&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 := 2024, root := ~, start := 1999, stop := 2023 ), rec( attributes := rec( ), content := [ rec( content := "Version 1.3.1", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Version", next := 2063, root := ~, start := 2027, stop := 2062 ), 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 := 2129, root := ~, start := 2092, stop := 2128 ), 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 := 2190, root := ~, start := 2134, stop := 2189 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2202, root := ~, start := 2067, stop := 2201 ), 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 := 2280, root := ~, start := 2238, stop := 2279 ), 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 := 2332, root := ~, start := 2285, stop := 2331 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2344, root := ~, start := 2206, stop := 2343 ), 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 := 2410, root := ~, start := 2378, stop := 2409 ), 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 := 2451, root := ~, start := 2415, stop := 2450 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2463, root := ~, start := 2348, stop := 2462 ), 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 := 2531, root := ~, start := 2494, stop := 2530 ), 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 := 2582, root := ~, start := 2536, stop := 2581 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2594, root := ~, start := 2467, stop := 2593 ), 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 := 2659, root := ~, start := 2625, stop := 2658 ), 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 := 2699, root := ~, start := 2664, stop := 2698 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2711, root := ~, start := 2598, stop := 2710 ), 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 := 2746, root := ~, start := 2715, stop := 2745 ), 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 := 2781, root := ~, start := 2750, stop := 2780 ), 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 := 2814, root := ~, start := 2785, stop := 2813 ), 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 := 2843, root := ~, start := 2818, stop := 2842 ), 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 := 2885, root := ~, start := 2847, stop := 2884 ), 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 := 2921, root := ~, start := 2889, stop := 2920 ), 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 := 2955, root := ~, start := 2925, stop := 2954 ), 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 := 2986, root := ~, start := 2959, stop := 2985 ), 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 := 3019, root := ~, start := 2990, stop := 3018 ), 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 := 3169, root := ~, start := 3165, stop := 3168 ), 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 := 3358, root := ~, start := 3272, stop := 3357 ), 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 := 3499, root := ~, start := 3023, stop := 3498 ), 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 := 3637, root := ~, start := 3503, stop := 3636 ), 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 := 3932, root := ~, start := 3641, stop := 3931 ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "TitlePage", next := 3946, root := ~, start := 1985, stop := 3945 ), rec( attributes := rec( ), content := 0, count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "TableOfContents", next := 3967, root := ~, start := 3949, stop := 3966 ), 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 := 4068, root := ~, start := 4019, stop := 4067 ), 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 := 4128, root := ~, start := 4071, stop := 4127 ), 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 := 4170, root := ~, start := 4139, stop := 4169 ), rec( content := "\n\n This is the manual\ for version 1.3.1 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 := 4346, root := ~, start := 4320, stop := 4345 ), 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 := 4384, root := ~, start := 4351, stop := 4383 ), 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 := 4420, root := ~, start := 4389, stop := 4419 ), 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 := 4460, root := ~, start := 4425, stop := 4459 ), 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 := 4493, root := ~, start := 4465, stop := 4492 ) ], count := [ 1, 1, 0, 4 ], mathmode := "Text", name := "List", next := 4503, root := ~, start := 4309, stop := 4502 ), 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 := 4586, root := ~, start := 4558, stop := 4585 ), 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 := 4619, root := ~, start := 4591, stop := 4618 ), 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 := 4650, root := ~, start := 4624, stop := 4649 ), 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 := 4677, root := ~, start := 4655, stop := 4676 ), 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 := 4717, root := ~, start := 4682, stop := 4716 ), 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 := 4751, root := ~, start := 4722, stop := 4750 ), 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 := 4783, root := ~, start := 4756, stop := 4782 ), 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 := 4816, root := ~, start := 4788, stop := 4815 ), 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 := 4849, root := ~, start := 4821, stop := 4848 ), 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 := 4880, root := ~, start := 4854, stop := 4879 ) ], count := [ 1, 1, 0, 16 ], mathmode := "Text", name := "List", next := 4890, root := ~, start := 4547, stop := 4889 ), 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 := 5152, root := ~, start := 5148, stop := 5151 ), 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 := 5334, root := ~, start := 5315, stop := 5333 ), 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 := 5433, root := ~, start := 5412, stop := 5432 ), 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 := 5489, root := ~, start := 5485, stop := 5488 ), 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 := 5528, root := ~, start := 5510, stop := 5527 ), 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 := 5890, root := ~, start := 5868, stop := 5889 ), 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 := 6028, root := ~, start := 5977, stop := 6027 ), 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 := 6055, root := ~, start := 6051, stop := 6054 ), 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 := 6536, root := ~, start := 6490, stop := 6535 ), 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 := 7167, root := ~, start := 7137, stop := 7166 ), 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 := 7270, root := ~, start := 7266, stop := 7269 ), 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 := 7292, root := ~, start := 7278, stop := 7291 ), 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 := 7315, root := ~, start := 7293, stop := 7314 ), 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 := 7360, root := ~, start := 7333, stop := 7359 ), 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 := 7402, root := ~, start := 7392, stop := 7401 ), 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 := 7436, root := ~, start := 7426, stop := 7435 ), 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 := 7485, root := ~, start := 7461, stop := 7484 ), 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 := 7515, root := ~, start := 7502, stop := 7514 ), 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 := 7532, root := ~, start := 7520, stop := 7531 ), 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 := 7626, root := ~, start := 7615, stop := 7625 ), 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 := 7681, root := ~, start := 7670, stop := 7680 ), 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 := 7766, root := ~, start := 7747, stop := 7765 ), 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 := 7849, root := ~, start := 7845, stop := 7848 ), 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 := 7877, root := ~, start := 7857, stop := 7876 ), 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 := 7989, root := ~, start := 7943, stop := 7988 ), 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 := 8024, root := ~, start := 