. Usi\ ng this definition in &Digraphs;, cliques\n and independent sets are both p\ ermitted, but not required, to contain vertices\n at which there is a loop.\ Another name for a clique is a complete\n subgraph.\n \n\ \n &Digraphs; provides extensive functionality for computing cliques and\n \ independent sets of a digraph, whether maximal or not. The fundamental\n \ algorithm used in most of the methods in &Digraphs; to calculate cliques\n \ and independent sets is a version of the Bron-Kerbosch algorithm.\n Calcu\ lating the cliques and independent sets of a digraph is a well-known\n and \ hard problem, and searching for cliques or independent sets in a\n digraph \ can be very lengthy, even for a digraph with a small number of\n vertices. \ &Digraphs; uses several strategies to increase the performance\n of these \ calculations.\n \n\n From the definition of cliques and independent \ sets, it follows that the\n presence of loops and multiple edges in a digra\ ph is irrelevant to the\n existence of cliques and independent sets in the \ digraph. See and for more\n information about these properties. Therefore given a\ digraph\n digraph, the cliques and independent sets of digraph are\n equal to the cliques and independent sets of the digraph:\n\n <\ List>\n \n DigraphRemoveLoops(DigraphRemoveAllMultipleEdg\ es(digraph)).\n \n \n\n See and for more information about these\n attributes. Furthermore, the cli\ ques of this digraph are equal to the\n cliques of the digraph formed by re\ moving any edge [u,v] for which\n the corresponding reverse edge \ [v,u] is not present. Therefore,\n overall, the cliques of digraph<\ /A> are equal to the cliques of the\n symmetric digraph:\n\n \n \ \n MaximalSymmetricSubdigraphWithoutLoops(digraph).\n \n \n\n See for more\n information about this. The of this symmetric dig\ raph contains the\n automorphism group of digraph as a subgroup. By\ performing the\n search for maximal cliques with the help of this larger a\ utomorphism group\n to reduce the search space, the computation time may be\ reduced. The\n functions and attributes which return representatives of c\ liques of\n digraph will return orbit representatives of cliques und\ er the\n action of the automorphism group of the maximal symmetric subdi\ graph\n without loops on sets of vertices. \n\n The independent\ sets of a digraph are equal to the independent sets of the\n . Therefore, overall, the independent\n sets of \ digraph are equal to the independent sets of the symmetric\n digraph\ :\n\n \n \n DigraphSymmetricClosure(DigraphRemov\ eLoops(DigraphRemoveAllMultipleEdges(\n digraph))).\n\ \n \n\n Again, the automorphism group of this symmet\ ric digraph contains the\n automorphism group of digraph. Since a se\ arch for independent sets\n is equal to a search for cliques in the , the\n methods used in &Digraphs; usually transform a \ search for independent sets\n into a search for cliques, as described above\ . The functions and\n attributes which return representatives of independe\ nt sets of\n digraph will return orbit representatives of independen\ t sets under\n the action of the automorphism group of the symmetric clo\ sure of the\n digraph formed by removing loops and multiple edges. \ \n\n Please note that in &Digraphs;, cliques and independent sets are not\n\ required to be maximal. Some authors use the word clique to mean\n m\ aximal clique, and some authors use the phrase independent set to\n mea\ n maximal independent set, but please note that &Digraphs; does\n no\ t use this definition.\n \n\n Finding cliques\n \n\n\ntrue or false.\n\n If digraph is a digraph and l is\ a duplicate-free list of\n vertices of digraph, then\n IsClique(<\ /C>digraph,l) returns true\n if l\ is a clique of digraph and false if it is\n not. Sim\ ilarly,\n IsMaximalClique(digraph,l) ret\ urns\n true if l is a maximal clique of digraph\ and\n false if it is not. \n\n A clique of a digraph\ is a set of mutually adjacent vertices of the\n digraph. A maximal cliq\ ue is a clique that is not properly\n contained in another clique. A cl\ ique is permitted, but not required, to\n contain vertices at which there i\ s a loop.\n D := CompleteDigraph(4);;\ngap> IsCliq\ ue(D, [1, 3, 2]);\ntrue\ngap> IsMaximalClique(D, [1, 3, 2]);\nfalse\ngap> IsMa\ ximalClique(D, DigraphVertices(D));\ntrue\ngap> D := Digraph([[1, 2, 4, 4], [1\ , 3, 4], [2, 1], [1, 2]]);\n\ \ngap> IsClique(D, [2, 3, 4]);\nfalse\ngap> IsMaximalClique(D, [1, 2, 4]);\ntr\ ue\ngap> D := CompleteDigraph(IsMutableDigraph, 4);;\ngap> IsClique(D, [1, 3, \ 2]);\ntrue]]>\n \n\n\n \n\ \n The argument user_param\ .\n \n This function finds cliques of the digraph <\ A>digraph subject to the\n conditions imposed by the other arguments as\ described below. Note\n that a clique is represented by the immutable list\ of the vertices that\n it contains.\n \n\n Let G denote t\ he automorphism group of the maximal symmetric\n subdigraph of digraph without loops (see and ).\n\ \n \n hook\n \n This argum\ ent should be a function or fail. \n\n If hook is a fu\ nction, then it should have two arguments\n user_param (see belo\ w) and a clique c. The function\n hook(user_param, c) is called every time a new\n clique c is found by \ CliquesFinder. \n\n If hook is fail, then a defaul\ t function is used that\n simply adds every new clique found by Cliq\ uesFinder to\n user_param, which must be a list in this case\ .\n \n\n user_param\n \n \ If hook is a function, then user_param can be any &GAP;\n \ object. The object user_param is used as the first argument for\n \ the function hook. For example, user_param might be a\n \ list, and hook(user_param, c) might add the size\n \ of the clique c to the list user_param. \n\n If \ the value of hook is fail, then the value of\n user_pa\ ram must be a list.\n \n\n limit\n \ \n This argument should be a positive integer or infinity.\n CliquesFinder will return after it has found\n l\ imit cliques or the search is complete.\n \n\n i\ nclude and exclude\n \n These arguments sh\ ould each be a (possibly empty) duplicate-free list\n of vertices of digraph (i.e. positive integers less than the\n number of vertices\ of digraph). \n\n CliquesFinder will only look for c\ liques containing all of the\n vertices in include and containin\ g none of the vertices in\n exclude. \n\n Note that t\ he search may be much more efficient if each of these lists\n is invari\ ant under the action of G on sets of vertices.\n \n\n \ max\n \n This argument should be true\ or false. If max is\n true then CliquesFinder will only search for maximal\n cliques. If max is fa\ lse then non-maximal cliques may be\n found.\n \n\n \ size\n \n This argument should be f\ ail or a positive integer.\n If size is a positive integer t\ hen CliquesFinder will\n only search for cliques that contain pr\ ecisely size vertices.\n If size is fail then cliq\ ues of any size may be found.\n \n\n reps\ \n \n This argument should be true or false.\n\n If reps is true then the arguments include \ and\n exclude are each required to be invariant under the action\ of\n G on sets of vertices. In this case, CliquesFinder\ will\n find representatives of the orbits of the desired cliques under\ the\n action of G, although representatives may be returned \ that\n are in the same orbit.\n If reps is false th\ en CliquesFinder will not take this into\n consideration. \n\ \n For a digraph such that G is non-trivial, the search for\n \ clique representatives can be much more efficient than the search for\n \ all cliques.\n \n \n\n This function uses a ver\ sion of the Bron-Kerbosch algorithm.\n D := Comple\ teDigraph(5);\n\ngap> user_param :\ = [];;\ngap> f := function(a, b) # Calculate size of clique\n> AddSet(user_\ param, Size(b));\n> end;;\ngap> CliquesFinder(D, f, user_param, infinity, [], \ [], false, fail,\n> true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFi\ nder(D, fail, [], 5, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 \ ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [], 2, [2, 4], [\ 3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> CliquesFinder(D, fai\ l, [], infinity, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5 ] ]\ngap> Cliques\ Finder(D, fail, [], infinity, [1, 3], [], false, 3, false);\n[ [ 1, 2, 3 ], [ \ 1, 3, 4 ], [ 1, 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], []\ , true, 3, false);\n[ ]\ngap> D := CompleteDigraph(IsMutableDigraph, 5);\n\ngap> user_param := [];;\ngap> f := f\ unction(a, b) # Calculate size of clique\n> AddSet(user_param, Size(b));\n>\ end;;\ngap> CliquesFinder(D, f, user_param, infinity, [], [], false, fail,\n>\ true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFinder(D, fail, [], 5\ , [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ], [ 2, 4, 5 ], [ 1\ , 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [], 2, [2, 4], [3], false, fail, fa\ lse);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> CliquesFinder(D, fail, [], infinity, []\ , [], true, 5, false);\n[ [ 1, 2, 3, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [],\ infinity, [1, 3], [], false, 3, false);\n[ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1, 3, \ 5 ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], [], true, 3, false);\ \n[ ]]]>\n \n\n\n \n \n \ \n An immutable list of positive integers, or fail.\ \n \n If digraph is a digraph, then these fu\ nctions returns a clique\n of digraph if one exists that satisfies t\ he arguments, else it\n returns fail. A clique is defined by the se\ t of vertices that it\n contains; see and . \n\n The optional arguments include and \ exclude must each be a\n (possibly empty) duplicate-free list of ver\ tices of digraph, and\n the optional argument size must be a \ positive integer. By default,\n include and exclude are empt\ y. These functions will search for\n a clique of digraph that includ\ es the vertices of include\n but does not include any vertices in exclude; if the argument\n size is supplied, then additionally \ the clique will be required to\n contain precisely size vertices. \n\n If include is not a clique, then these functions return fai\ l.\n Otherwise, the functions behave in the following way, depending on\ the\n number of arguments:\n\n \n One or two arguments\ \n \n If one or two arguments are supplied, then D\ igraphClique and\n DigraphMaximalClique greedily enlarge the\ clique include\n until it can not continue. The result is guar\ anteed to be a maximal\n clique. This may or may not return an answer m\ ore quickly than using\n . with a\ limit of 1.\n \n Three arguments\n \ \n If three arguments are supplied, then DigraphClique greedily\ \n enlarges the clique include until it can not continue, althou\ gh\n this clique may not be maximal. \n\n Given three argumen\ ts, DigraphMaximalClique returns the maximal\n clique returned b\ y DigraphMaximalCliques(digraph,\n include<\ C>, exclude, 1) if one exists,\n else fail.\n \ \n Four arguments\n \n If four\ arguments are supplied, then DigraphClique returns the\n clique\ returned by DigraphCliques(digraph,\n includ\ e, exclude, 1,\n size) if one ex\ ists, else fail. This clique may\n not be maximal. Given fou\ r arguments, DigraphMaximalClique\n returns the maximal clique r\ eturned by\n DigraphMaximalCliques(digraph, inc\ lude,\n exclude, 1, size) if one\ exists, else\n fail.\n \n \n\n D := Digraph([[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\n\ gap> DigraphClique(D);\n[ 5 ]\ngap> DigraphMaximalClique(D);\n[ 5 ]\ngap> Digr\ aphClique(D, [1, 2]);\n[ 1, 2, 3 ]\ngap> DigraphMaximalClique(D, [1, 3]);\n[ 1\ , 3, 2 ]\ngap> DigraphClique(D, [1], [2]);\n[ 1, 4 ]\ngap> DigraphMaximalCliqu\ e(D, [1], [3, 4]);\nfail\ngap> DigraphClique(D, [1, 5]);\nfail\ngap> DigraphCl\ ique(D, [], [], 2);\n[ 1, 2 ]\ngap> D := Digraph(IsMutableDigraph,\n> \ [[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> DigraphClique(D);\n[\ 5 ]]]>\n \n\n\n \n \n \n \n \ \n \n \n An immutable l\ ist of lists of positive integers.\n \n If digra\ ph is digraph, then these functions and attributes use to return cliques of digraph. A\n clique is define\ d by the set of vertices that it contains; see \ and . \n\n The optional arguments inc\ lude and exclude must each be a\n (possibly empty) list of verti\ ces of digraph, the optional argument\n limit must be either \ a positive integer or infinity, and the\n optional argument size<\ /A> must be a positive integer. If not\n specified, then include an\ d exclude are chosen to be empty\n lists, and limit is set to\ infinity. \n\n The functions will return as many suitable cliqu\ es as possible, up to the\n number limit. These functions will find\ cliques that contain all\n of the vertices of include but do not co\ ntain any of the\n vertices of exclude. The argument size re\ stricts the search\n to those cliques that contain precisely size ve\ rtices.\n If the function or attribute has Maximal in its name, then\ only\n maximal cliques will be returned; otherwise non-maximal cliques may\ be\n returned. \n\n Let G denote the automorphism group of m\ aximal symmetric subdigraph\n of digraph without loops (see and ).\n\n \n Dist\ inct cliques\n \n DigraphMaximalCliques and DigraphCliques each return a\n duplicate-free list of at most l\ imit cliques of digraph\n that satisfy the arguments. \n\ \n The computation may be significantly faster if include and\n \ exclude are invariant under the action of G\n on s\ ets of vertices.