8014, stop := 8023 ), 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 := 8046, root := ~, start := 8036, stop := 8045 ), 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 := 8071, root := ~, start := 8057, stop := 8070 ), 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 := 8101, root := ~, start := 8075, stop := 8100 ), 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 := 8200, root := ~, start := 8177, stop := 8199 ), 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 := 8248, root := ~, start := 8226, stop := 8247 ), 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 := 8277, root := ~, start := 8256, stop := 8276 ), 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 := 8366, root := ~, start := 8350, stop := 8365 ), 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 := 8373, root := ~, start := 8369, stop := 8372 ), 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 := 8396, root := ~, start := 8383, stop := 8395 ), 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 := 8468, root := ~, start := 8416, stop := 8467 ), 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 := 8491, root := ~, start := 8481, stop := 8490 ), 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 := 8603, root := ~, start := 8557, stop := 8602 ), 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 := 8679, root := ~, start := 8669, stop := 8678 ), 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 := 8706, root := ~, start := 8696, stop := 8705 ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8710, root := ~, start := 8654, stop := 8709 ), 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 := 8775, root := ~, start := 8765, stop := 8774 ), 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 := 8800, root := ~, start := 8792, stop := 8799 ), 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 := 8826, root := ~, start := 8812, stop := 8825 ), 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 := 8947, root := ~, start := 8938, stop := 8946 ), 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 := 8972, root := ~, start := 8951, stop := 8971 ), 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 := 9044, root := ~, start := 9040, stop := 9043 ), rec( content := "\n\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 1 ], mathmode := "Text", name := "Subsection", next := 9061, root := ~, start := 7105, stop := 9060 ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 9074, root := ~, start := 4130, stop := 9073 ), rec( content := "\n", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 9085, root := ~, start := 3978, stop := 9084 ), 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 := 9184, root := ~, start := 9144, stop := 9183 ), 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 := 9270, root := ~, start := 9231, stop := 9269 ), 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 := 9360, root := ~, start := 9356, stop := 9359 ), 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 \ 4.10.0 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 := 9626, root := ~, start := 9590, stop := 9625 ), 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 := 9631, root := ~, start := 9627, stop := 9630 ), 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 := 9922, root := ~, start := 9761, stop := 9921 ), 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 := 10202, root := ~, start := 9930, stop := 10201 ), 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 := 10331, root := ~, start := 10210, stop := 10330 ), 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 := 10383, root := ~, start := 10354, stop := 10382 ), 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 := 10523, root := ~, start := 10489, stop := 10522 ), 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 := 10845, root := ~, start := 10811, stop := 10844 ), 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 := 10874, root := ~, start := 10339, stop := 10873 ), 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 := 10932, root := ~, start := 10903, stop := 10931 ), 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 := 11491, root := ~, start := 11479, stop := 11490 ), 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 := 11520, root := ~, start := 10888, stop := 11519 ), 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-1.3.1.tar.gz", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "F", next := 11599, root := ~, start := 11572, stop := 11598 ), 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://digraphs.github.\ io/Digraphs/", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "URL", next := 11694, root := ~, start := 11613, stop := 11693 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], name := "Item", next := 11709, root := ~, start := 11528, stop := 11708 ), 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-1.3.1", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "F", next := 11824, root := ~, start := 11804, stop := 11823 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 19 ], name := "Item", next := 11839, root := ~, start := 11717, stop := 11838 ), 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 := 11883, root := ~, start := 11873, stop := 11882 ), 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 := 11968, root := ~, start := 11958, stop := 11967 ), 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 := 11980, root := ~, start := 11970, stop := 11979 ), 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-1.3.1", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12039, root := ~, start := 12019, stop := 12038 ), 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 := 12059, root := ~, start := 12049, stop := 12058 ), rec( content := " directo\ ry.\n ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], name := "Item", next := 12084, root := ~, start := 11847, stop := 12083 ), 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-1.3.1", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "F", next := 12201, root := ~, start := 12177, stop := 12200 ), 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 := 12274, root := ~, start := 12226, stop := 12273 ), 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 := 12392, root := ~, start := 12351, stop := 12391 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], name := "Item", next := 12407, root := ~, start := 12092, stop := 12406 ), 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 := 12480, root := ~, start := 12470, stop := 12479 ), 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 := 12516, root := ~, start := 12415, stop := 12515 ), 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 := 12575, root := ~, start := 12544, stop := 12574 ), rec( content := "\n \ ", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 29 ], name := "Item", next := 12589, root := ~, start := 12524, stop := 12588 ) ], count := [ 2, 1, 0, 6 ], mathmode := "Text", name := "List", next := 12601, root := ~, start := 9747, stop := 12600 ), 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 := 12768, root := ~, start := 12729, stop := 12767 ), rec( content := ".\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 12782, root := ~, start := 9188, stop := 12781 ), 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 := 13047, root := ~, start := 12999, stop := 13046 ), 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 := 13342, root := ~, start := 13338, stop := 13341 ), 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 := 13510, root := ~, start := 13474, stop := 13509 ), 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 := 13655, root := ~, start := 13636, stop := 13654 ) , 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 := 13686, root := ~, start := 13630, stop := 13685 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Attr := "AsGraph" ), content := 0, count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "Ref", next := 13722, root := ~, start := 13701, stop := 13721\ ), rec( content := " with a digraph argument", count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 7 ], name := "Item", next := 13753, root := ~, start := 13695, stop := 13752 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Oper := "Digraph" ), content := 0, count := [ 2, 2, 1, 9 ], mathmode := \ "Text", name := "Ref", next := 13789, root := ~, start := 13768, stop := 13788\ ), 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 := 13826, root := ~, start := 13762, stop := 13825 ) ], count := [ 2, 2, 1, 4 ], mathmode := "Text", name := "List", next := 13840, root := ~, start := 13615, stop := 13839 ), 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 := 13997, root := ~, start := 13429, stop := 13996 ), rec( content := "\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 14010, root := ~, start := 12947, stop := 14009 ), 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 := 14271, root := ~, start := 14225, stop := 14270 ), 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 := 14451, root := ~, start := 14447, stop := 14450 ), 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 := 14531, root := ~, start := 14527, stop := 14530 ), 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.\ 1", count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "F", next := 14604, root := ~, start := 14580, stop := 14603 ), 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 := 14674, root := ~, start := 14625, stop := 14673 ), 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 := 14683, root := ~, start := 14679, stop := 14682 ), 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 := 15061, root := ~, start := 15057, stop := 15060 ), 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 := 15709, root := ~, start := 14175, stop := 15708 ), 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 := 15972, root := ~, start := 15925, stop := 15971 ), 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 := 16249, root := ~, start := 16220, stop := 16248 ), 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 := 16308, root := ~, start := 16271, stop := 16307 ), 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 := 16337, root := ~, start := 16311, stop := 16336 ), 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 := 16467, root := ~, start := 16340, stop := 16466 ) ], count := [ 2, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 16481, root := ~, start := 16256, stop := 16480 ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 16495, root := ~, start := 15874, stop := 16494 ), 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 := 16752, root := ~, start := 16708, stop := 16751 ), 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 := 16931, root := ~, start := 16898, stop := 16930 ), 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 := 17002, root := ~, start := 16968, stop := 17001 ), 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 := 17008, root := ~, start := 17004, stop := 17007 ), 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 := 17126, root := ~, start := 17086, stop := 17125 ), 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/digraphs/Digraphs/issues", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "URL", next := 17317, root := ~, start := 17236, stop := 17316 ), 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 := 17420, root := ~, start := 17379, stop := 17419 ), 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 := 17443, root := ~, start := 17432, stop := 17442 ), 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 := 17459, root := ~, start := 17447, stop := 17458 ), rec( content := ".", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 17470, root := ~, start := 17423, stop := 17469 ), 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 := 17781, root := ~, start := 17747, stop := 17780 ), rec( content := ".\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 17799, root := ~, start := 17473, stop := 17798 ) ], count := [ 2, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 17813, root := ~, start := 17364, stop := 17812 ), 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 := 17876, root := ~, start := 17834, stop := 17875 ), 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 := 17899, root := ~, start := 17888, stop := 17898 ), 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 := 17915, root := ~, start := 17903, stop := 17914 ), rec( content := ".", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 17926, root := ~, start := 17879, stop := 17925 ), 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 := 18112, root := ~, start := 18108, stop := 18111 ), 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 := 18214, root := ~, start := 18177, stop := 18213 ), 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 := 18284, root := ~, start := 18247, stop := 18283 ), rec( content := ".\n ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 18302, root := ~, start := 17929, stop := 18301 ) ], count := [ 2, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 18316, root := ~, start := 17819, stop := 18315 ), 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 := 18378, root := ~, start := 18337, stop := 18377 ), 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 := 18408, root := ~, start := 18381, stop := 18407 ), 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 := 18684, root := ~, start := 18680, stop := 18683 ), 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 := 18759, root := ~, start := 18740, stop := 18758 ), 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 := 18816, root := ~, start := 18801, stop := 18815 ), 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 := 18871, root := ~, start := 18852, stop := 18870 ), 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 := 18946, root := ~, start := 18411, stop := 18945 ) ], count := [ 2, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 18960, root := ~, start := 18322, stop := 18959 ), 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 := 18975, root := ~, start := 16660, stop := 18974 ), 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 := 19149, root := ~, start := 9104, stop := 19148 ), 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 := 19244, root := ~, start := 19208, stop := 19243 ), 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 := 19303, root := ~, start := 19299, stop := 19302 ), 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 := 19352, root := ~, start := 19316, stop := 19351 ), 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 := 19412, root := ~, start := 19372, stop := 19411 ), 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 := 19511, root := ~, start := 19495, stop := 19510 ), 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 := 19601, root := ~, start := 19572, stop := 19600 ), 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 := 19629, root := ~, start := 19603, stop := 19628 ), 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 := 19666, root := ~, start := 19639, stop := 19665 ), rec( content := ".\n " , count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 19686, root := ~, start := 19415, stop := 19685 ) ], count := [ 3, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 19700, root := ~, start := 19357, stop := 19699 ), 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 := 19768, root := ~, start := 19721, stop := 19767 ), 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 := 19812, root := ~, start := 19789, stop := 19811 ), 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 := 19854, root := ~, start := 19830, stop := 19853 ), 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 := 19902, root := ~, start := 19859, stop := 19901 ), 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 := 20081, root := ~, start := 20030, stop := 20080 ), 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 := 20091, root := ~, start := 20087, stop := 20090 ), 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 := 20233, root := ~, start := 20191, stop := 20232 ), 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 := 20272, root := ~, start := 19771, stop := 20271 ) ], count := [ 3, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 20286, root := ~, start := 19706, stop := 20285 ), 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 := 20356, root := ~, start := 20307, stop := 20355 ), 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 := 20402, root := ~, start := 20377, stop := 20401 ), 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 := 20447, root := ~, start := 20423, stop := 20446 ), 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 := 20499, root := ~, start := 20474, stop := 20498 ), 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 := 20623, root := ~, start := 20572, stop := 20622 ), 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 := 20629, root := ~, start := 20625, stop := 20628 ), 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 := 