\n \n\n Orbit representatives of clique\ s\n \n To use DigraphMaximalCliquesReps or \ DigraphCliquesReps,\n the arguments include and exclude must each be invariant\n under the action of G on sets of ver\ tices. \n\n If this is the case, then DigraphMaximalCliquesReps and\n DigraphCliquesReps each return a duplicate-free list of\ at most\n limit orbits representatives (under the action of \ G on\n sets vertices) of cliques of digraph that satisfy the\ \n arguments. \n\n The representatives are not guaranteed to \ be in distinct orbits.\n However, if fewer than lim results are \ returned, then there\n will be at least one representative from each or\ bit of maximal cliques.\n \n \n\n D := Digraph([\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]]\ );\n\ngap> IsSymmetricDigraph(D);\ \ntrue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4)\ ])\ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]\ng\ ap> DigraphMaximalCliquesReps(D);\n[ [ 1, 2, 3 ], [ 3, 4 ] ]\ngap> Orbit(G, [1\ , 2, 3], OnSets);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]\ngap> Orbit(G, [3, 4], OnSets)\ ;\n[ [ 3, 4 ] ]\ngap> DigraphMaximalCliquesReps(D, [3, 4], [], 1);\n[ [ 3, 4 ]\ ]\ngap> DigraphMaximalCliques(D, [1, 2], [5, 6], 1, 2);\n[ ]\ngap> DigraphCl\ iques(D, [1], [5, 6], infinity, 2);\n[ [ 1, 2 ], [ 1, 3 ] ]\ngap> D := Digraph\ (IsMutableDigraph, [\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]])\ ;\n\ngap> IsSymmetricDigraph(D);\nt\ rue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ \ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]]]>\n \n\n\n \n \n A non-negative integer.\ \n \n If digraph is a digraph, then CliqueNumber(digraph)\n returns the largest integer n such that digra\ ph contains a\n clique with n vertices as an induced subdigraph.\ \n \n\n A clique of a digraph is a set of mutually adjacent v\ ertices of the\n digraph. Loops and multiple edges are ignored for the purp\ ose of\n determining the clique number of a digraph. \n D := CompleteDigraph(4);;\ngap> CliqueNumber(D);\n4\ngap> D := Digrap\ h([[1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);\n\ngap> CliqueNumber(D);\n3\ngap> D := CompleteDigraph(IsMut\ ableDigraph, 4);;\ngap> CliqueNumber(D);\n4]]>\n\n\n\n \n\n Finding independent sets\n \n \n \n <\ K>true or false.\n \n If digraph \ is a digraph and l is a duplicate-free list of\n vertices of digr\ aph, then\n IsIndependentSet(digraph,l)<\ /C> returns\n true if l is an independent set of di\ graph and\n false if it is not. Similarly,\n IsMaximalInd\ ependentSet(digraph,l)\n returns true if l is a maximal independent set of\n digraph and <\ K>false if it is not. \n\n An independent set of a digraph \ is a set of mutually non-adjacent\n vertices of the digraph. A maximal i\ ndependent set is an independent\n set that is not properly contained i\ n another independent set. An\n independent set is permitted, but not requi\ red, to contain vertices at\n which there is a loop.\n D := CycleDigraph(4);;\ngap> IsIndependentSet(D, [1]);\ntrue\ngap> IsM\ aximalIndependentSet(D, [1]);\nfalse\ngap> IsIndependentSet(D, [1, 4, 3]);\nfa\ lse\ngap> IsIndependentSet(D, [2, 4]);\ntrue\ngap> IsMaximalIndependentSet(D, \ [2, 4]);\ntrue\ngap> D := CycleDigraph(IsMutableDigraph, 4);;\ngap> IsIndepend\ entSet(D, [1]);\ntrue]]>\n \n\n\n \n \n \n An immutable l\ ist of positive integers, or fail.\n \n If <\ A>digraph is a digraph, then these functions returns an independent\n s\ et of digraph if one exists that satisfies the arguments, else it\n \ returns fail. An independent set is defined by the set of vertices\n \ that it contains; see and . \n\n The optional arguments include and exclude must each be a\n (possibly empty) duplicate-free list \ of vertices of digraph, and the\n optional argument size must\ be a positive integer. By default,\n include and exclude ar\ e empty. These functions will search\n for an independent set of digraph\ that includes the vertices of\n include but does not include an\ y vertices in exclude;\n if the argument size is supplied, th\ en additionally the independent\n set will be required to contain precisely\ size vertices. \n\n If include is not an independent set,\ then these functions return\n fail. Otherwise, the functions behav\ e in the following way,\n depending on the number of arguments:\n\n \n One or two arguments\n \n If one or t\ wo arguments are supplied, then DigraphIndependentSet\n and D\ igraphMaximalIndependentSet greedily enlarge the\n independent set \ include until it can not continue. The result\n is guaranteed t\ o be a maximal independent set. This may or may not\n return an answer \ more quickly than using . with a limit of 1.\n \n Three arguments\n \ \n If three arguments are supplied, then DigraphIndependen\ tSet\n greedily enlarges the independent set include until i\ t can not\n continue, although this independent set may not be maximal.\ \n\n Given three arguments, DigraphMaximalIndependentSet ret\ urns the\n maximal independent set returned by\n DigraphMaxim\ alIndependentSets(digraph,\n include, <\ A>exclude, 1) if one exists,\n else fail.\n \n Four arguments\n \n If four argument\ s are supplied, then DigraphIndependentSet\n returns the indepen\ dent set returned by\n DigraphIndependentSets(digraph,\ include,\n exclude, 1, size)\ if one exists, else\n fail. This independent set may not be\ maximal. Given four\n arguments, DigraphMaximalIndependentSet returns the maximal\n independent set returned by\n Digrap\ hMaximalIndependentSets(digraph,\n include, exclude, 1,\n size) if one exists, e\ lse fail.\n \n \n D :\ = ChainDigraph(6);\n\ngap> DigraphInd\ ependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphMaximalIndependentSet(D);\n[ 6, 4, 2\ ]\ngap> DigraphIndependentSet(D, [2, 4]);\n[ 2, 4, 6 ]\ngap> DigraphMaximalIn\ dependentSet(D, [1, 3]);\n[ 1, 3, 6 ]\ngap> DigraphIndependentSet(D, [2, 4], [\ 6]);\n[ 2, 4 ]\ngap> DigraphMaximalIndependentSet(D, [2, 4], [6]);\nfail\ngap>\ DigraphIndependentSet(D, [1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximalIndepende\ ntSet(D, [1], [], 2);\nfail\ngap> DigraphMaximalIndependentSet(D, [1], [], 3);\ \n[ 1, 3, 5 ]\ngap> D := ChainDigraph(IsMutableDigraph, 6);\n\ngap> DigraphIndependentSet(D);\n[ 6, 4, 2 ]\ngap> D\ igraphMaximalIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphIndependentSet(D, [2\ , 4]);\n[ 2, 4, 6 ]\ngap> DigraphMaximalIndependentSet(D, [1, 3]);\n[ 1, 3, 6 \ ]\ngap> DigraphIndependentSet(D, [2, 4], [6]);\n[ 2, 4 ]\ngap> DigraphMaximalI\ ndependentSet(D, [2, 4], [6]);\nfail\ngap> DigraphIndependentSet(D, [1], [], 2\ );\n[ 1, 3 ]\ngap> DigraphMaximalIndependentSet(D, [1], [], 2);\nfail\ngap> Di\ graphMaximalIndependentSet(D, [1], [], 3);\n[ 1, 3, 5 ]]]>\n \n\n\n \n \n \ \n \n \n \n \n \ An immutable list of lists of positive integers.\n \n If digraph is digraph, then these functions and attributes\ use to return independent sets of dig\ raph. An\n independent set is defined by the set of vertices that it co\ ntains; see\n and . \n\n The optional arguments include and <\ A>exclude must each be a\n (possibly empty) list of vertices of digr\ aph, the optional argument\n limit must be either a positive int\ eger or infinity, and the\n optional argument size must be a \ positive integer. If not\n specified, then include and exclude are chosen to be empty\n lists, and limit is set to infinity. \n\n The functions will return as many suitable independent sets as\ possible, up\n to the number limit. These functions will find inde\ pendent sets\n that contain all of the vertices of include but do no\ t\n contain any of the vertices of exclude The argument size\ \n restricts the search to those cliques that contain precisely size\ \n vertices. If the function or attribute has Maximal in its name,\ \n then only maximal independent sets will be returned; otherwise non-maxim\ al\n independent sets may be returned.\n \n\n Let G denote\ the \n of the of the digraph formed from\n digrap\ h by removing loops and ignoring the multiplicity of edges.\n\n \ \n Distinct independent sets\n \n Digra\ phMaximalIndependentSets and DigraphIndependentSets\n each r\ eturn a duplicate-free list of at most limit independent\n sets \ of digraph that satisfy the arguments. \n\n The computation m\ ay be significantly faster if include and\n exclude are i\ nvariant under the action of G on sets of\n vertices.\n \n\n Representatives of distinct orbits of independent sets\n \n To use DigraphMaximalIndependentSetsReps or\ \n DigraphIndependentSetsReps, the arguments include and\ \n exclude must each be invariant under the action of G o\ n\n sets of vertices. \n\n If this is the case, then Digra\ phMaximalIndependentSetsReps and\n DigraphIndependentSetsReps each return a list of\n at most limit orbits representatives (\ under the action of\n G on sets of vertices) of independent sets\ of digraph\n that satisfy the arguments. \n\n The re\ presentatives are not guaranteed to be in distinct orbits.\n However, i\ f lim is not specified, or fewer than lim\n results are r\ eturned, then there will be at least one representative\n from each orb\ it of maximal independent sets.\n \n \n\n D := CycleDigraph(5);\n\ \ngap> DigraphMaximalIndependentSetsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndepe\ ndentSetsReps(D);\n[ [ 1 ], [ 1, 3 ] ]\ngap> Set(DigraphMaximalIndependentSets\ (D));\n[ [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaxim\ alIndependentSets(D, [1]);\n[ [ 1, 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSet\ s(D, [], [4, 5]);\n[ [ 1 ], [ 2 ], [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentS\ ets(D, [], [4, 5], 1, 2);\n[ [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigra\ ph, 5);\n\ngap> DigraphMaximalIndepe\ ndentSetsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSetsReps(D);\n[ [ 1 ],\ [ 1, 3 ] ]\ngap> Set(DigraphMaximalIndependentSets(D));\n[ [ 1, 3 ], [ 1, 4 ]\ , [ 2, 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndependentSets(D, [1]);\ \n[ [ 1, 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5]);\n[ [ 1 ]\ , [ 2 ], [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5], 1, 2);\ \n[ [ 1, 3 ] ]]]>\n \n\n\n \n\ \n \n Visualising and IO\n\n Visualising a digraph\n \n \n\ Nothing.\n \n This function attemp\ ts to convert the string str into a pdf\n document and open this d\ ocument, i.e. to splash it all over your monitor.\n\n The string s\ tr must correspond to a valid dot or\n LaTeX text file \ and you must have have GraphViz and\n pdflatex installed on\ your computer. For details about these file\n formats, see http://\ www.latex-project.org and\n http://www.graphviz.org. \ \n\n This function is provided to allow convenient, immediate viewing of \ the\n pictures produced by the function . \n\ \n The optional second argument opts should be a record with\n \ components corresponding to various options, given below.\n\n \n \ path\n \n this should be a string re\ presenting the path to the directory where\n you want Splash t\ o do its work. The default value of this\n option is \"~/\".\n\ \n\n directory\n \n t\ his should be a string representing the name of the directory in\n path where you want Splash to do its work. This function\n \ will create this directory if does not already exist. \n\n The \ default value of this option is \"tmp.viz\" if the option\n path is present, and the result of\n is used otherwise.\n \n\n \ filename\n \n this should be a string representin\ g the name of the file where\n str will be written. The defau\ lt value of this option is\n \"vizpicture\".\n \ \n\n viewer\n \n this should be a st\ ring representing the name of the program which\n should open the fil\ es produced by GraphViz or pdflatex.\n \n\n \ type\n \n this option can be used to speci\ fy that the string str contains\n a &LaTeX; or dot docu\ ment. You can specify this option in\n str directly by making \ the first line \"%latex\" or\n \"//dot\". There is no \ default value for this option, this\n option must be specified in \ str or in opt.type.\n \n\n engine\ \n \n this option can be used to specify the GraphViz en\ gine to use\n to render a dot document. The valid choices are \ \"dot\",\n \"neato\", \"circo\", \"twopi\", \"fdp\",\n \"sfdp\", and \"patchwork\". Pleas\ e refer to the\n GraphViz documentation for details on these engines.\ \n The default value for this option is \"dot\", and it\n \ must be specified in opt.engine.\n \n\n filetype\n \n this should be a string representi\ ng the type of file which\n Splash should produce. For &LaTeX;\ files, this option is\n ignored and the default value \"pdf\"\ is used.\n \n \n\n This function was written b\ y Attila Egri-Nagy and Manuel Delgado with some\n minor changes by J. D. \ Mitchell. \n\n Splash(DotDigraph(RandomDigraph(4)\ ));\n]]>\n \n \n\n \n \ \n \n A string.\n \n DotDigraph produces a graphical representation of the digra\ ph\n digraph. Vertices are displayed as circles, numbered consiste\ ntly\n with digraph. Edges are displayed as arrowed lines between\ \n vertices, with the arrowhead of each line pointing towards the range\n\ of the edge. \n\n DotVertexLabelledDigraph differs from <\ C>DotDigraph only in\n that the values in are used to label\n the vertices in the produced picture rather \ than the numbers 1 to\n the number of vertices of the digraph. \n\n The output is in dot format (also known as GraphViz)\ \n format. For details about this file format, and information about how \ to\n display or edit this format see http://www.graphviz.org. \ \n\n The string returned by DotDigraph or\n DotVertexL\ abelledDigraph can be written to a file using\n the command . \n\n adj :\ = List([1 .. 