20749, root := ~, start := 20724, stop := 20748 ), 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 := 20807, root := ~, start := 20765, stop := 20806 ), 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 := 20827, root := ~, start := 20823, stop := 20826 ), 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 := 20879, root := ~, start := 20847, stop := 20878 ), 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 := 21061, root := ~, start := 20359, stop := 21060 ) ], count := [ 3, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 21075, root := ~, start := 20292, stop := 21074 ), 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 := 21142, root := ~, start := 21096, stop := 21141 ), 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 := 21185, root := ~, start := 21163, stop := 21184 ), 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 := 21222, root := ~, start := 21206, stop := 21221 ), 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 := 21346, root := ~, start := 21319, stop := 21345 ), 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 := 21429, root := ~, start := 21145, stop := 21428 ) ], count := [ 3, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 21443, root := ~, start := 21081, stop := 21442 ), 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 := 21525, root := ~, start := 21464, stop := 21524 ), 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 := 21583, root := ~, start := 21546, stop := 21582 ), 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 := 21620, root := ~, start := 21604, stop := 21619 ), 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 := 21658, root := ~, start := 21644, stop := 21657 ), 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 := 21741, root := ~, start := 21528, stop := 21740 ) ], count := [ 3, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 21755, root := ~, start := 21449, stop := 21754 ), 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 := 21816, root := ~, start := 21776, stop := 21815 ), rec( attributes := rec( Name := "DigraphFamily" ), content := 0, count := [ 3, 1, 6, 4 ], mathmode := "Text", name := "Fam", next := 21846, root := ~, start := 21819, stop := 21845 ), 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 := 21928, root := ~, start := 21895, stop := 21927 ), 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 := 21984, root := ~, start := 21964, stop := 21983 ), rec( content := ".\n ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "Description", next := 22002, root := ~, start := 21849, stop := 22001 ) ], count := [ 3, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 22016, root := ~, start := 21761, stop := 22015 ), 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 := 22098, root := ~, start := 22037, stop := 22097 ), 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 := 22191, root := ~, start := 22101, stop := 22190 ), 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 := 22308, root := ~, start := 22194, stop := 22307 ), 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 := 22340, root := ~, start := 22311, stop := 22339 ), 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 := 22403, root := ~, start := 22392, stop := 22402 ), 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 := 22549, root := ~, start := 22538, stop := 22548 ), 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 := 22583, root := ~, start := 22553, stop := 22582 ), 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 := 22651, root := ~, start := 22640, stop := 22650 ), 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 := 22691, root := ~, start := 22655, stop := 22690 ), 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 := 22776, root := ~, start := 22765, stop := 22775 ), 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 := 22834, root := ~, start := 22798, stop := 22833 ), 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 := 22920, root := ~, start := 22872, stop := 22919 ), 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 := 22955, root := ~, start := 22945, stop := 22954 ) , 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 := 23030, root := ~, start := 23022, stop :=\ 23029 ), 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 := 23054, root\ := ~, start := 23044, stop := 23053 ), rec( content := ")", count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "C", next := 23059, root := ~, start := 23\ 034, stop := 23058 ), 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 := 23145, root := ~, start := 23130, stop := 23144 ), 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 := 23171, root := ~, start := 23161, stop := 231\ 70 ), rec( content := ")]", count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "C", next := 23177, root := ~, start := 23145, stop := 23176 ), 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 := 23239, root := ~, start := 23229, stop := 23\ 238 ), 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 := 23245, root := ~, star\ t := 23241, stop := 23244 ), 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 :=\ 23308, root := ~, start := 23300, stop := 23307 ), 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 := 23320, root := ~, start := 23312, stop :=\ 23319 ), 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 := 23352, root := ~, start := 23344, stop := 23351 ), 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 := 23372, root := ~, start := 23362, stop := 233\ 71 ), rec( content := "[i]", count := [ 3, 1, 7, 10 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 23379, root := ~, start := 23359, stop := 23378 ), 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 := 23424, root := ~, start := 23416, stop := 23423 ), 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 :=\ 23450, root := ~, start := 23442, stop := 23449 ), 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 := 23463, root := ~, start := 23455, stop :=\ 23462 ), 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 := 23495, root := \ ~, start := 23485, stop := 23494 ), rec( content := "[i]", count := [ 3, 1, 7,\ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7\ , 10 ], mathmode := "Text", name := "C", next := 23502, root := ~, start := 23\ 482, stop := 23501 ), 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 := 23529, root := ~, start := 23521, stop := 23\ 528 ), 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 := 23554, root := ~, start := 23546, stop := 23553 ), 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 :=\ 23582, root := ~, start := 23574, stop := 23581 ), 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 := 23594, root := ~, start := 23586, stop :=\ 23593 ), rec( content := ".\n ", count := [ 3, 1, 7, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], name := "Item\ ", next := 23609, root := ~, start := 22927, stop := 23608 ), 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 := 23645, root := ~, start := 23617, stop := 23644 ), 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 := 23680, root := ~, start := 23670, stop := 2367\ 9 ), 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 := 23724, root := ~, start := 23711, stop := 23\ 723 ), 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 := 23749, root := ~, start := 23737, stop := 23748 ), 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 := 23854, root := ~, st\ art := 23844, stop := 23853 ), rec( content := ")]", count := [ 3, 1, 7, 12 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ] , math\ mode := "Text", name := "C", next := 23860, root := ~, start := 23828, stop :=\ 23859 ), 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 \ := 23944, root := ~, start := 23929, stop := 23943 ), 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 := 23970, root := ~, start := 23960, stop := 23969 ), re\ c( content := ")]", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C" , next\ := 23976, root := ~, start := 23944, stop := 23975 ), 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 := 24013, root := ~, start\ := 24000, stop := 24012 ), rec( content := ")", count := [ 3, 1, 7, 12 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], math\ mode := "Text", name := "C", next := 24018, root := ~, start := 23990, stop :=\ 24017 ), 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 := 24044, root := ~, start := 24036, stop := 24043 ), 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 := 2408\ 5, root := ~, start := 24072, stop := 24084 ), rec( content := ")]", count := \ [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next := 24091, root := ~, star\ t := 24056, stop := 24090 ), 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 :\ = 24156, root := ~, start := 24140, stop := 24155 ), 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 := 24189, root := \ ~, start := 24174, stop := 24188 ), 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 := 24230, root := ~, start := 2420\ 3, stop := 24229 ), 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 := 24261, root := ~, start := 24235, stop := 24260 ), 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 := 24275, root := ~, start :=\ 24271, stop := 24274 ), 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 := 24367, root := ~, start := 24357, stop := 24366 ), rec( conte\ nt := ".\n ", count := [ 3, 1, 7, 13 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 7, 12 ], name := "Item", next := 24382, root := ~, start := 236\ 52, stop := 24381 ), 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\ 2, root := ~, start := 24390, stop := 24431 ), 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\ 4467, root := ~, start := 24457, stop := 24466 ), 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 := 24500, root := \ ~, start := 24487, stop := 24499 ), 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 := 24525, root := ~, start := 2451\ 3, stop := 24524 ), 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 := 24623, root := ~, start := 24613, stop := 24622 ), rec( content := "]", c\ ount := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next := 24628, root := \ ~, start := 24604, stop := 24627 ), 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 := 24712, root := ~, start := 24697, stop := 24711 ), 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 := 24731, root := ~, start := 24721, stop := 247\ 30 ), rec( content := "]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name\ := "C", next := 24736, root := ~, start := 24712, stop := 24735 ), 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 := 24773, r\ oot := ~, start := 24760, stop := 24772 ), rec( content := ")", count := [ 3, \ 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3,\ 1, 7, 15 ], mathmode := "Text", name := "C", next := 24778, root := ~, start \ := 24750, stop := 24777 ), 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 := 24804, root := ~, start := 24796, stop :=\ 24803 ), 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 := 24845, root := ~, start := 24832, stop := 24844 ), rec( conten\ t := ")]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next :=\ 24851, root := ~, start := 24816, stop := 24850 ), 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 := 24916, root := ~, start := 24900, stop := 24915 ), 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\ := 24949, root := ~, start := 24934, stop := 24948 ), 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 := 24990, root :\ = ~, start := 24963, stop := 24989 ), 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 := 25021, root := ~, start := 24995, stop \ := 25020 ), rec( content := ".\n ", count := [ 3, 1, 7, 15 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], name := "Ite\ m", next := 25036, root := ~, start := 24439, stop := 25035 ), 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 := 25082, root := ~, start := 25044, stop := 25081 ), 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 := 25118, root := ~, start := 25107, stop := 251\ 17 ), 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 := 25144, root := ~, start := 25133, stop := 25143 ), 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 := 25218, root := ~, start := 25207 , stop\ := 25217 ), 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 := 25235, root := ~, start := 25224, stop := 25234 ), 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 :=\ 25263, root := ~, start := 25252, stop := 25262 ), 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 := 25279, root := ~, start := 2526\ 7, stop := 25278 ), 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 := 2536\ 8, root := ~, start := 25357, stop := 25367 ), rec( content := ")]", count := \ [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 25374, root := ~, star\ t := 25341, stop := 25373 ), 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 := 25415, root := ~, star\ t := 25407, stop := 25414 ), 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 := 25434, root := ~, start := 25426, stop := 25433\ ), 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 := 25472, root :=\ ~, start := 25461, stop := 25471 ), 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 := 25484, root := ~, start := 25473, stop := 25\ 483 ), 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 := 25500, root := ~, start := 25489, stop := 25499 ), rec( content := "[j\ ])", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 25508 , root\ := ~, start := 25458, stop := 25507 ), 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 := 25528, root := ~, start := 25517, stop :=\ 25527 ), rec( content := ".\n ", count := [ 3, 1, 7, 17 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], name := "Ite\ m", next := 25543, root := ~, start := 25089, stop := 25542 ), 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 := 25602, root := ~, start := 25551, stop := 25601 ), 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 := 25670, root := ~, star\ t := 25646, stop := 25669 ), 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\ 5676, root := ~, start := 25672, stop := 25675 ), 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 := 25698, root := ~, star\ t := 25690, stop := 25697 ), 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 := 25746, root := ~ , star\ t := 25735, stop := 25745 ), 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 := 25780, root := ~, start := 25770, stop\ := 25779 ), 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 := 25800, root := ~, start := 25790, stop := 25799 ), 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 := 25858, root := ~, start := 25847, stop := 25\ 857 ), 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 := 25897, root := ~, start := 25886, stop := 25896 ), 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 := \ 25921, root := ~, start := 25909, stop := 25920 ), 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 := 25946, root := ~, star\ t := 25936, stop := 25945 ), 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\ := 26015, root := ~, start := 26004, stop := 26014 ), 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 := 26072, root := ~, start := 26064, stop := 26071 ), 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\ 172, root := ~, start := 26161, stop := 26171 ), 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 := 26205, root := \ ~, start := 26192, stop := 26204 ), 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 := 26233, root := ~, start := 26\ 219, stop := 26232 ), 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 := 26248, root := ~, start := 26237, stop := 26247 ), 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 := 26254, root := ~, start := 26250, stop :=\ 26253 ), 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 := 26296, root := ~, start := 26288, stop :=\ 26295 ), 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 := 26326, root := ~, start := 26315, stop := 26325 ), 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 := 26388, root := ~, start := 26362, stop := 26387 ), 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 := 26467, root := ~, start := 26438, stop \ := 26466 ), rec( content := ".\n ", count := [ 3, 1, 7, 21 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 19 ], name := "Ite\ m", next := 26482, root := ~, start := 25609, stop := 26481 ), 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 := 26528, root := ~, start := 26490, stop := 26527 ), 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 := 26563, root := ~, start := 26553, stop := 2656\ 2 ), 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 := 26651, root := \ ~, start := 26637, stop := 26650 ), 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 := 26671, root := ~, start := 26660, stop :=\ 26670 ), 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 := 26743, root := \ ~, start := 26733, stop := 26742 ), rec( content := ".