4], x -> [1, 1, 1, 1]);\n[ [ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ], [ 1, 1\ , 1, 1 ], [ 1, 1, 1, 1 ] ]\ngap> adj[1][3] := 0;\n0\ngap> gr := DigraphByAdjac\ encyMatrix(adj);\n\ngap> FileStri\ ng(\"dot/k4.dot\", DotDigraph(gr));\n154]]>\n \n \n\n \n \n A string.\n \n This fu\ nction produces a graphical representation of the symmetric\n digraph \ digraph. DotSymmetricDigraph will return an\n error if digr\ aph is not a symmetric digraph. See\n . \n\n Vertices are displayed as circles, numbered consistently wi\ th\n digraph. Since digraph is symmetric, for every non-loo\ p\n edge there is a complementary edge with opposite source and range.\n \ DotSymmetricDigraph displays each pair of complementary edges\n \ as a single line between the relevant vertices, with no arrowhead. \n\n \ The output is in dot format (also known as GraphViz)\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n http://www.graphviz.org. \ \n\n The string returned by DotSymmetricDigraph can be written\ to a\n file using the command . \n\n star := Digraph([[2, 2, 3, 4], [1, 1],\ [1], [1, 4]]);\n\ngap> IsSym\ metricDigraph(star);\ntrue\ngap> FileString(\"dot/star.dot\", DotSymmetricDigr\ aph(gr));\n83]]>\n \n \n\n \n \n A string.\n \n This function produces a graphi\ cal representation of a partial order\n digraph digraph. DotPar\ tialOrderDigraph will return an error\n if digraph is not a pa\ rtial order digraph. See . \n\ \n Since digraph is a partial order, it is both reflexive and\n \ transitive. The output of DotPartialOrderDigraph is the\n of the\n of digraph. \n\n The output is in dot format (\ also known as GraphViz)\n format. For details about this file form\ at, and information about how\n to display or edit this format see\n \ http://www.graphviz.org. \n\n The string returned by D\ otPartialOrderDigraph can be written to a\n file using the command . \n\n poset := Digraph([[1, 4], [2], [2, 3, 4], [4]);\ngap> IsPartialOrderDigraph(\ gr);\ntrue\ngap> FileString(\"dot/poset.dot\", DotPartialOrderDigraph(gr));\n8\ 3]]>\n \n \n\n \n \n \n A string.\n \n This function produces a graphical representation of a preorder\n \ digraph digraph. DotPreorderDigraph will return an error\n \ if digraph is not a preorder digraph. See . \n\n A preorder digraph is reflexive and transitiv\ e but in general it is\n not anti-symmetric and may have strongly connect\ ed components\n containing more than one vertex. The \n Q obtained by forming the quotient of digraph\ by the\n partition of its vertices into the strongly connected component\ s\n satisfies . Thus every vertex of\ \n Q corresponds to a strongly connected component of digraph.\n The output of DotPreorderDigraph displays the \n of Q with\n vertices\ displayed as rounded rectangles labelled by all of the vertices\n of \ digraph in the corresponding strongly connected component. \n\n T\ he output is in dot format (also known as GraphViz)\n forma\ t. For details about this file format, and information about how\n to dis\ play or edit this format see\n http://www.graphviz.org. \n\ \n The string returned by DotPreorderDigraph can be written to a\n\ file using the command .\n\n preset := Digraph([[1, 2, 4, 5], [1, 2, 4, 5]\ , [3, 4], [4], [1, 2, 4, 5]);\ngap> IsPreorderDigraph(gr);\ntrue\ngap> FileStr\ ing(\"dot/preset.dot\", DotProrderDigraph(gr));\n83]]>\n \n \n\n \n \n A string.\n \n DotHighlightedDigraph produces a grap\ hical represenation of the\n digraph digraph, where the vertices i\ n the list verts, and\n edges between them, are drawn with colour \ colour1 and all other\n vertices and edges in digraph are d\ rawn with colour colour2.\n If colour1 and colour2 a\ re not given then\n DotHighlightedDigraph uses black and grey resp\ ectively. \n\n Note that DotHighlightedDigraph does not valida\ te the colours\n colour1 and colour2 - consult the GraphViz\ documentation to\n see what is available.\n\n See for more details on the output. \n\n di\ graph := Digraph([[2, 3], [2], [1, 3]]);\n\n\ gap> FileString(\"dot/my_digraph.dot\",\n> DotHighlightedDigraph(digraph, [1, \ 2], \"red\", \"black\"));\n264]]>\n \n \n\ \n \n\n Reading and writing digraphs to a file\n This section describes different ways to store and read graphs\n\ from a file in the &Digraphs; package.\n \n Graph6 \n \n\t Graph6 is a graph data format for storing undi\ rected graphs\n\twith no multiple edges nor loops of size up to 2^{36} - 1\ in\n\tprintable chracters. The format consists of two parts. The first p\ art\n\tuses one to eight bytes to store the number of vertices. And the second\ \n\tpart is the upper half of the adjacency matrix converted into ASCII\n\tcha\ racters. For a more detail description see \n\thttp://cs.an\ u.edu.au/~bdm/data/formats.txt.\n \n\n Sparse6 <\ /Mark>\n \n\t Sparse6 is a graph data format for storing un\ directed graphs\n\twith possibly multiple edges or loops. The maximal number o\ f vertices\n\tallowed is 2^{36} - 1 . The format consists of two parts\ . The\n\tfirst part uses one to eight bytes to store the number of vertices.\n\ \tAnd the second part only stores information about the edges.\n\tTherefore, t\ he Sparse6 format return a more compact encoding\n\tthen Graph6 <\ /E> for sparse graph, i.e. graphs where the number\n\tof edges is much less t\ han the number of vertices squared. For a more\n\tdetail description see \n http://cs.anu.edu.au/~bdm/data/formats.txt.\n \ \n\n Digraph6 \n \n\tDigraph6\ is a new format based on Graph6 , but designed\n\tfor digraphs. The e\ ntire adjacency matrix is stored, and\n\ttherefore there is support for direct\ ed edges and single-vertex loops.\n\tHowever, multiple edges are not supported\ .\n \n\n DiSparse6 \n \n\tDiSpars\ e6 is a new format based on Sparse6 , but\n\tdesigned for digraphs\ . In this format the list of edges is\n\tpartitioned into inceasing and decrea\ sing edges, depending whether the\n\tedge has its source bigger than the range\ . Then both sets of edges are\n\twritten separetly in Sparse6 format \ with a separation symbol\n\tin between.\n \n \n \n \n \n A string.\n \n Ret\ urns a string string such that EvalString(string)\n is equal \ to digraph, and has the same mutability.\n See .\n \n\n The methods installed for String<\ /C> make some attempts to\n ensure that string has as short a length\ as possible, but\n there may exist shorter strings that also evaluate to <\ A>digraph.\n \n\n It is possible that string may contain \ escaped special\n characters. To obtain a representation of digraph \ that\n can be entered as GAP input, please use\n .\n Note that Print for a digraph delegates to\n \ PrintString, which delegates to String.\n D := CycleDigraph(3);\n\ngap> Prin\ t(D);\nCycleDigraph(3);\ngap> G := PetersenGraph(IsMutableDigraph);\n\ngap> String(G);\n\"DigraphFromGraph6Strin\ g(IsMutableDigraph, \\\"IheA@GUAo\\\");\"\ngap> Print(last);\nDigraphFromGraph\ 6String(IsMutableDigraph, \"IheA@GUAo\");\ngap> DigraphFromGraph6String(IsMuta\ bleDigraph, \"IheA@GUAo\");\n\n]]>\ \n \n\n\n \n \n \n \n \n A digraph.\n \n If s\ tr is a string encoding a graph in Graph6, Digraph6, Sparse6 or\n DiSpa\ rse6 format, then the corresponding function returns a digraph. In the\n ca\ se of either Graph6 or Sparse6, formats which do not support directed\n edg\ es, this will be a digraph such that for every edge, the edge going in\n th\ e opposite direction is also present. \n\n Each of these functions takes\ an optional first argument filt,\n which should be either \n or ,\n and w\ hich specifies whether the output digraph shall\n be mutable or immutable.\ \n If no first argument is provided, then an immutable\n digraph is retu\ rned by default.\n DigraphFromGraph6String(\"?\");\ \n\ngap> DigraphFromGraph6String(\"C]\ \");\n\ngap> DigraphFromGraph6Stri\ ng(\"H?AAEM{\");\n\ngap> DigraphF\ romDigraph6String(\"&?\");\n\ngap> Di\ graphFromDigraph6String(IsMutableDigraph, \"&DOOOW?\");\n\ngap> DigraphFromDigraph6String(\"&CQFG\");\n\ngap> DigraphFromDigraph6String(\"&IM[SrKLc~\ lhesbU[F_\");\n\ngap> DigraphFro\ mDiSparse6String(\".CaWBGA?b\");\n\n]]>\n \n\n\n \n \ \n \n \n \n A string.\n\ \n These four functions return a highly compressed string fu\ lly describing the\n digraph digraph. \n\n Graph6 and Digraph\ 6 are formats best used on small, dense graphs, if\n applicable. For larger\ , sparse graphs use Sparse6 and\n Disparse6 (this latter also\ preserves multiple edges). \n\n See .\n \ gr := Digraph([[2, 3], [1], [1]]);\n\ngap> Sparse6String(gr);\n\":Bc\"\ngap> DiSparse\ 6String(gr);\n\".Bc{f\"\n]]>\n \n\n\n \ \n \n An IO file object.\n \n If filena\ me is a string representing the name of a file, then\n DigraphFile returns an &IO; package file object for that file.\n \n\n If the \ optional argument coder is specified\n and is a function which eithe\ r encodes a digraph as a string, or decodes a\n string into a digraph, then\ this function will be used when reading or\n writing to the returned file \ object. If the optional argument coder\n is not specified, then the \ encoding of the digraphs in the returned file\n object must be specified in\ the the file extension. The file extension must\n be one of: .g6, <\ C>.s6, .d6, .ds6, .txt,\n .p, or .pickle\ ; more details of these file formats is given\n below. \n\n If t\ he optional argument mode is specified, then it must be one of:\n \"w\" (for write), \"a\" (for append), or \"r\" (for read).\ \n If mode is not specified, then \"r\" is used by default. <\ P/>\n\n If filename ends in one of: .gz, .bz2, or\n \ .xz, then the digraphs which are read from, or written to, the\n re\ turned file object are decompressed, or compressed, appropriately.\n \n\ \n The file object returned by DigraphFile can be given as the first\ \n argument for either of the functions or . The purpose of this is to reduce the overhe\ ad of\n recreating the file object inside the functions \n or when, for example, reading or \ writing many\n digraphs in a loop.\n \n\n The currently supported\ file formats, and associated filename extensions,\n are:\n \n \ graph6 (.g6)\n \n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n \n sparse6 (.s6)\n \n\tUnlike graph6, sparse6 has support for \ loops and multiple edges.\n\tHowever, its use is still limited to symmetric gr\ aphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those with few \ edges per vertex.\n \n digraph6 (.d6)\n \n\tThis format is based on graph6, but stores direction information -\n\tt\ herefore is not limited to symmetric graphs. Loops are allowed, but\n\tmultip\ le edges are not. Best compression with \"dense\" graphs.\n \n \ disparse6 (.ds6)\n \n\tAny type of digraph can be e\ ncoded in disparse6: directions, loops, and\n\tmultiple edges are all allowed.\ Similar to sparse6, this has the best\n\tcompression rate with \"sparse\" gr\ aphs.\n \n plain text (.txt)\n \n\tTh\ is is a human-readable format which stores graphs in the form\n 0 7 \ 0 8 1 7 2 8 3 8 4 8 5 8 6 8 i.e. pairs of vertices\n describ\ ing edges in a graph. More specifically, the vertices making up\n one e\ dge must be separated by a single space, and pairs of vertices must\n b\ e separated by two spaces. \n\n See for a more flexible way to store\n digraphs in a plain text file.\ \n \n\n pickled (.p or .pickle)\n \n Digraphs are pickled using the &IO; package. This is\ particularly good\n when the is non-trivi\ al.\n \n \n\n filename := Co\ ncatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap> file := DigraphFil\ e(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> WriteDigraphs(file, Digrap\ h([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Close(file);;\ngap> file := Digraph\ File(filename, \"r\");;\ngap> ReadDigraphs(file, 9);\n]]>\n \n\n\n \n \n\ A digraph, or a list of digraphs.\n \n I\ f filename is a string containing the name of a file containing\n en\ coded digraphs or an &IO; file object created using , then ReadDigraphs returns the digraphs\n encoded in the fi\ le as a list. Note that if filename is a\n compressed file, which h\ as been compressed appropriately to give a filename\n extension of .gz, .bz2, or .xz, then\n ReadDigraphs can read file\ name without it first needing to be\n decompressed.\n \n\n If\ the optional argument decoder is specified\n and is a function whic\ h decodes a string into a digraph,\n then ReadDigraphs will use d\ ecoder to decode the digraphs\n contained in filename. \n\n \ If the optional argument n is specified, then ReadDigraphs\n \ returns the nth digraph encoded in the file filename.\n \n\n If the optional argument decoder is not specified, then\n <\ C>ReadDigraphs will deduce which decoder to use based on the filename\n \ extension of filename (after removing the compression-related\n fil\ ename extensions .gz, .bz2, and .xz). For example,\n \ if the filename extension is .g6, then ReadDigraphs will use\n \ the graph6 decoder .\n \n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n \n graph6 (.g6)\n \n\tA standar\ d and widely-used format for undirected graphs, with no support\n\tfor loops o\ r multiple edges. Only symmetric graphs are allowed -- each\n\tedge is combin\ ed with its converse edge to produce a single undirected\n\tedge. This format\ is best used for \"dense\" graphs -- those with many\n\tedges per vertex.\n \ \n sparse6 (.s6)\n \n\tUnlike graph6, \ sparse6 has support for loops and multiple edges.\n\tHowever, its use is still\ limited to symmetric graphs. This format is\n\tbetter-suited to \"sparse\" g\ raphs -- those with few edges per vertex.\n \n digraph6\ (.d6)\n \n\tThis format is based on graph6, but stores dire\ ction information -\n\ttherefore is not limited to symmetric graphs. Loops ar\ e allowed, but\n\tmultiple edges are not. Best compression with \"dense\" gra\ phs.\n \n disparse6 (.ds6)\n \n\tAny \ type of digraph can be encoded in disparse6: directions, loops, and\n\tmultipl\ e edges are all allowed. Similar to sparse6, this has the best\n\tcompression\ rate with \"sparse\" graphs.\n \n plain text (.txt)\n \n\tThis is a human-readable format which stores graphs in t\ he form\n 0 7 0 8 1 7 2 8 3 8 4 8 5 8 6 8 i.e. pairs of v\ ertices\n describing edges in a graph. More specifically, the vertices \ making up\n one edge must be separated by a single space, and pairs of \ vertices must\n be separated by two spaces. \n\n See for a more flexible way to store\n digrap\ hs in a plain text file. \n\n \n\n pickled (.p or .pickle<\ /F>)\n \n Digraphs are pickled using the &IO; package\ . This is particularly good\n when the is \ non-trivial.\n \n \n\n ReadD\ igraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 10);\n\ngap> ReadDigraphs(\n> Concatenation(D\ IGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 17);\n\ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/data/tre\ e9.4.txt\"));\n[ , \n , \n , \n , \n , \n ,\ \n , \n , \n , \n <\ immutable digraph with 9 vertices, 8 edges>, \n , \n , \n , \n ]]]>\n \n\n\n \n \n \n If digraphs is a list of digraphs or a digra\ ph and filename\n is a string or an &IO; file object created using <\ Ref Func=\"DigraphFile\"/>,\n then WriteDigraphs writes the digraphs\ to the file represented by\n filename. If the supplied filename en\ ds in one of the extensions\n .gz, .bz2, or .xz, then \ the file will be compressed\n appropriately. Excluding these extensions, i\ f the file ends with an\n extension in the list below, the corresponding gr\ aph format will be used to\n encode it. If such an extension is not includ\ ed, an appropriate format\n will be chosen intelligently, and an extension \ appended, to minimise file\n size.\n \n\n \n\n For more verbose information on the progress of the function, \ set the info\n level of InfoDigraphs to 1 or higher, using SetInf\ oLevel. \n\n The currently supported file formats are:\n \n\ graph6 (.g6)\n \n\tA standard and widely-used fo\ rmat for undirected graphs, with no support\n\tfor loops or multiple edges. O\ nly symmetric graphs are allowed -- each\n\tedge is combined with its converse\ edge to produce a single undirected\n\tedge. This format is best used for \"\ dense\" graphs -- those with many\n\tedges per vertex.\n \n <\ Mark>sparse6 (.s6)\n \n\tUnlike graph6, sparse6 has support \ for loops and multiple edges.\n\tHowever, its use is still limited to symmetri\ c graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those with \ few edges per vertex.\n \n digraph6 (.d6)\n \ \n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n \ \n disparse6 (.ds6)\n \n\tAny type of digraph can\ be encoded in disparse6: directions, loops, and\n\tmultiple edges are all all\ owed. Similar to sparse6, this has the best\n\tcompression rate with \"sparse\ \" graphs.\n \n\n plain text (.txt)\n \n\tThis is a human-readable format which stores graphs in the form\n \ 0 7 0 8 1 7 2 8 3 8 4 8 5 8 6 8 i.e. pairs of vertices\n \ describing edges in a graph. More specifically, the vertices making up\n \ one edge must be separated by a single space, and pairs of vertices must\n \ be separated by two spaces. \n\n See for a more flexible way to store\n digraphs in a plain tex\ t file. \n\n \n pickled (.p or .pickle)\n \n \ Digraphs are pickled using the &IO; package. This is particularly good\ \n when the is non-trivial.\n \ \n \n\n grs := [];;\ngap> grs[1] := Digr\ aph([]);\n\ngap> grs[2] := Digraph([[\ 1, 3], [2], [1, 2]]);\n\ngap> grs[\ 3] := Digraph([\n> [6, 7], [6, 9], [1, 3, 4, 5, 8, 9],\n> [1, 2, 3, 4, 5, 6, 7\ , 10], [1, 5, 6, 7, 10], [2, 4, 5, 9, 10],\n> [3, 4, 5, 6, 7, 8, 9, 10], [1, 3\ , 5, 7, 8, 9], [1, 2, 5],\n> [1, 2, 4, 6, 7, 8]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/ou\ t/man.d6.gz\");;\ngap> WriteDigraphs(filename, grs, \"w\");\nIO_OK\ngap> ReadD\ igraphs(filename);\n[ , \n , \n ]]]>\n \n\n\n \n \n \ An iterator.\n \n If filename is a \ string representing the name of a file containing\n encoded digraphs, then \ IteratorFromDigraphFile returns an iterator\n for which the value of\ is the\n next digraph encode\ d in the file.\n \n\n If the optional argument decoder is spe\ cified and is a function\n which decodes a string into a digraph, then I\ teratorFromDigraphFile\n will use decoder to decode the digraphs\ contained in filename.\n \n\n The purpose of this function i\ s to easily allow looping over digraphs\n encoded in a file when loading al\ l of the encoded digraphs would require\n too much memory. \n\n To s\ ee what file types are available, see .\n\n filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/ma\ n.d6.gz\");;\ngap> file := DigraphFile(filename, \"w\");;\ngap> for i in [1 ..\ 10] do\n> WriteDigraphs(file, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap>\ IO_Close(file);;\ngap> iter := IteratorFromDigraphFile(filename);\n\ \ngap> for x in iter do od;\n]]>\n \n\n\ \n \n \n \n A string.\n \n These two functions return a function which encodes or decode\ s a digraph in\n a plain text format. \n\n DigraphPlainTextLineEnc\ oder returns a function which takes a single\n digraph as an argument. \ The function returns a string describing the edges\n of that digraph; each\ edge is written as a pair of integers separated by the\n string delimit\ er2, and the edges themselves are separated by the\n string delimite\ r1. DigraphPlainTextLineDecoder returns the\n corresponding dec\ oder function, which takes a string argument in this format\n and returns a\ digraph. \n\n If only one delimiter is passed as an argument to\n DigraphPlainTextLineDecoder, it will return a function which decodes\n \ a single edge, returning its contents as a list of integers. \n\n The a\ rgument offset should be an integer, which will describe a\n number \ to be added to each vertex before it is encoded, or after it is\n decoded. \ This may be used, for example, to label vertices starting at 0\n instead o\ f 1. \n\n Note that the number of vertices of a digraph is not stored, a\ nd so vertices\n which are not connected to any edge may be lost.\n\n gr := Digraph([[2, 3], [1], [1]]);\n\ngap> enc := DigraphPlainTextLineEncoder(\" \", \" \ \", -1);;\ngap> dec := DigraphPlainTextLineDecoder(\" \", \" \", 1);;\ngap> e\ nc(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> dec(last);\n\n]]>\n \n\n\n \n \n A dig\ raph.\n \n This function takes a string str,\ decodes it, and then returns the\n tournament [see ] which it defines, according to\n the following rules. \n\n Th\ e characters of the string str represent the entries in the upper\n \ triangle of a tournament's adjacency matrix. The number of vertices\n n<\ /C> will be detected from the length of the string and will be as\n large a\ s possible. \n\n The first character represents the possible edge 1\ -> 2, the second\n represents 1 -> 3 and so on until 1 -> n<\ /C>; then the following\n character represents 2 -> 3, and so on up \ to the character which\n represents the edge n-1 -> n. \n\n I\ f a character of the string with corresponding edge i -> j is equal\n \ to 1, then the edge i -> j is present in the tournament.\n \ Otherwise, the edge i -> j is present instead. In this way, all the\n \ possible edges are encoded one-by-one.\n\n gr :\ = TournamentLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 1, 2, 4 ], [ 1, 2 ] ]\n]]>\n \n\n\n \n \n A digraph.\n \n This function takes a string str, decode\ s it, and then returns the\n topologically sorted digraph [see ]\n which it defines, according to the following \ rules. \n\n The characters of the string str represent the entri\ es in the upper\n triangle of a digraph's adjacency matrix. The number of v\ ertices n\n will be detected from the length of the string and will \ be as large as\n possible. \n\n The first character represents the \ possible edge 1 -> 2, the second\n represents 1 -> 3 and so o\ n until 1 -> n; then the following\n character represents 2 -> 3<\ /C>, and so on up to the character which\n represents the edge n-1 -> n<\ /C>. If a character of the string with\n corresponding edge i -> j \ is equal to 1, then this edge is\n present in the digraph. Otherwise\ , it is not present. In this way, all the\n possible edges are encoded one\ -by-one. \n\n In particular, note that there exists no edge [i, j] if j \\leq\n i. In order words, the digraph will be topologica\ lly sorted.\n\n gr := AdjacencyMatrixUpperTriangle\ LineDecoder(\"100001\");\n\ngap> O\ utNeighbours(gr);\n[ [ 2 ], [ ], [ 4 ], [ ] ]\ngap> gr := AdjacencyMatrixUpp\ erTriangleLineDecoder(\"111111x111\");\n\ngap> OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 3, 4 ], [ 4, 5 ], [ 5 ], \ [ ] ]\n]]>\n \n\n\n \n \ \n A digraph.\n \ \n If str is a string consisting of at least two non-ne\ gative integers\n separated by spaces, then this function will attempt to r\ eturn the digraph\n which it defines as a TCode string. \n\n The fir\ st integer of the string defines the number of vertices v in\n the d\ igraph, and the second defines the number of edges e. The\n followi\ ng 2e integers should be vertex numbers in the range [0 ..\n v-\ 1]. These integers are read in pairs and define the digraph's\n edges. \ This function will return an error if str has fewer than\n 2e+2<\ /C> entries. \n\n Note that the vertex numbers will be incremented by 1\ in the digraph\n returned. Hence the string fragment 0 6 will desc\ ribe the edge\n [1,7].\n gr := TCodeDeco\ der(\"3 2 0 2 2 1\");\n\ngap> OutN\ eighbours(gr);\n[ [ 3 ], [ ], [ 2 ] ]\ngap> gr := TCodeDecoder(\"12 3 0 10 5 \ 2 8 8\");\n\ngap> OutNeighbours(g\ r);\n[ [ 11 ], [ ], [ ], [ ], [ ], [ 3 ], [ ], [ ], [ 9 ], [ ], \n [ \ ], [ ] ]\n]]>\n \n\n\n \ \n \n \n A string.\n \n PlainTextString takes a single digraph, and returns a string\n \ describing the edges of that digraph. DigraphFromPlainTextString\n \ takes such a string and returns the digraph which it describes. Each edge\ \n is written as a pair of integers separated by a single space. The edges\ \n themselves are separated by a double space. Vertex numbers are reduced \ by\n 1 when they are encoded, so that vertices in the string are labelled\n\ starting at 0. \n\n Note that the number of vertices of a digraph is\ not stored, and so vertices\n which are not connected to any edge may be l\ ost. \n\n \n The operation DigraphFromPlainTextString\n tak\ es an optional first argument \n or , which specifies whether the output digraph shall\ \n be mutable or immutable. If no first argument is provided, then an immut\ able\n digraph is returned by default.\n\n gr :\ = Digraph([[2, 3], [1], [1]]);\n\n\ gap> PlainTextString(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> DigraphFromPlainTextSt\ ring(last);\n\n]]>\n \n\n\n \n \n \n \n These functions write and read a single digraph in a human-readable pl\ ain\n text format as follows: each line contains a single edge, and each ed\ ge is\n written as a pair of integers separated by the string delimiter<\ /A>. \n\n filename should be the name of a file which will be wri\ tten to or\n read from, and offset should be an integer which is add\ ed to each\n vertex number as it is written or read. For example, if\n \ WritePlainTextDigraph is called with offset -1, then\n \ the vertices will be numbered in the file starting from 0 instead of 1 -\n \ ReadPlainTextDigraph would then need to be called with offset\n\ 1 to convert back to the original graph. \n\n ignore s\ hould be a list of characters which will be ignored when\n reading the grap\ h.\n gr := Digraph([[1, 2, 3], [1, 1], [2]]);\n\ngap> filename := Concatenation\ (DIGRAPHS_Dir(), \"/tst/out/plain.txt\");;\ngap> WritePlainTextDigraph(filenam\ e, gr, \",\", -1);\ngap> ReadPlainTextDigraph(filename, \",\", 1, ['/', '%']);\ \n\n]]>\n \n\n\n \n \n \n \n These operations write or read the single symmetri\ c digraph digraph\n to or from a file in DIMACS format, as appropria\ te. The operation\n WriteDIMACSDigraph records the vertices and edge\ s of digraph.\n The vertex labels of digraph will be recorded\ only if they are\n integers. See and . \n\n The first argument fil\ ename should be the name of the file which will\n be written to or read\ from. A file can contain one symmetric digraph in\n DIMACS format. If \ filename ends in one of .gz, .bz2,\n or .xz, then \ the file is compressed, or decompressed, appropriately.\n \n\n The D\ IMACS format is described as follows. Each line in the DIMACS file has\n o\ ne of four types:\n \n \n A line beginning with \ c and followed by any number of characters\n is a comment line, and\ is ignored.\n \n \n A line beginning with p<\ /C> defines the numbers of vertices and edges\n the digraph. This line\ has the format p edge <nr_vertices>\n <nr_edges>,\ where <nr_vertices> and\n <nr_edges> are rep\ laced by the relevant integers. There\n must be exactly one such line i\ n the file, and it must occur before any\n of the following kinds of li\ ne. \n\n Although it is required to be present, the value of\n \ <nr_edges> will be ignored. The correct number of edges\n \ will be deduced from the rest of the information in the file.\n \ \n \n A line of the form e <v> <w>, where\ <v>\n and <w> are integers in the range [\ 1 ..\n <nr_vertices>], specifies that there is a (symmetric\ ) edge\n in the digraph between the vertices <v> and\n \ <w>. A symmetric edge only needs to be defined once; an\n \ additional line e <v> <w>, or e <w>\n \ <v>, will be interpreted as an additional, multiple, edge.\n \ Loops are permitted.\n \n \n A line of the form\ n <v> <label>, where\n <v> is an integ\ er in the range [1 ..\n <nr_vertices>] and <label&\ gt; is an integer,\n signifies that the vertex <v> has\ the label\n <label> in the digraph. If a label is not spe\ cified for a\n vertex, then ReadDIMACSDigraph will assign the la\ bel\n 1, according to the DIMACS specification.\n \n\ \n\n A detailed definition of the DIMACS format can be found at\ \n http://mat.gsia.cmu.edu/COLOR/general/ccformat.ps, in Section\ \n 2.1. Note that optional descriptor lines, as described in Section 2.1, w\ ill\n be ignored.\n \n\n\n gr := Digraph\ ([[2], [1, 3, 4], [2, 4], [2, 3]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(),\n> \ \"/tst/out/dimacs.dimacs\");;\ngap> WriteDIMACSDigraph(filename, gr);;\n\ gap> ReadDIMACSDigraph(filename);\n]]>\n \n\n\n \n\n \ \n\n\n\n \n Grape to Dig\ raphs Command Map\n \n Below is a table of &Grape; commands with t\ he &Digraphs; counterparts.\n The sections in this chapter correspond to the \ chapters in the &Grape;\n manual.\n\n \n \n Functi\ ons to construct and modify graphs\n \n \n \ \n The table in this section contains more information when view\ ed in\n html format.\n \n \n\n \n \n \n \n &Grape; comma\ nd\n \n \n &Digraphs; command\n \n \n \n Comments\n \ \n \n \n \n\n \ \n \n Graph\n \n \n \ \n \n \n \n Used with the arguments of Graph in &Grape;\ , will return an equivalent digraph. \ \n \n \n \n \n\n <\ Row>\n \n EdgeOrbitsGraph\n \n \ \n \n \n \ \n \n The operation behaves the same as\n EdgeOrbitsDigraph \ in &Grape;.\n \n \n \n \ \n\n \n \n NullGraph\n \n\ \n \n \n or\n \n \n \n \n \n \ Note that, unlike in &Grape;, the arguments of and are just\n non-n\ egative integers. Since digraphs in &Digraphs; are immutable,\n the\ re is no reason (although it is possible) to create a null digraph\n \ and then add edges, as you might be used to doing in &Grape;.\n \n \n \n \n\n \n <\ Item>\n CompleteGraph\n \n \n \ \n \n \n \ \n Note that, unlike in &Grape;, the argument \ to is simply a non-negative int\ eger\n indicating the number of vertices. By default a complete dig\ raph in\n &Digraphs; does not have any loops. If you want a complet\ e digraph\n with loops at every vertex, then use .\n \n \n \n \n\n \n \n JohnsonGrap\ h\n \n \n \n \n \n \n \ The &Grape; function JohnsonGraph is equivalent to\n .\n \n \n \n \ \n\n \n \n CayleyGraph\n\ \n \n \n \ \n \n \n The op\ eration behaves the same as\n Cayl\ eyGraph in &Grape;.\n \n \n \n \ \n\n \n \n AddEdgeOrbit\n \ \n \n \n\ \n \n \n No\ te that, unlike in &Grape;, digraphs in &Digraphs; are immutable,\n \ and this operation returns a new digraph, i.e. it does not modify the\n \ digraph given as an argument.\n \n \n \ \n \n\n \n \n RemoveEd\ geOrbit\n \n \n \n \n \n \n Note that, unlike in &Grape;, digraphs in &Digraphs; are immu\ table,\n and this operation returns a new digraph, i.e. it does not\ modify the\n digraph given as an argument.\n \n \ \n \n \n\n \n \n \ AssignVertexNames\n \n \n <\ Ref Oper=\"SetDigraphVertexLabels\"/>\n \n \ and\n \n \n\ \n \n \n Us\ e to set an individual\n ver\ tex label and to set all of\n \ the vertex labels.\n \n \n \n <\ HorLine/>\n \n \n\n \n \n Fun\ ctions to inspect graphs, vertices and edges\n \n \n \n The table in this section contains more information\ when viewed in\n html format.\n \n \n\n \n \n\n \n \n &\ Grape; command\n \n \n &Digraphs; command\ \n \n \n \n \ Comments\n \n \n \n \n\ \n \n \n IsGraph\n \n \ \n \n \n \n \n is a \ category and not a function like\n IsGraph.\n \n \n \n \n\n \n \n OrderGraph\n \n \n <\ Ref Attr=\"DigraphNrVertices\"/>\n \n \n <\ Alt Only=\"HTML\">\n The operation behaves the same as\n OrderGraph in &Grape;.\n \n \n \n \n\n \n \ \n IsVertex(graph, v)\n \n \ \n v in DigraphVertices(digraph)\n \ \n \n \n The \ attribute can be used to return\n t\ he list of vertices of a digraph, or if v is known to be\n a\ positive integer, then you can simply do\n v <= Digra\ phNrVertices(digraph)\n \n \n \n \n\n \n \n VertexName<\ /C>\n \n \n \n \n \n \n \ Unlike VertexName in &Grape;, the return value of\n is mutable.\n \n \ \n \n \n\n \n \n \ VertexNames\n \n \n \n \n \n \n Unlike VertexNames in &Grape;, the return value of\n \ is mutable.\n \n \ \n \n \n\n \n \ \n Vertices\n \n \n \n \n \n \n The vertices of a digraph in &Digraphs; is always the\ range [1\n .. n] where n is the number of vertices\ of the\n digraph. The operation b\ ehaves the\n same as Vertices in &Grape;.\n \ \n \n \n \n\n \n \n VertexDegree\n \n \n \ \n \n \ (see also: )\n \n \n \n \n The operation behaves the same as\n VertexDegr\ ee in &Grape;.\n \n \n \n \n\n \n \n VertexDegrees\n \ \n \n \n <\ Alt Only=\"HTML\">\n (see also:\n ,\n ,\n ,\n ,\n <\ Ref Attr=\"InDegreeSet\"/>)\n \n \n \ \n \n The operation behaves the same as\n VertexDegrees in &Grape;.\n \ \n \n \n \n\n \n \n IsLoopy\n \n \n\ \n \n \n \ \n The operation behaves the same as\n IsLoopy in &Grape;.\n \ \n \n \n \n\n \n \ \n IsSimpleGraph\n \n \n \ \n \n\ and not \n \n \ \n \n \ \n \n \n \n \ The &Grape; function IsSimpleGraph is equivalent to\n \ \n and not .\n \n \n \n \n\n \n \n Adjacency\n \n \n \n \ \n (see also )\n \n \n \n \n The operation behav\ es the same\n as Adjacency in &Grape;.\n \n \ \n \n \n\n \n \n\ IsEdge\n \n \n \n \n <\ Item>\n \n The arguments of\n \ \n ca\ n be a digraph and an edge, or a digraph and two vertices. In the\n \ first form the operation\n \n behaves the same as IsEdge in &Grape\ ;.\n \n \n \n \n\n \ \n \n DirectedEdges\n \n \ \n \n \n \n \n The attribute behaves the same\n as DirectedEdges in &Grape;\ .\n \n \n \n \n\n \ \n \n UndirectedEdges\n \n \ \n None\n \n \n \n There is no analogous function in &Digraphs;, and no\ plans to\n include one.\n \n \n \ \n \n\n \n \n Distance\ \n \n \n \n <\ /Item>\n \n \n The operati\ on behaves the same as\n Distance in &Grape;\ .\n \n \n \n \n\n \ \n \n Diameter\n \n \n \n \n \n \n The attribute behaves the same\n as Diameter in &Grape;.\n \ \n \n \n \n\n \ \n \n Girth\n \n \n \ \n \n \ \n \n The attribute behaves the same\n as Girth in &Grape; e\ xcept that in the case that the\n argument is a forest, infinity\ is returned by &Digraphs;\n whereas -1 is returned by &\ Grape;.\n \n \n \n \n\n\ \n \n IsConnectedGraph\n \n \n \n \ \n \n \n The att\ ribute behaves the\n sam\ e as IsConnectedGraph in &Grape;.\n \n \n \ \n \n\n \n \n IsB\ ipartite\n \n \n \n \n and \n \n \n \n \ \n The &Grape; function IsBipartite is\ equivalent to\n and .\n \n \n \ \n \n\n \n \n IsNu\ llGraph\n \n \n \n \n or \n \n \n \n \n The &Grape; function IsNullGraph is equivalent\ to\n and its synonym .\n \n \n \n \n\n \n \n IsCompleteGr\ aph\n \n \n \n \n \n \n \ The &Grape; function IsCompleteGraph is roughly\n equ\ ivalent to\n except that in &Digr\ aphs;\n a complete digraph has no loops.\n \n\ \n \n \n\n\n \n \ \n\n \n \n Functions to determine regularity proper\ ties of graphs\n \n \n \n The\ table in this section contains more information when viewed in\n html \ format.\n \n \n\n \n \n\n \n \n &Grape; command\n \n \n &Digraphs; command\n \n \n \n Comments\n \n \ \n \n \n\n \n \n \ IsRegularGraph\n \n \n \n \n (\ see also\n \n and\n \ )\n \n \ \n \n \n The &Grape; funct\ ion IsRegularGraph is\n equivalent to .\n \n \n \n \n\n \n \n LocalParameters\n \ \n \n None\n \n \n \ \n The &Grape; function LocalParameters<\ /C> has no equivalent\n function in &Digraphs;, although this might\ change in the future.\n \n \n \n \ \n\n \n \n GlobalParameters\ \n \n \n None\n \n \n \n The &Grape; function Global\ Parameters has no equivalent\n function in &Digraphs;, although\ this might change in the future.\n \n \n \n \n\n \n \n IsDistance\ Regular\n \n \n \n \n \n \n The &Grape; function IsDistanceRegular is\n \ equivalent to .\n \n \ \n \n \n\n \n \ \n CollapsedAdjacencyMat\n \n \n \ None\n \n \n \n \ The &Grape; function CollapsedAdjacencyMat has no equivalent\ \n function in &Digraphs;, although this might change in the future\ .\n \n \n \n \n\n \ \n \n OrbitalGraphColadjMats\n \n \n None\n \n \n <\ Alt Only=\"HTML\">\n The &Grape; function OrbitalGraphColadjMats\ has no equivalent\n function in &Digraphs;, although this migh\ t change in the future.\n \n \n \n \ \n\n \n \n VertexTransitiveDRGs\ \n \n \n None\n \n \ \n \n The &Grape; function Ve\ rtexTransitiveDRGs has no equivalent\n function in &Digraphs;, \ although this might change in the future.\n \n \n\ \n \n\n \n\n \n\n \ \n \n Some special vertex subsets of a graph\n \ \n \n \n The table in this section contain\ s more information when viewed in\n html format.\n \n \n\n \n \n\n \n <\ Item>\n &Grape; command\n \n \n &\ Digraphs; command\n \n \n \n Comments\n \n \n \n \ \n\n \n \n ConnectedComponent\ \n \n \n \n \n \n \n\ When applied to a simple di/graph and a vertex,\n Co\ nnectedComponent in &Grape; is equivalent to\n .\n can also be applied to\n non-simple graphs (in the language of\ &Grape;).\n \n \n \n \ \n\n \n \n ConnectedComponents\n \ \n \n \ \n \n \n \n \ When applied to a simple di/graph,\n ConnectedComponents in \ &Grape; is equivalent to\n .\n can also be applie\ d to\n non-simple graphs (in the language of &Grape;).\n <\ /Alt>\n \n \n \n\n \n \ \n Bicomponents\n \n \n \ \n \n \n \ \n When applied to a simple bipartite di/\ graph,\n Bicomponents in &Grape; is equivalent to\n \ .\n The definition of does not require\n its argument to be s\ imple (in the language of &Grape;), i.e. the\n bipartite graphs in \ &Grape; are a proper subset of the bipartite\n digraphs in &Digraph\ s;.\n \n \n \n \n\n \ \n \n DistanceSet\n \n \ \n \n \n \n \ \n DistanceSet in &Grape; is equiv\ alent to\n in &Digraphs;.\n \n \ \n \n \n\n \n \n \ Layers\n \n \n \n \n \n \n Layers in &Grape; is equivalent to\n in &Digraphs;.\n \n \n \ \n \n\n \n \n Ind\ ependentSet\n \n \n \n \n \n \n IndependentSet in &Grape; is equivalent to\n \ in &Digraphs;.\n \n \ \n \n \n\n \n\n \ \n\n \n \n Functions to construct new graphs from o\ ld\n \n \n \n The table in th\ is section contains more information when viewed in\n html format.\n \ \n \n\n \n \n\n \ \n \n &Grape; command\n \n <\ Item>\n &Digraphs; command\n \n \n \ \n Comments\n \n \ \n \n \n\n \n \n \ InducedSubgraph\n \n \n \n \n \n \n in &Digraphs; does not al\ low a third\n argument G specifying a group of automorphisms\ on the\n resultant graph. This may be included in a future release\ .\n \n \n \n \n\n \ \n \n DistanceSetInduced\n \n \ \n None\n \n \n \n There is no analogous function in &Digraphs;, and\ no plans to\n include one.\n \n \n \ \n \n\n \n \n Dista\ nceGraph\n \n \n \n \n \n \ \n DistanceGraph in &Grape; is eq\ uivalent to\n in\n &Digraphs;.\n \n \n \ \n \n\n \n \n Comple\ mentGraph\n \n \n \n \n \n \n \ in &Digraphs; is equivalent to\n \ ComplementGraph in &Grape; with the optional argument\n \ comploops set to true (loops/nonloops are\n complemented\ ).\n \n \n \n \n\n \ \n \n PointGraph\n \n \ \n None\n \n \n \n There is no analogous function in &Digraphs;, and no pl\ ans to\n include one.\n \n \n \n \n\n \n \n EdgeGraph\n \n \n \n \n \n \n \ EdgeGraph in &Grape; is equivalent to\n in &Digraphs;.\n \n \n \ \n \n\n \n \n Switc\ hedGraph\n \n \n None\n \ \n \n \n There is no analo\ gous function in &Digraphs;, and no plans to\n include one.\n \ \n \n \n \n\n \n \ \n UnderlyingGraph\n \n \n \n \n \ \n \n UnderlyingGraph in \ &Grape; is equivalent to\n \ in &Digraphs;.\n \n \n \n \n\n \n \n QuotientGraph\n \n \n \n \n \n \n in &Digraphs; takes a partition\n p of the\ vertices of the digraph, rather than a set of\n generating pairs. \ Hence, the equivalence might not be\n group-invariant. Also note \ that in &Digraphs; multiple edges may be\n created, whereas in &Gra\ pe; they are effectively combined into a\n single edge.\n \ \n \n \n \n\n \n \ \n BipartiteDouble\n \n \n \ \n \n \n \n BipartiteDouble in &Grap\ e; is equivalent to\n in &Di\ graphs;.\n \n \n \n \n\ \n \n \n GeodesicsGraph\n \n \n None\n \n \n \ \n There is no analogous function in &Digraphs;,\ and no plans to\n include one.\n \n \ \n \n \n\n \n \n \ CollapsedIndependentOrbitsGraph\n \n \n \ None\n \n \n \n \ There is no analogous function in &Digraphs;, although this feature\n \ might be implemented in the future.\n \n \n \n \n\n \n \n CollapsedCompleteOrbitsGraph\n \n \n \ None\n \n \n \n \ There is no analogous function in &Digraphs;, although this feature\n \ might be implemented in the future.\n \n \ \n \n \n\n \n \n \ NewGroupGraph\n \n \n None\n \n \n \n There is no \ analogous function in &Digraphs;. However, the same\n results woul\ d be possible by calling on\n the digra\ ph to produce a copy gr, and then applying\n SetDigraphGr\ oup(gr, G).\n \n \n \n \n\n \n\n \n\n \n \n Vert\ ex-Colouring and Complete Subgraphs\n \n \n \ \n The table in this section contains more information when vie\ wed in\n html format.\n \n \n\n \n \n\n \n \n &Grape; co\ mmand\n \n \n &Digraphs; command\n \ \n \n \n Comments\n\ \n \n \n \n\n \n \n VertexColouring\n \n \ \n \n \n \n \n \ VertexColouring in &Grape; is equivalent to in &Digraphs;,\n \ except it returns a transformation rather than a list of vertex\n \ colors.\n \n \n \n \n\ \n \n \n CompleteSubgraphs\n \n \n \n \n \n \n The functionalit\ y of CompleteSubgraphs in &Grape; is provided\n by and \n i\ n &Digraphs;.\n \n \n \n \n\n \n \n CompleteSubgraphsOfGivenSize\n \n \n \ \n \n \n \n \ Much of the functionality of CompleteSubgraphsOfGivenSize in\n \ &Grape; is provided by and in &Digraphs;, except that the\n \ functionality relating to vertex weights and partial colourings is\n \ not implemented.\n \n \n \n <\ HorLine/>\n\n \n \n\n \n \n A\ utomorphism groups and isomorphism testing for graphs\n \n \n \n The table in this section contains more in\ formation when viewed in\n html format.\n \n \n\n \ \n \n\n \n \n \ &Grape; command\n \n \n &Digraphs;\ command\n \n \n \n \ Comments\n \n \n \n \n\n \n \n AutGroupGraph\n \ \n \n \n \n \n \ \n AutGroupGraph in &Grape; is equivalent to\n in &Digraphs;.\n \ \n \n \n \n\n \n \ \n GraphIsomorphism\n \n \n \n \ \n \n \n Gra\ phIsomorphism in &Grape; is equivalent to in &Digraphs;.\n Note howev\ er that GraphIsomorphism's optional argument\n firstunbin\ dcanon is not applicable, since the canonical\n labellings of a\ digraph are not portable between sessions.\n \n \ \n \n \n\n \n \n \ IsIsomorphicGraph\n \n \n \n \n \n \n IsIsomorphicGraph in &Gra\ pe; is equivalent to in &Digraphs;.\n Note however that IsIsomorphicGr\ aph's optional argument\n firstunbindcanon is not applic\ able, 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hurr\ y", Homomorphisms := "Homomorphisms", ("Installing Digraphs") := "Installing Digraphs", ("Isomorphisms and canonical labellings") := "Is\ omorphisms and canonical labellings", Operators := "Operators", ("Optional package dependencies") := "Optional p\ ackage dependencies", ("Properties of digraphs") := "Properties of dig\ raphs", ("Rebuilding the documentation") := "Rebuilding \ the documentation", ("Testing your installation") := "Testing your i\ nstallation", ("The Digraphs package") := "\n The Digraphs package\n ", ("The Grape package") := "The Grape package", ("Visualising and IO") := "Visualising and IO" ), linelength := 76, mathmode := "MathJax", name := "WHOLEDOCUMENT", next := 474555, root := ~, six := [ [ "Title page", ".", [ 0, 0, 0 ], 1, 1, "title page", "X7D2C85EC87DD46E5" ], [ "Abstract", ".-2", [ 0, 0, 2 ], 56, 2, "abstract", "X7AA6C5737B711C89" ], [ "Copyright", ".-1", [ 0, 0, 1 ], 62, 2, "copyright", "X81488B807F2A1CF1" ], [ "Acknowledgements", ".-3", [ 0, 0, 3 ], 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DigraphsTestInstall\n   2.5-2 DigraphsTestStandard\n   \ 2.5-3 DigraphsTestExtreme\n \n
\n3 Cr\ eating digraphs\n 3\ .1 Creating digraphs\n\n\n &nb\ sp;3.1-1 IsDigraph\n   3.1-2 IsMutableDigraph\n   3.1-3 IsImmutableDigraph\n   3.1-4 IsCayleyDigraph\n   3.1-5 IsDigraphWithAdjacencyFunction\n   3.1-6 DigraphByOutNeighboursType\n   3.1-7 Digraph\n   3.1-8 DigraphByAdjacencyMatrix\n   3.1-9 DigraphByEdges\n \   3.1-10 EdgeOrbitsDigraph\n   3.1-11 DigraphByInNeighbours\n   3.1-12 CayleyDigraph\n \n 3.2 Changing representations\n\n\n   3.2-1 AsBinaryRelation\n   3.2\ -2 AsDigraph\n &n\ bsp; 3.2-3 Graph\n   <\ a href=\"chap3_mj.html#X7C4F13E080EC16B0\">3.2-4 AsGraph\n   3.2-5 AsTransformation\n \n 3.3 New\ digraphs from old\n\n\n   3.3-1 DigraphImmutableCopy\n   3.3-2 DigraphImmutableCopyIfImmutable\n   3.3-3 InducedSubdigraph\n   3.3-4 ReducedDigraph\n   3.3-5 MaximalSymmetricSubdigraph\n   3.3-6 MaximalAntiSymmetricSubdigraph\n   3.3-7 UndirectedSpanningForest\n   3.3-8 QuotientDigraph\n   3.3-9 DigraphReverse\n  3.3-10 DigraphDual\n   \ 3.3-11 DigraphSymmetricClosure\n   3.3-12 DigraphTransitiveClosure\n   3.3-13 DigraphTransitiveReduction\n \   3.3-14 DigraphAddVertex\n   3.3-15 DigraphAddVertices\n   3.3\ -16 DigraphAddEdge\n   3.3-17 Digr\ aphAddEdgeOrbit\n   3.3-18 Digraph\ AddEdges\n \ 3.3-19 DigraphRemoveV\ ertex\n &nb\ sp;3.3-20 DigraphRemoveVert\ ices\n &nbs\ p;3.3-21 DigraphRemoveEdge<\ /a>\n   3.3-22 DigraphRemoveEdgeOrbit<\ /a>\n   3.3-23 DigraphRemoveEdges<\ /span>\n   \ 3.3-24 DigraphRemoveLoops\n   3.3-25 DigraphRemoveAllMultipleEdges\n   3.3-26 DigraphReverseEdges<\ /span>\n   \ 3.3-27 DigraphDisjointUnion\n   3.3-28 DigraphEdgeUnion\n\   3.3-29 DigraphJoin\n   3.3-30 DigraphCartesianProduct\n   3.3-31 DigraphDirectProduct\n   3.3-32 DigraphCartesianProductProjections\n\   3.3-33 DigraphDirectProductProjections\ \n   3.3-34 LineDigraph\n   3.3-35 LineUndirectedDigraph\n   3.3-36 DoubleDigraph\n   3.3-37 BipartiteDoubleDigraph\n   3.3-38 DigraphAddAllLoops\n   3.3-39 DistanceDigraph\n\   3.3-40 DigraphClosure\n   3.3-41 DigraphMycielskian\n \n 3.4 Random digraphs\ \n\n\n   3.4-1 RandomDigraph\n   3.4-2 R\ andomMultiDigraph\n   3.4-3 Random\ Tournament\n &nbs\ p; 3.4-4 RandomLattice\ \n \n 3.\ 5 Standard examples\n\n\n   \ ;3.5-1 ChainDigraph\n   3.5-2 CompleteDigraph\n   3.5-3 CompleteBipartiteDigraph\n   3.5-4 CompleteMultipartiteDigraph\ \n   3.5-5 CycleDigraph\n   3.5-6 EmptyDigraph\n   3.5-7 JohnsonDigraph\n \   3.5-8 PetersenGraph\n   3.5\ -9 GeneralisedPetersenGraph\n \n
\n4 \ Operators\n 4.1 Operators for digraphs\n\n\n   \ ;4.1-1 IsSubdigraph\n   4.1-2 IsUndirectedSpanningTree\n \n \n5 Attributes and operations\n&nbs\ p;5.1 Vertices and edges\n\n\n   5.1-1 DigraphVertices\n   5.1-2 DigraphNrVertices\n   5.1-3 DigraphEdges\n   5.1-4 DigraphNrEdges\n   5.1-\ 5 DigraphSinks\n \   5.1-6 DigraphSo\ urces\n &nb\ sp;5.1-7 DigraphTopological\ Sort\n &nbs\ p;5.1-8 DigraphVertexLabel<\ /a>\n   5.1-9 DigraphVertexLabels<\ /span>\n   \ 5.1-10 DigraphEdgeLabel\ \n   5.1-11 DigraphEdgeLabels\n   5.1-12 DigraphInEdges\n   5.1-13 DigraphOutEdges\n   5.1-14 IsDigraphEdge\n   5.1-15 IsMatching\n   5.1\ -16 DigraphMaximalMatching\n   5.1\ -17 DigraphMaximumMatching\n \n<\ span class=\"tocline\"> 5.2 Neighbours and degree\n\n\n   5.2-1 AdjacencyMatrix\n   5.2-\ 2 CharacteristicPolynomial\n   5.2\ -3 BooleanAdjacencyMatrix\n   5.2-\ 4 DigraphAdjacencyFunction\n   5.2\ -5 DigraphRange\n   5.2-6 OutNeigh\ bours\n &nb\ sp;5.2-7 InNeighbours\n   <\ a href=\"chap5_mj.html#X7F5ACE807D1BC2E2\">5.2-8 OutDegrees\n   5.2-9 InDegrees\n   5.2-10 OutDegreeOfVertex\n  5.2-11 OutNeighboursOfVertex\n  5.2-12 InDegreeOfVertex\n   5.2-13 InNeighboursOfVertex\n   5.2-14 DigraphLoops\n   5.2-15 \ PartialOrderDigraphMeetOfVertices\n   5.2-16 DegreeMatrix\n   5.2-1\ 7 LaplacianMatrix\n \n 5.3 Reachability and connectivity\n\n\n   5.3-1 DigraphDiameter\n   5.3-2\ DigraphShortestDistance\n   5.3-3\ DigraphShortestDistances\n   5.3-\ 4 DigraphLongestDistanceFromVertex\n   5.3-5 DigraphDistanceSet\n   \ 5.3-6 DigraphGirth\n   5.3-7 Digra\ phOddGirth\n &nbs\ p; 5.3-8 DigraphUndire\ ctedGirth\n \ ; 5.3-9 DigraphConnect\ edComponents\n &n\ bsp; 5.3-10 DigraphCon\ nectedComponent\n   5.3-11 Digraph\ StronglyConnectedComponents\n   5.\ 3-12 DigraphStronglyConnectedComponent\n   5.3-13 DigraphBicomponents\n <\ span class=\"nocss\">  5.3-14 ArticulationPoints\n   5.3-15 Bridges\n   5.3-16 Stron\ gOrientation\n &n\ bsp; 5.3-17 DigraphPer\ iod\n   \ ;5.3-18 DigraphFloydWarshal\ l\n   <\ /span>5.3-19 IsReachable\n   5.3-20 DigraphPath\n   5.3-21 DigraphShortestPath\n   5.3-22 IteratorOfPaths\n   5.3-23 DigraphAllSimpleCircuits\n   5.3-24 DigraphLongestSimpleCircuit\n   5.3-25 DigraphLayers\n   5.3-26 DigraphDegeneracy\n  5.3-27 DigraphDegeneracyOrdering\n   5.3-28 HamiltonianPath\n \   5.3-29 NrSpanningTrees\n   \ 5.3-30 DigraphDijkstra\n \n 5.4 Cayley graphs of groups\ \n\n\n   5.4-1 GroupOfCayleyDigraph\n   \ 5.4-2 GeneratorsOfCayleyDigraph\n \n 5.5 Associated semigroups\ \n\n\n   5.5-1 AsSemigroup\n   5.5\ -2 AsSemigroup\n \n 5.6 Planarity\n\n\n &nbs\ p; 5.6-1 KuratowskiPla\ narSubdigraph\n &\ nbsp; 5.6-2 Kuratowski\ OuterPlanarSubdigraph\n   5.6-3 Pl\ anarEmbedding\n &\ nbsp; 5.6-4 OuterPlana\ rEmbedding\n &nbs\ p; 5.6-5 SubdigraphHom\ eomorphicToK23\n \n \n6 Properties of\ digraphs\n 6.1 Edge properties\n\n\n   6.1-1 DigraphHasLoops\ \n   6.1-2 IsAntiSymmetricDigraph\ \n   6.1-3 IsBipartiteDigraph\n   6.1-4 IsCompleteBipartiteDigraph\n\   6.1-5 IsCompleteDigraph\n   6.1-6 IsCompleteMultipartiteDigraph\n\   6.1-7 IsEmptyDigraph\n   6.1-8 IsEquivalenceDigraph\n   6.1-9 IsFunctionalDigraph\n   6.1-10 IsMultiDigraph\n<\ br />  6.1-11 IsReflexiveDigraph\n   6.1-12 IsSymmetricDigraph\n   6.1-13 IsTournament\n   6.1-1\ 4 IsTransitiveDigraph\n   6.1-15 I\ sPreorderDigraph\n   6.1-16 IsPar\ tialOrderDigraph\n   6.1-17 IsMee\ tSemilatticeDigraph\n \n 6.2 Regularity\n\n<\ div class=\"ContSSBlock\">\n   6.2-1 IsInRegu\ larDigraph\n &nbs\ p; 6.2-2 IsOutRegularD\ igraph\n &n\ bsp;6.2-3 IsRegularDigraph<\ /a>\n   6.2-4 IsDistanceRegularDigraph\ \n \n 6.\ 3 Connectivity and cycles\n\n\n \ ; 6.3-1 IsAcyclicDigra\ ph\n   \ 6.3-2 IsChainDigraph\n   6.3-3 IsConnectedDigraph\ \n   6.3-4 IsBiconnectedDigraph\n<\ span class=\"ContSS\">   6.3-5 IsBridgelessDigraph\n   6.3-6 IsStronglyConnectedDigraph\n<\ span class=\"ContSS\">   6.3-7 IsAperiodicDigraph\n   6.3-8 IsDirectedTree\n   6.3-9 IsUndirectedTree\n   6.3-10 IsEulerianDigraph\n  6.3-11 IsHamiltonianDigraph\n   6.3-12 IsCycleDigraph\n \n 6.4 Planarity\n\ \n\n   6.4-\ 1 IsPlanarDigraph\n   6.4-2 IsOute\ rPlanarDigraph\n \n 6.5 Homomorphisms and transformations\n\n\n   6.5-1 IsDigraphCore\n   6.5-2\ IsEdgeTransitive\n   6.5-3 IsVert\ exTransitive\n \n
\n7 Homomorphisms\n 7.1 Acting on digraphs\n\n\ \n   7.1-1 OnDigraphs\n   7.1-2 OnMultiDigraphs\n \n \ 7.2 Isomor\ phisms and canonical labellings\n\n\n   7.2-1 DigraphsUseNauty\n   7.2-2 AutomorphismGroup\n   7.2-3 BlissAutomorphismGroup\n   7.2-4 NautyAutomorphismGroup\n   7.2-5 AutomorphismGroup\n   7.2-6 AutomorphismGroup\n   7.2-7 BlissCanonicalLabelling\n   7.2-8 BlissCanonicalLabelling\n   7.2-9 BlissCanonicalDigraph\n<\ br />  7.2-10 DigraphGroup\n   7.2-11 DigraphOrbits\n   7.2-1\ 2 DigraphOrbitReps\n   7.2-13 Digr\ aphSchreierVector\n   7.2-14 Digra\ phStabilizer\n &n\ bsp; 7.2-15 IsIsomorph\ icDigraph\n \ ; 7.2-16 IsIsomorphicD\ igraph\n &n\ bsp;7.2-17 IsomorphismDigra\ phs\n   \ ;7.2-18 IsomorphismDigraphs\ \n   7.2-19 RepresentativeOutNeigh\ bours\n &nb\ sp;7.2-20 IsDigraphIsomorph\ ism\n   \ ;7.2-21 IsDigraphColouring<\ /a>\n \n 7.3\ Homomorphisms of digraphs\n\n\n &nbs\ p; 7.3-1 HomomorphismD\ igraphsFinder\n &\ nbsp; 7.3-2 DigraphHom\ omorphism\n \ ; 7.3-3 HomomorphismsD\ igraphs\n &\ nbsp;7.3-4 DigraphMonomorph\ ism\n   \ ;7.3-5 MonomorphismsDigraph\ s\n   <\ /span>7.3-6 DigraphEpimorphism\ \n   7.3-7 EpimorphismsDigraphs\n   7.3-8 DigraphEmbedding\n<\ span class=\"ContSS\">   7.3-9 EmbeddingsDigraphs\n   7.3-10 IsDigraphHomomorphism\n   7.3-11 IsDigraphEmbedding\n   7.3-12 DigraphsRespectsColouring\n   7.3-13 GeneratorsOfEndomorphismMonoid\ \n   7.3-14 DigraphColouring\n   7.3-15 DigraphGreedyColouring\n   7.3-16 DigraphWelshPowellOrder\n   7.3-17 ChromaticNumber\n   7.3-18 DigraphCore\n \n\ \n8 Finding cliques and independent sets\n 8.1 Finding cliq\ ues\n\n\n<\ br />  8.1-1 IsClique\n   8.1\ -2 CliquesFinder\n   8.1-3 Digrap\ hClique\n &\ nbsp;8.1-4 DigraphMaximalCl\ iques\n &nb\ sp;8.1-5 CliqueNumber\n \n 8.2 Finding independent sets\n\n\n &nb\ sp;8.2-1 IsIndependentSet\n   8.2-2 DigraphIndependentSet\ \n   8.2-3 DigraphMaximalIndependentSe\ ts\n \n \n9 Visualising and IO\ \n&n\ bsp;9.1 Visualising a digraph\n\n\n<\ span class=\"ContSS\">   9.1-1 Splash\n   9.1-2 DotDigraph\n <\ span class=\"nocss\">  9.1-3 DotSymmetricDigraph\n   9.1-4 DotPartialOrderDigraph\n   9.1-5 DotPreorderDigraph\n   9.\ 1-6 DotHighlightedDigraph\n \n 9.2 Reading and writing digraph\ s to a file\n\n\n   9.2-1 String\n   9.2-2 DigraphFromGraph6String\n   9.2-3 Graph6String\n   9.2-4 Di\ graphFile\n \ ; 9.2-5 ReadDigraphs\n   9.2-6 WriteDigraphs\ \n   9.2-7 IteratorFromDigraphFile\ \n   9.2-8 DigraphPlainTextLineEncoder\n   9.2-9 TournamentLineDecoder\n   9.2-10 AdjacencyMatrixUpperTriangleLin\ eDecoder\n \ 9.2-11 TCodeDecoder\n   9.2-12 PlainTextString\n   9.2-13 WritePlainTextDigraph\n   9.2-14 WriteDIMACSDigraph\n<\ /div> \n \nA \n Grape to Digraphs Command Map\n \n A.1 \n Functions to construct and modify graphs\n \n<\ /span>\n \n A.2 \n Functions to inspect graphs, vertices and edges\n \n\n \n A\ .3 \n Functions to determine regularity propertie\ s of graphs\n \n\n \n A.4 \n Some special vertex subs\ ets of a graph\n \n\n \n A.5 \n Functions to construc\ t new graphs from old\n \n\n \n A.6 \n Vertex-Colouri\ ng and Complete Subgraphs\n \n\n \n A.7 \n Automorphi\ sm groups and isomorphism testing for graphs\n \n\n\ \n \nReferences \nIndex \n1 \n \ The Digraphs package\n \n 1.1 Introduc\ tion\n\n\n\   1.1-1 Definitions\n\n \n \n2 Installing Digraphs<\ /strong>\n 2.1 For those in a hurry\n\n \n 2.2 Optional \ package dependencies\n\n\n   2.2-1 The Grape package\n\n \n 2.3 Compiling the kernel module\n\n \n 2.4 Rebuilding the documentation\n\n\n   2.4-1 DigraphsMakeDoc\n \n 2.5 Testing your installation\n\n\n   \ ;2.5-1 DigraphsTestInstall<\ /a>\n   2.5-2 DigraphsTestStandard\ \n   2.5-3 DigraphsTestExtreme\n \n \n3 Creating digraphs\n \ 3.1 Creati\ ng digraphs\n\n\n   3.1-1 IsDigraph\n <\ span class=\"nocss\">  3.1-2 IsMutableDigraph\n   3\ .1-3 IsImmutableDigraph\n   3.1-4\ IsCayleyDigraph\n   3.1-5 IsDigr\ aphWithAdjacencyFunction\n   3.1-6\ DigraphByOutNeighboursType\n   3.\ 1-7 Digraph\n &nb\ sp; 3.1-8 DigraphByAdj\ acencyMatrix\n &n\ bsp; 3.1-9 DigraphByEd\ ges\n   \ ;3.1-10 EdgeOrbitsDigraph\n   3.1-11 DigraphByInNeighbours\n   3.1-12 CayleyDigraph\ \n \n 3.2 Changing representations\n\n\n   3.2-1 AsBinaryRelation\n   3.2-2 AsDigraph\n   3.2-3 Graph\n   3.2-4 AsGraph\n   3.2-5 A\ sTransformation\n \n 3.3 New digraphs from old\n\n\n   3.3-1 D\ igraphImmutableCopy\n   3.3-2 Digr\ aphImmutableCopyIfImmutable\n   3.\ 3-3 InducedSubdigraph\n   3.3-4 Re\ ducedDigraph\n &n\ bsp; 3.3-5 MaximalSymm\ etricSubdigraph\n   3.3-6 MaximalA\ ntiSymmetricSubdigraph\n   3.3-7 U\ ndirectedSpanningForest\n   3.3-8\ QuotientDigraph\n   3.3-9 Digrap\ hReverse\n \ 3.3-10 DigraphDual\n   3.3-11 DigraphSymmetricClosure\n   3.3-12 DigraphTransitiveClosure\ \n   3.3-13 DigraphTransitiveReduc\ tion\n &nbs\ p;3.3-14 DigraphAddVertex\n   3.3-15 DigraphAddVertices\n   <\ a href=\"chap3_mj.html#X7E58CC4880627658\">3.3-16 DigraphAddEdge\n<\ span class=\"ContSS\">   3.3-17 DigraphAddEdgeOrbit\n   3.3-18 DigraphAddEdges\n   3.3-19 DigraphRemoveVertex\n   3.3-20 DigraphRemoveVertices\n   3.3-21 DigraphRemoveEdge\n   3.3-22 DigraphRemoveEdgeOrbit\n   3.3-23 DigraphRemoveEdges\n   3.3-24 DigraphRemoveLoops\n<\ br />  3.3-25 DigraphRemoveAllMultipleEdges\n   3.3-26 DigraphReverseEdges\n   3.3-27 DigraphDisjointUnion\n   3.3-28 DigraphEdgeUnion\n   3.3-29 DigraphJoin\n   3.3\ -30 DigraphCartesianProduct\n   3.\ 3-31 DigraphDirectProduct\n   3.3-\ 32 DigraphCartesianProductProjections\n   3.3-33 DigraphDirectProductProjections\n   3.3-34 LineDigraph\n <\ span class=\"nocss\">  3.3-35 LineUndirectedDigraph\n   3.3-36 DoubleDigraph\n   3.3-\ 37 BipartiteDoubleDigraph\n   3.3-\ 38 DigraphAddAllLoops\n   3.3-39 D\ istanceDigraph\n \   3.3-40 DigraphC\ losure\n &n\ bsp;3.3-41 DigraphMycielski\ an\n \n<\ span class=\"nocss\"> \ 3.4 Random digraphs\n\n\n   \ ;3.4-1 RandomDigraph\n   3.4-2 RandomMultiDigraph\ \n   3.4-3 RandomTournament\n   3.4-4 RandomLattice\n\n<\ div class=\"ContSect\"> 3.5 St\ andard examples\n\n\n   3.5-1 ChainDigraph\n\   3.5-2 CompleteDigraph\n   3.5-3 CompleteBipartiteDigraph\n <\ span class=\"nocss\">  3.5-4 CompleteMultipartiteDigraph\n  3.5-5 CycleDigraph\n   3\ .5-6 EmptyDigraph\n   3.5-7 Johnso\ nDigraph\n \ 3.5-8 PetersenGraph\n   3.5-9 GeneralisedPetersenGraph<\ /a>\n \n \n4 Operators\n <\ a href=\"chap4_mj.html#X84E0B5B88358C96B\">4.1 Operato\ rs for digraphs\n\n\n   4.1-1 IsSubdigraph\n\   4.1-2 IsUndirectedSpanningTree\n \n\n\ 5 Attributes and operations\n\ 5.1 Vertices and edges\ \n\n\n   5.1-1 DigraphVertices\n   5.1-2\ DigraphNrVertices\n   5.1-3 Digra\ phEdges\n &\ nbsp;5.1-4 DigraphNrEdges\n   5.1-5 DigraphSinks\n\   5.1-6 DigraphSources\n   5.1-7 DigraphTopologicalSort\n   5.1-8 DigraphVertexLabel\n   5.1-9 DigraphVertexLabels\n   5.1-10 DigraphEdgeLabel\n  5.1-11 DigraphEdgeLabels\n <\ span class=\"nocss\">  5.1-12 DigraphInEdges\n   5.\ 1-13 DigraphOutEdges\n   5.1-14 Is\ DigraphEdge\n &nb\ sp; 5.1-15 IsMatching<\ /a>\n   5.1-16 DigraphMaximalMatching<\ /a>\n   5.1-17 DigraphMaximumMatching<\ /a>\n \n 5.2\ Neighbours and degree\n\n\n &n\ bsp;5.2-1 AdjacencyMatrix\n   5.2-2 CharacteristicPolynomial<\ /a>\n   5.2-3 BooleanAdjacencyMatrix\n   5.2-4 DigraphAdjacencyFunction<\ /a>\n   5.2-5 DigraphRange\ \n   5.2-6 OutNeighbours\n   5.2-7 InNeighbours\n   5.2-8 OutDegrees\n   5.2-9 InDegrees\n   5.2-10 Out\ DegreeOfVertex\n \   5.2-11 OutNeigh\ boursOfVertex\n &\ nbsp; 5.2-12 InDegreeO\ fVertex\n &\ nbsp;5.2-13 InNeighboursOfV\ ertex\n &nb\ sp;5.2-14 DigraphLoops<\ /span>\n   \ 5.2-15 PartialOrderDigraphMeetOfVe\ rtices\n &n\ bsp;5.2-16 DegreeMatrix\ \n   5.2-17 LaplacianMatrix\ \n \n 5.3 Reachability and connectivity\n\n\n &nb\ sp;5.3-1 DigraphDiameter\n   5.3-2 DigraphShortestDistance\n   5.3-3 DigraphShortestDistances\n   5.3-4 DigraphLongestDistanceFro\ mVertex\n &\ nbsp;5.3-5 DigraphDistanceS\ et\n   \ 5.3-6 DigraphGirth\n   5.3-7 DigraphOddGirth\n   5.3-8 DigraphUndirectedGirth\n   5.3-9 DigraphConnectedComponents\n   5.3-10 DigraphConnectedComponent\ \n   5.3-11 DigraphStronglyConnectedComponent\ s\n   <\ /span>5.3-12 DigraphStronglyConnec\ tedComponent\n &n\ bsp; 5.3-13 DigraphBic\ omponents\n \ ; 5.3-14 ArticulationP\ oints\n &nb\ sp;5.3-15 Bridges\n   5.3-16 StrongOrientation\n   5.3-17 DigraphPeriod\n   5.3-18 DigraphFloydWarshall\n   5.3-19 IsReachable\n\   5.3-20 DigraphPath\n   5.3-21 DigraphShortestPath\n   \ 5.3-22 IteratorOfPaths\n   5.3-23 \ DigraphAllSimpleCircuits\n   5.3-2\ 4 DigraphLongestSimpleCircuit\n   \ 5.3-25 DigraphLayers\n   5.3-26 Di\ graphDegeneracy\n   5.3-27 Digraph\ DegeneracyOrdering\n   5.3-28 Hami\ ltonianPath\n &nb\ sp; 5.3-29 NrSpanningT\ rees\n &nbs\ p;5.3-30 DigraphDijkstra\n \n 5.4 <\ span class=\"Heading\">Cayley graphs of groups\n\n\n &n\ bsp;5.4-1 GroupOfCayleyDigr\ aph\n   \ ;5.4-2 GeneratorsOfCayleyDi\ graph\n \n 5.5 Associated semigroups\n\n\n &\ nbsp; 5.5-1 AsSemigrou\ p\n   <\ /span>5.5-2 AsSemigroup\ \n \n 5.6 Planarity\n\n\n   5.6-1 KuratowskiPlanarSubdigraph\n\   5.6-2 KuratowskiOuterPlanarSubdigraph<\ /span>\n   \ 5.6-3 PlanarEmbedding\n\   5.6-4 OuterPlanarEmbedding\n   5.6-5 SubdigraphHomeomorphicToK23\ \n \n \n6 Properties of digraphs\n 6.1 Edge\ properties\n\n\n   6.1-1 DigraphHasLoops\n<\ br />  6.1-2 IsAntiSymmetricDigraph\n  6.1-3 IsBipartiteDigraph\n <\ span class=\"nocss\">  6.1-4 IsCompleteBipartiteDigraph\n   6.1-5 IsCompleteDigraph\n   6.1-6 IsCompleteMultipartiteDigraph\n   6.1-7 IsEmptyDigraph\n   \ 6.1-8 IsEquivalenceDigraph\n   6.1\ -9 IsFunctionalDigraph\n   6.1-10 \ IsMultiDigraph\n \   6.1-11 IsReflex\ iveDigraph\n &nbs\ p; 6.1-12 IsSymmetricD\ igraph\n &n\ bsp;6.1-13 IsTournament\ \n   6.1-14 IsTransitiveDigraph\n   6.1-15 IsPreorderDigraph\ \n   6.1-16 IsPartialOrderDigraph\ \n   6.1-17 IsMeetSemilatticeDigraph\n \n 6.2 Regularity\n\n\n\   6.2-1 IsInRegularDigraph\n   6.2-2 IsOutRegularDigraph\n   6.2-3 IsRegularDigraph\n   6.2-4 IsDistanceRegularDigraph\n\n<\ div class=\"ContSect\"> 6.3 Co\ nnectivity and cycles\n\n\n   6.3-1 IsAcyclicDigraph\n   6.3-2 IsChainDigraph\n   6.3-3 IsConnectedDigraph\n  6.3-4 IsBiconnectedDigraph\n   6.3-5 IsBridgelessDigraph\n   6.3-6 IsStronglyConnectedDigraph\n \   6.3-7 IsAperiodicDigraph\n   6.3-8 IsDirectedTree\n   6.3-9\ IsUndirectedTree\n   6.3-10 IsEul\ erianDigraph\n &n\ bsp; 6.3-11 IsHamilton\ ianDigraph\n &nbs\ p; 6.3-12 IsCycleDigra\ ph\n
\n<\ span class=\"nocss\"> \ 6.4 Planarity\n\n\n   6.4-1 IsPlanarDigraph\ \n   6.4-2 IsOuterPlanarDigraph\n<\ /div> \n 6.5 Homomorphisms and transformations\n\n\n &nb\ sp;6.5-1 IsDigraphCore<\ /span>\n   \ 6.5-2 IsEdgeTransitive\ \n   6.5-3 IsVertexTransitive\n \n \n7 Homomorphisms\n 7.1 Acting on digraphs\ \n\n\n   7.1-1 OnDigraphs\n   7.1-\ 2 OnMultiDigraphs\n \n 7.2 Isomorphisms and canonical labellin\ gs\n\n\n  7.2-1 DigraphsUseNauty\n   7.2-2 AutomorphismGroup\n   7.\ 2-3 BlissAutomorphismGroup\n   7.2\ -4 NautyAutomorphismGroup\n   7.2-\ 5 AutomorphismGroup\n   7.2-6 Auto\ morphismGroup\n &\ nbsp; 7.2-7 BlissCanon\ icalLabelling\n &\ nbsp; 7.2-8 BlissCanon\ icalLabelling\n &\ nbsp; 7.2-9 BlissCanon\ icalDigraph\n &nb\ sp; 7.2-10 DigraphGrou\ p\n   <\ /span>7.2-11 DigraphOrbits\n   7.2-12 DigraphOrbitReps\n<\ span class=\"ContSS\">   7.2-13 DigraphSchreierVector\n<\ span class=\"ContSS\">   7.2-14 DigraphStabilizer\n   7.2-15 IsIsomorphicDigraph\n   7.2-16 IsIsomorphicDigraph\n   7.2-17 IsomorphismDigraphs\n   7.2-18 IsomorphismDigraphs\n   7.2-19 RepresentativeOutNeighbours\n   7.2-20 IsDigraphIsomorphism\n   7.2-21 IsDigraphColouring\n \n 7.3 Homom\ orphisms of digraphs\n\n\n   7.3-1 HomomorphismDigraphsFinder\n   7.3-2 DigraphHomomorphism\n   7.3-3 HomomorphismsDigraphs\n   7.3-4 DigraphMonomorphism\n   7.3-5 MonomorphismsDigraphs\n   7.3-6 DigraphEpimorphism\n   7.3-7 EpimorphismsDigraphs\n\   7.3-8 DigraphEmbedding\n <\ span class=\"nocss\">  7.3-9 EmbeddingsDigraphs\n   7.3-10 IsDigraphHomomorphism\n   \ 7.3-11 IsDigraphEmbedding\n   7.3-\ 12 DigraphsRespectsColouring\n   7\ .3-13 GeneratorsOfEndomorphismMonoid\n \   7.3-14 DigraphColouring\n   7.3-15 DigraphGreedyColouring\n   7.3-16 DigraphWelshPowellOrder\n   7.3-17 ChromaticNumber\n   7.3-\ 18 DigraphCore\n \n \n8 Finding cliqu\ es and independent sets\n 8.1 Finding cliques\n\n\n &nbs\ p; 8.1-1 IsClique<\ /span>\n   \ 8.1-2 CliquesFinder\n   8.1-3 DigraphClique\n   8.1-4 DigraphMaximalCliques\n   8.1-5 CliqueNumber\n \n 8.2 Finding inde\ pendent sets\n\n\n   8.2-1 IsIndependentSet\n   8.2-2 DigraphIndependentSet\n<\ br />  8.2-3 DigraphMaximalIndependentSets\n \n\n9 Visualising and IO\n 9.1 Visualising a digraph\ \n\n\n   9.1-1 Splash\n   9.1-2 DotDigra\ ph\n   \ 9.1-3 DotSymmetricDigraph\n   9.1-4 DotPartialOrderDigraph\n   9.1-5 DotPreorderDigraph\n   9.1-6 DotHighlightedDigraph\n \n 9.2 Reading and writing digraphs to a file\n\n<\ div class=\"ContSSBlock\">\n   9.2-1 String\n   9.2-2 DigraphFromGraph6String\n   9.2-3 Graph6String\n\   9.2-4 DigraphFile\n   9.2-5 ReadDigraphs\n\   9.2-6 WriteDigraphs\n   9.2-7 IteratorFromDigraphFile\n   9.2-8 DigraphPlainTextLineEncoder\n \   9.2-9 TournamentLineDecoder\n   9.2-10 AdjacencyMatrixUpperTriangleLineDecoder\n   9.2-11 TCodeDecoder\n<\ br />  9.2-12 PlainTextString\n   9.2-13 WritePlainTextDigraph\n   9.2-14 WriteDIMACSDigraph\n \n \nA \n Grape to Digraphs Command Map\n \n A.1 \n Functions to con\ struct and modify graphs\n \n\n \n A.2 \n Functions t\ o inspect graphs, vertices and edges\n \n\n \n <\ a href=\"chapA_mj.html#X78A5A34783C70CD7\">A.3 \n \ Functions to determine regularity properties of graphs\n \n\n \n A.4 \n Some special vertex subsets of a graph\n \n<\ /span>\n \n A.5 \n Functions to construct new graphs from old\n \ \n\n \n A.6 \n Vertex-Colouring and Complete Subgraphs\n \n\n \n A.\ 7 \n Automorphism groups and isomorphism testing \ for graphs\n \n\n \n \n\ References\nIndex\n" ) gap> gap> ───────────────────────────────────────────────────────────────────────────── Loading datastructures 0.2.5 (datastructures - GAP Data Structures) by Markus Pfeiffer (http://www.morphism.de/~markusp), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/), and Steve Linton (http://sl4.host.cs.st-andrews.ac.uk/). Homepage: https://gap-packages.github.io/datastructures Report issues at https://github.com/gap-packages/datastructures/issues ───────────────────────────────────────────────────────────────────────────── ────────────────────────────────────────────────────────────────────��──────── Loading orb 4.8.2 (Methods to enumerate orbits) by Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske). maintained by: Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske), and Max Horn (https://www.quendi.de/math). Homepage: https://gap-packages.github.io/orb Report issues at https://github.com/gap-packages/orb/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.3 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape Report issues at https://github.com/gap-packages/grape/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.3.0 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (https://jdbm.me), Michael Torpey (https://mtorpey.github.io), Maria Tsalakou (https://mariatsalakou.github.io/), and Wilf Wilson (http://wilf.me). with contributions by: Stuart Burrell (sb235@st-andrews.ac.uk), Reinis Cirpons (rc234@st-andrews.ac.uk), Luke Elliott (le27@st-andrews.ac.uk), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk), Markus Pfeiffer (https://www.morphism.de/~markusp), Christopher Russell (cr66@st-andrews.ac.uk), Finn Smith (fls3@st-andrews.ac.uk), and Murray Whyte (mw231@st-andrews.ac.uk). maintained by: James Mitchell (https://jdbm.me) and Wilf Wilson (http://wilf.me). Homepage: https://gap-packages.github.io/Digraphs Report issues at https://github.com/gap-packages/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── #I File: /builddir/build/BUILD/digraphs-1.3.0/../pkg/digraphs-1.3.0/doc/manual.lab written. gap> gap> + rm -fr ../pkg + RPM_EC=0 ++ jobs -p + exit 0 Executing(%install): /bin/sh -e /var/tmp/rpm-tmp.jfJo9d + umask 022 + cd /builddir/build/BUILD + '[' /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64 '!=' / ']' + rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64 ++ dirname /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64 + mkdir -p /builddir/build/BUILDROOT + mkdir /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64 + cd digraphs-1.3.0 + mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.0/doc + cp -a bin data gap tst VERSIONS PackageInfo.g init.g makedoc.g read.g /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.0 + cp -p doc/digraphs.bib doc/lefttoc.css doc/manual.css doc/nocolorprompt.css doc/ragged.css doc/times.css doc/toggless.css doc/chap0.html doc/chap0_mj.html doc/chap1.html doc/chap1_mj.html doc/chap2.html doc/chap2_mj.html doc/chap3.html doc/chap3_mj.html doc/chap4.html doc/chap4_mj.html doc/chap5.html doc/chap5_mj.html doc/chap6.html doc/chap6_mj.html doc/chap7.html doc/chap7_mj.html doc/chap8.html doc/chap8_mj.html doc/chap9.html doc/chap9_mj.html doc/chapA.html doc/chapA_mj.html doc/chapBib.html doc/chapBib_mj.html doc/chapInd.html doc/chapInd_mj.html doc/chooser.html doc/manual.js doc/rainbow.js doc/toggless.js doc/manual.lab doc/manual.pdf doc/manual.six doc/chap0.txt doc/chap1.txt doc/chap2.txt doc/chap3.txt doc/chap4.txt doc/chap5.txt doc/chap6.txt doc/chap7.txt doc/chap8.txt doc/chap9.txt doc/chapA.txt doc/chapBib.txt doc/chapInd.txt doc/attr.xml doc/cliques.xml doc/constructors.xml doc/digraph.xml doc/display.xml doc/examples.xml doc/grahom.xml doc/grape.xml doc/io.xml doc/isomorph.xml doc/labels.xml doc/main.xml doc/oper.xml doc/orbits.xml doc/planar.xml doc/prop.xml doc/utils.xml doc/z-appA.xml doc/z-chap0.xml doc/z-chap1.xml doc/z-chap2.xml doc/z-chap3.xml doc/z-chap4.xml doc/z-chap5.xml doc/z-chap6.xml doc/z-chap8.xml doc/z-chap9.xml doc/z-title.xml /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.0/doc + /usr/lib/rpm/find-debuginfo.sh -j4 --strict-build-id -m -i --build-id-seed 1.3.0-2.fc33 --unique-debug-suffix -1.3.0-2.fc33.riscv64 --unique-debug-src-base gap-pkg-digraphs-1.3.0-2.fc33.riscv64 --run-dwz --dwz-low-mem-die-limit 10000000 --dwz-max-die-limit 50000000 -S debugsourcefiles.list /builddir/build/BUILD/digraphs-1.3.0 explicitly decompress any DWARF compressed ELF sections in /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.0/bin/riscv64-redhat-linux-gnu-default64-kv7/digraphs.so extracting debug info from /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.0/bin/riscv64-redhat-linux-gnu-default64-kv7/digraphs.so original debug info size: 5608kB, size after compression: 5460kB /usr/lib/rpm/sepdebugcrcfix: Updated 1 CRC32s, 0 CRC32s did match. 983 blocks + /usr/lib/rpm/check-buildroot + /usr/lib/rpm/redhat/brp-ldconfig + /usr/lib/rpm/brp-compress + /usr/lib/rpm/redhat/brp-strip-lto /usr/bin/strip + /usr/lib/rpm/brp-strip-static-archive /usr/bin/strip + /usr/lib/rpm/redhat/brp-python-bytecompile '' 1 0 + /usr/lib/rpm/brp-python-hardlink + /usr/lib/rpm/redhat/brp-mangle-shebangs *** WARNING: ./usr/src/debug/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/extern/bliss-0.73/graph.cc is executable but has no shebang, removing executable bit Executing(%check): /bin/sh -e /var/tmp/rpm-tmp.cFXUYd + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.3.0 + export LC_ALL=C.UTF-8 + LC_ALL=C.UTF-8 + cd tst + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/lib/gap;/usr/lib/gap' ┌───────┐ GAP 4.11.0 of 29-Feb-2020 │ GAP │ https://www.gap-system.org └───────┘ Architecture: riscv64-redhat-linux-gnu-default64-kv7 Configuration: gmp 6.2.0, GASMAN, readline Loading the library and packages ... #I autpgrp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I alnuth package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I crisp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I ctbllib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I factint package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I fga package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I irredsol package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I laguna package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polenta package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polycyclic package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I resclasses package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I sophus package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I tomlib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') Packages: GAPDoc 1.6.4, IO 4.7.0, PrimGrp 3.4.1, SmallGrp 1.4.1, TransGrp 3.0 Try '??help' for help. See also '?copyright', '?cite' and '?authors' gap> ───────────────────────────────────────────────────────────────────────────── Loading datastructures 0.2.5 (datastructures - GAP Data Structures) by Markus Pfeiffer (http://www.morphism.de/~markusp), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/), and Steve Linton (http://sl4.host.cs.st-andrews.ac.uk/). Homepage: https://gap-packages.github.io/datastructures Report issues at https://github.com/gap-packages/datastructures/issues ───────────────────────────────────────────────────────────────────────────── ──────────────────────────────────────────────────────��────────────────────── Loading orb 4.8.2 (Methods to enumerate orbits) by Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske). maintained by: Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske), and Max Horn (https://www.quendi.de/math). Homepage: https://gap-packages.github.io/orb Report issues at https://github.com/gap-packages/orb/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.3 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape Report issues at https://github.com/gap-packages/grape/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.3.0 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (https://jdbm.me), Michael Torpey (https://mtorpey.github.io), Maria Tsalakou (https://mariatsalakou.github.io/), and Wilf Wilson (http://wilf.me). with contributions by: Stuart Burrell (sb235@st-andrews.ac.uk), Reinis Cirpons (rc234@st-andrews.ac.uk), Luke Elliott (le27@st-andrews.ac.uk), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk), Markus Pfeiffer (https://www.morphism.de/~markusp), Christopher Russell (cr66@st-andrews.ac.uk), Finn Smith (fls3@st-andrews.ac.uk), and Murray Whyte (mw231@st-andrews.ac.uk). maintained by: James Mitchell (https://jdbm.me) and Wilf Wilson (http://wilf.me). Homepage: https://gap-packages.github.io/Digraphs Report issues at https://github.com/gap-packages/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: testinstall.tst msecs: 5305 true gap> Digraphs package: standard/attr.tst msecs: 32151 true gap> Digraphs package: standard/cliques.tst msecs: 8428 true gap> Digraphs package: standard/constructors.tst msecs: 1058 true gap> Digraphs package: standard/display.tst msecs: 217 true gap> Digraphs package: standard/examples.tst msecs: 251 true gap> Digraphs package: standard/grahom.tst msecs: 8549 true gap> Digraphs package: standard/grape.tst msecs: 311 true gap> Digraphs package: standard/io.tst msecs: 25051 true gap> Digraphs package: standard/isomorph.tst msecs: 8073 true gap> Digraphs package: standard/labels.tst msecs: 98 true gap> Digraphs package: standard/oper.tst msecs: 9198 true gap> Digraphs package: standard/orbits.tst msecs: 36 true gap> Digraphs package: standard/planar.tst msecs: 830 true gap> Digraphs package: standard/prop.tst msecs: 45205 true gap> gap> + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/lib/gap;/usr/lib/gap' ┌───────┐ GAP 4.11.0 of 29-Feb-2020 │ GAP │ https://www.gap-system.org └───────┘ Architecture: riscv64-redhat-linux-gnu-default64-kv7 Configuration: gmp 6.2.0, GASMAN, readline Loading the library and packages ... #I autpgrp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I alnuth package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I crisp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I ctbllib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I factint package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I fga package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I irredsol package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I laguna package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polenta package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polycyclic package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I resclasses package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I sophus package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I tomlib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') Packages: GAPDoc 1.6.4, IO 4.7.0, PrimGrp 3.4.1, SmallGrp 1.4.1, TransGrp 3.0 Try '??help' for help. See also '?copyright', '?cite' and '?authors' gap> ─────────────��─────────────��──────────────────────────��────────────────────── Loading datastructures 0.2.5 (datastructures - GAP Data Structures) by Markus Pfeiffer (http://www.morphism.de/~markusp), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/), and Steve Linton (http://sl4.host.cs.st-andrews.ac.uk/). Homepage: https://gap-packages.github.io/datastructures Report issues at https://github.com/gap-packages/datastructures/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading orb 4.8.2 (Methods to enumerate orbits) by Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske). maintained by: Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske), and Max Horn (https://www.quendi.de/math). Homepage: https://gap-packages.github.io/orb Report issues at https://github.com/gap-packages/orb/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.3 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape Report issues at https://github.com/gap-packages/grape/issues ───────────────────────────────────────────────────────────────────────────── ─────────────��─────────────��──────────────────────────��────────────────────── Loading Digraphs 1.3.0 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (https://jdbm.me), Michael Torpey (https://mtorpey.github.io), Maria Tsalakou (https://mariatsalakou.github.io/), and Wilf Wilson (http://wilf.me). with contributions by: Stuart Burrell (sb235@st-andrews.ac.uk), Reinis Cirpons (rc234@st-andrews.ac.uk), Luke Elliott (le27@st-andrews.ac.uk), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk), Markus Pfeiffer (https://www.morphism.de/~markusp), Christopher Russell (cr66@st-andrews.ac.uk), Finn Smith (fls3@st-andrews.ac.uk), and Murray Whyte (mw231@st-andrews.ac.uk). maintained by: James Mitchell (https://jdbm.me) and Wilf Wilson (http://wilf.me). Homepage: https://gap-packages.github.io/Digraphs Report issues at https://github.com/gap-packages/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: standard/digraph.tst msecs: 6492 gap> + find /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.0 -size 0 -delete + RPM_EC=0 ++ jobs -p + exit 0 Processing files: gap-pkg-digraphs-1.3.0-2.fc33.riscv64 Executing(%doc): /bin/sh -e /var/tmp/rpm-tmp.cYuHbe + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.3.0 + DOCDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/share/doc/gap-pkg-digraphs + export LC_ALL=C + LC_ALL=C + export DOCDIR + /usr/bin/mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/share/doc/gap-pkg-digraphs + cp -pr CHANGELOG.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/share/doc/gap-pkg-digraphs + cp -pr README.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/share/doc/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Executing(%license): /bin/sh -e /var/tmp/rpm-tmp.1xF5Ib + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.3.0 + LICENSEDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/share/licenses/gap-pkg-digraphs + export LC_ALL=C + LC_ALL=C + export LICENSEDIR + /usr/bin/mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/share/licenses/gap-pkg-digraphs + cp -pr GPL /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/share/licenses/gap-pkg-digraphs + cp -pr LICENSE /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64/usr/share/licenses/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Provides: bundled(bliss) = 0.73 gap-pkg-digraphs = 1.3.0-2.fc33 gap-pkg-digraphs(riscv-64) = 1.3.0-2.fc33 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Requires: ld-linux-riscv64-lp64d.so.1()(64bit) ld-linux-riscv64-lp64d.so.1(GLIBC_2.27)(64bit) libc.so.6()(64bit) libc.so.6(GLIBC_2.27)(64bit) libgcc_s.so.1()(64bit) libgcc_s.so.1(GCC_3.0)(64bit) libgcc_s.so.1(GCC_3.3.1)(64bit) libgcc_s.so.1(GCC_3.4)(64bit) libgcc_s.so.1(GCC_4.2.0)(64bit) libm.so.6()(64bit) libplanarity.so.0()(64bit) libstdc++.so.6()(64bit) libstdc++.so.6(CXXABI_1.3)(64bit) libstdc++.so.6(CXXABI_1.3.9)(64bit) libstdc++.so.6(GLIBCXX_3.4)(64bit) rtld(GNU_HASH) Recommends: gap-pkg-grape(riscv-64) gap-pkg-nautytracesinterface(riscv-64) Processing files: gap-pkg-digraphs-doc-1.3.0-2.fc33.noarch Provides: gap-pkg-digraphs-doc = 1.3.0-2.fc33 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Processing files: gap-pkg-digraphs-debugsource-1.3.0-2.fc33.riscv64 Provides: gap-pkg-digraphs-debugsource = 1.3.0-2.fc33 gap-pkg-digraphs-debugsource(riscv-64) = 1.3.0-2.fc33 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Processing files: gap-pkg-digraphs-debuginfo-1.3.0-2.fc33.riscv64 Provides: debuginfo(build-id) = ebe60adf0a02a4f2925d111318e3ea1a41c7c5f1 gap-pkg-digraphs-debuginfo = 1.3.0-2.fc33 gap-pkg-digraphs-debuginfo(riscv-64) = 1.3.0-2.fc33 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Recommends: gap-pkg-digraphs-debugsource(riscv-64) = 1.3.0-2.fc33 Checking for unpackaged file(s): /usr/lib/rpm/check-files /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64 Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debugsource-1.3.0-2.fc33.riscv64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-1.3.0-2.fc33.riscv64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debuginfo-1.3.0-2.fc33.riscv64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-doc-1.3.0-2.fc33.noarch.rpm Executing(%clean): /bin/sh -e /var/tmp/rpm-tmp.UUAREb + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.3.0 + /usr/bin/rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.0-2.fc33.riscv64 + RPM_EC=0 ++ jobs -p + exit 0 Child return code was: 0