\n ", count := [ 3,\ 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], name := "Ite\ m", next := 26758, root := ~, start := 26535, stop := 26757 ), 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 := 26800, root := ~, start := 26766, stop := 26799 ), 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 := 26835, root := ~, start := 26825, stop := 2683\ 4 ), 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 := 26886, root := ~, start := 26871, stop :=\ 26885 ), 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 := 26930, root := ~, start := 26922, stop :=\ 26929 ), 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 := 27000, root := \ ~, start := 26990, stop := 26999 ), 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 := 27074, root := ~, start := 27066, stop := 27073 ), 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 := 27127, root := ~, start := 27119, stop := 27126 ), 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 :=\ 27154, root := ~, start := 27146, stop := 27153 ), 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 := 27182, root := \ ~, start := 27170, stop := 27181 ), 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 := 2723\ 4, root := ~, start := 27224, stop := 27233 ), rec( content := ".\n ", coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], name := "Ite\ m", next := 27249, root := ~, start := 26807, stop := 27248 ) ], count := [ 3, 1, 7, 8 ], mathmode := "Text", name := "List", next := 27261, root := ~, start := 22859, stop := 27260 ), 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 := 28003, root := ~, start := 27267, stop := 28002 ), 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 := 28270, root := ~, start := 28251, stop := 28269 ), 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 := 28388, root := ~, start := 28362, stop := 28387 ), 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 := 29358, root := ~, start := 28390, stop := 29357 ), rec( content := "\n ", count := [ 3, 1, 7, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Description", next := 29375, root := ~, start := 22343, stop := 29374 ) ], count := [ 3, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 29389, root := ~, start := 22022, stop := 29388 ), 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 := 29468, root := ~, start := 29410, stop := 29467 ), 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 := 29500, root := ~, start := 29471, stop := 29499 ), 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 := 29563, root := ~, start := 29552, stop := 29562 ), 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 := 29709, root := ~, start := 29698, stop := 29708 ), 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 := 29743, root := ~, start := 29713, stop := 29742 ), 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 := 29811, root := ~, start := 29800, stop := 29810 ), 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 := 29851, root := ~, start := 29815, stop := 29850 ), 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 := 29936, root := ~, start := 29925, stop := 29935 ), 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 := 29994, root := ~, start := 29958, stop := 29993 ), 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 := 30018, root := ~, start := 30014, stop := 30017 ), 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 := 30038, root := ~, start := 30027, stop := 30037 ), 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 := 30125, root := ~, start := 30096, stop := 30124 ), 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 := 30202, root := ~, start := 30191, stop := 30201 ), 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 := 30208, root := ~, start := 30204, stop := 30207 ), 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 := 30243, root := ~, start := 30232, stop := 30242 ), 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 := 30335, root := ~, start := 30321, stop := 30334 ), 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 := 30346, root := ~, start := 30335, stop := 30345 ), 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 := 30354, root := ~, start := 30346, stop := 30353 ), 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 := 30399, root := ~, start := 30387, stop := 30398 ), 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 := 30430, root := ~, start := 30419, stop := 30429 ), 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 := 30443, root := ~, start := 30430, stop := 30442 ), 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 := 30458, root := ~, start := 30447, stop := 30457 ), 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 := 30464, root := ~, start := 30460, stop := 30463 ), 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 := 31060, root := ~, start := 30470, stop := 31059 ), rec( content := "\n ", count := [ 3, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 31077, root := ~, start := 29503, stop := 31076 ) ], count := [ 3, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 31091, root := ~, start := 29395, stop := 31090 ), 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 := 31165, root := ~, start := 31112, stop := 31164 ), 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 := 31197, root := ~, start := 31168, stop := 31196 ), 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 := 31260, root := ~, start := 31249, stop := 31259 ), 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 := 31406, root := ~, start := 31395, stop := 31405 ), 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 := 31440, root := ~, start := 31410, stop := 31439 ), 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 := 31508, root := ~, start := 31497, stop := 31507 ), 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 := 31548, root := ~, start := 31512, stop := 31547 ), 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 := 31633, root := ~, start := 31622, stop := 31632 ), 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 := 31691, root := ~, start := 31655, stop := 31690 ), 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 := 31715, root := ~, start := 31711, stop := 31714 ), 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 := 31735, root := ~, start := 31724, stop := 31734 ), 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 := 31862, root := ~, start := 31854, stop := 31861 ), 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 := 31903, root := ~, start := 31892, stop := 31902 ), 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 := 31908, root := ~, start := 31904, stop := 31907 ), 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 := 31954, root := ~, start := 31946, stop := 31953 ), 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 := 31997, root := ~, start := 31989, stop := 31996 ) , rec( content := "\ >= m", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "C", next := 32006, root := ~, start := 31986, stop := 32005 ), 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 := 32021, root := ~, start := 32013, stop := 32020 ), 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 := 32097, root := ~, start := 32089, stop := 32096 ), 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 := 32131, root := ~, start := 32120, stop := 32130 ), 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 := 32137, root := ~, start := 32133, stop := 32136 ), 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 := 32173, root := ~, start := 32147, stop := 32172 ), 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 := 32703, root := ~, start := 32180, stop := 32702 ), rec( content := "\n ", count := [ 3, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 32720, root := ~, start := 31200, stop := 32719 ) ], count := [ 3, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 32734, root := ~, start := 31097, stop := 32733 ), 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 := 32807, root := ~, start := 32755, stop := 32806 ), 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 := 32858, root := ~, start := 32810, stop := 32857 ), 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 := 32890, root := ~, start := 32882, stop := 32889 ), 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 := 32927, root := ~, start := 32915, stop := 32926 ), 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 := 32973, root := ~, start := 32965, stop := 32972 ), 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 := 33018, root := ~, start := 33010, stop := 33017 ), 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 := 33046, root := ~, start := 33038, stop := 33045 ) , rec( content := "\ ]", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "C", next := 33051, root := ~, start := 33029, stop := 33050 ), 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 := 33127, root := ~, start := 33119, stop := 33126 ), 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 := 33199, root := ~, start := 33185, stop := 33198 ), 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 := 33254, root := ~, start := 33246, stop := 33253 ), 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 := 33332, root := ~, start := 33328, stop := 33331 ), 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 := 33377, root := ~, start := 33369, stop := 33376 ), 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 := 33460, root := ~, start := 33452, stop := 33459 ), 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 := 33484, root := ~, start := 33480, stop := 33483 ), 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 := 33842, root := ~, start := 33490, stop := 33841 ), rec( content := "\n ", count := [ 3, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 33859, root := ~, start := 32861, stop := 33858 ) ], count := [ 3, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 33873, root := ~, start := 32740, stop := 33872 ), 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 := 33949, root := ~, start := 33894, stop := 33948 ), rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByInNeig\ hbors" ), content := 0, count := [ 3, 1, 11, 4 ], mathmode := "Text", name := "Oper", next := 34006, root := ~, start := 33952, stop := 34005 ), 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 := 34038, root := ~, start := 34009, stop := 34037 ), 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 := 34099, root := ~, start := 34088, stop := 34098 ), 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 := 34241, root := ~, start := 34230, stop := 34240 ), 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 := 34275, root := ~, start := 34245, stop := 34274 ), 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 := 34341, root := ~, start := 34330, stop := 34340 ), 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 := 34379, root := ~, start := 34345, stop := 34378 ), 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 := 34462, root := ~, start := 34451, stop := 34461 ), 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 := 34518, root := ~, start := 34484, stop := 34517 ), 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 := 34542, root := ~, start := 34538, stop := 34541 ), 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 := 34560, root := ~, start := 34549, stop := 34559 ), 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 := 34645, root := ~, start := 34634, stop := 34644 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34651, root := ~, start := 34618, stop := 34650 ), 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 := 34719, root := ~, start := 34708, stop := 34718 ), 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 := 34746, root := ~, start := 34735, stop := 34745 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34752, root := ~, start := 34719, stop := 34751 ), 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 := 34809, root := ~, start := 34798, stop := 34808 ), 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 := 34873, root := ~, start := 34865, stop := 34872 ), 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 := 34899, root := ~, start := 34891, stop := 34898 ), 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 := 34911, root := ~, start := 34903, stop := 34910 ), 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 := 34936, root := ~, start := 34925, stop := 34935 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34943, root := ~, start := 34922, stop := 34942 ), 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 := 34949, root := ~, start := 34945, stop := 34948 ), 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 := 34964, root := ~, start := 34956, stop := 34963 ), 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 := 34991, root := ~, start := 34980, stop := 34990 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 34998, root := ~, start := 34977, stop := 34997 ), 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 := 35025, root := ~, start := 35017, stop := 35024 ), 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 := 35052, root := ~, start := 35044, stop := 35051 ), 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 := 35081, root := ~, start := 35073, stop := 35080 ), 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 := 35093, root := ~, start := 35085, stop := 35092 ), 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 := 35099, root := ~, start := 35095, stop := 35098 ), 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 := 35133, root := ~, start := 35107, stop := 35132 ), 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 := 35669, root := ~, start := 35140, stop := 35668 ), rec( content := "\n ", count := [ 3, 1, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Description", next := 35686, root := ~, start := 34041, stop := 35685 ) ], count := [ 3, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 35700, root := ~, start := 33879, stop := 35699 ), 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 := 35765, root := ~, start := 35721, stop := 35764 ), 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 := 35808, root := ~, start := 35768, stop := 35807 ), 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 := 35841, root := ~, start := 35833, stop := 35840 ), 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 := 35874, root := ~, start := 35863, stop := 35873 ), 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 := 35912, root := ~, start := 35904, stop := 35911 ), 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 := 36010, root := ~, start := 36002, stop := 36009 ), 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 := 36039, root := ~, start := 36028, stop := 36038 ), 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 := 36082, root := ~, start := 36074, stop := 36081 ), 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 := 36133, root := ~, start := 36125, stop := 36132 ), 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 := 36156, root := ~, start := 36148, stop := 36155 ), 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 := 36209, root := ~, start := 36201, stop := 36208 ), 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 := 36224, root := ~, start := 36213, stop := 36223 ), 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 := 36251, root := ~, start := 36235, stop := 36250 ), 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 := 36257, root := ~, start := 36253, stop := 36256 ), 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 := 36384, root := ~, start := 36373, stop := 36383 ), 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 := 36436, root := ~, start := 36428, stop := 36435 ), 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 := 36461, root := ~, start := 36457, stop := 36460 ), 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 := 36564, root := ~, start := 36529, stop := 36563 ), 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 := 36584, root := ~, start := 36576, stop := 36583 ), 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 := 36663, root := ~, start := 36629, stop := 36662 ), 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 := 36699, root := ~, start := 36688, stop := 36698 ), 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 := 36761, root := ~, start := 36722, stop := 36760 ), 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 := 36766, root := ~, start := 36762, stop := 36765 ), 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 := 37322, root := ~, start := 36839, stop := 37321 ), rec( content := "\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 37339, root := ~, start := 35811, stop := 37338 ) ], count := [ 3, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 37353, root := ~, start := 35706, stop := 37352 ), rec( content := "\n\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 37367, root := ~, start := 19307, stop := 37366 ), 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 := 37423, root := ~, start := 37380, stop := 37422 ), 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 := 37488, root := ~, start := 37443, stop := 37487 ), 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 := 37528, root := ~, start := 37491, stop := 37527 ), 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 := 37566, root := ~, start := 37552, stop := 37565 ), 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 := 37627, root := ~, start := 37619, stop := 37626 ), 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 := 37729, root := ~, start := 37716, stop := 37728 ), 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 := 37756, root := ~, start := 37744, stop := 37755 ), 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 := 37814, root := ~, start := 37802, stop := 37813 ), 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 := 37843, root := ~, start := 37829, stop := 37842 ), 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 := 37853, root := ~, start := 37849, stop := 37852 ), 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 := 38035, root := ~, start := 37854, stop := 38034 ), rec( content := "\n ", count := [ 3, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 38052, root := ~, start := 37531, stop := 38051 ) ], count := [ 3, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 38066, root := ~, start := 37428, stop := 38065 ), 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 := 38136, root := ~, start := 38087, stop := 38135 ), 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 := 38173, root := ~, start := 38162, stop := 38172 ), rec( content := ".", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 38184, root := ~, start := 38139, stop := 38183 ), 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 := 38247, root := ~, start := 38236, stop := 38246 ), 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 := 38393, root := ~, start := 38382, stop := 38392 ), 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 := 38427, root := ~, start := 38397, stop := 38426 ), 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 := 38495, root := ~, start := 38484, stop := 38494 ), 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 := 38535, root := ~, start := 38499, stop := 38534 ), 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 := 38620, root := ~, start := 38609, stop := 38619 ), 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 := 38678, root := ~, start := 38642, stop := 38677 ), 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 := 38702, root := ~, start := 38698, stop := 38701 ), 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 := 38723, root := ~, start := 38711, stop := 38722 ), 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 := 38757, root := ~, start := 38749, stop := 38756 ), 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 := 38829, root := ~, start := 38817, stop := 38828 ), 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 := 38850, root := ~, start := 38842, stop := 38849 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 38855, root := ~, start := 38833, stop := 38854 ), 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 := 38905, root := ~, start := 38897, stop := 38904 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 38910, root := ~, start := 38888, stop := 38909 ), 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 := 38933, root := ~, start := 38917, stop := 38932 ), 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 := 38982, root := ~, start := 38974, stop := 38981 ), 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 := 39019, root := ~, start := 39007, stop := 39018 ), 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 := 39058, root := ~, start := 39025, stop := 39057 ), 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 := 39068, root := ~, start := 39064, stop := 39067 ), 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 := 39128, root := ~, start := 39112, stop := 39127 ), 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 := 39142, root := ~, start := 39134, stop := 39141 ), 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 := 39178, root := ~, start := 39170, stop := 39177 ), 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 := 39199, root := ~, start := 39191, stop := 39198 ) , rec( content := "\ ]", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39204, root := ~, start := 39182, stop := 39203 ), 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 := 39253, root := ~, start := 39245, stop := 39252 ), 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 := 39292, root := ~, start := 39280, stop := 39291 ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39296, root := ~, start := 39275, stop := 39295 ), 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 := 39306, root := ~, start := 39302, stop := 39305 ), 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 := 39352, root := ~, start := 39344, stop := 39351 ), 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 := 39424, root := ~, start := 39412, stop := 39423 ), 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 := 39484, root := ~, start := 39472, stop := 39483 ), 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 := 39505, root := ~, start := 39497, stop := 39504 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 39510, root := ~, start := 39488, stop := 39509 ), 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 := 39568, root := ~, start := 39560, stop := 39567 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 39573, root := ~, start := 39551, stop := 39572 ), 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 := 39605, root := ~, start := 39593, stop := 39604 ) , 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 := 39615, root := ~, start := 39607, stop := 39614 ) , rec( content := "\ )", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 39620, root := ~, start := 39580, stop := 39619 ), 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 := 39644, root := ~, start := 39633, stop := 39643 ), 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 := 40137, root := ~, start := 39651, stop := 40136 ), rec( content := "\n ", count := [ 3, 2, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 40154, root := ~, start := 38187, stop := 40153 ) ], count := [ 3, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 40168, root := ~, start := 38072, stop := 40167 ), 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 := 40223, root := ~, start := 40189, stop := 40222 ), 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 := 40269, root := ~, start := 40226, stop := 40268 ), 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 := 40307, root := ~, start := 40293, stop := 40306 ), 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 := 40458, root := ~, start := 40444, stop := 40457 ), 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 := 40464, root := ~, start := 40460, stop := 40463 ), 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 := 40487, root := ~, start := 40473, stop := 40486 ), 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 := 40755, root := ~, start := 40712, stop := 40754 ), 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 := 40769, root := ~, start := 40755, stop := 40768 ), 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 := 40778, root := ~, start := 40769, stop := 40777 ), 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 := 41431, root := ~, start := 40785, stop := 41430 ), rec( content := "\n ", count := [ 3, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 41448, root := ~, start := 40272, stop := 41447 ) ], count := [ 3, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 41462, root := ~, start := 40174, stop := 41461 ), 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 := 41519, root := ~, start := 41483, stop := 41518 ), 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 := 41565, root := ~, start := 41522, stop := 41564 ), 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 := 41603, root := ~, start := 41589, stop := 41602 ), 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 := 41678, root := ~, start := 41659, stop := 41677 ), 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 := 41709, root := ~, start := 41695, stop := 41708 ), 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 := 41799, root := ~, start := 41785, stop := 41798 ), 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 := 41846, root := ~, start := 41831, stop := 41845 ), 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 := 41860, root := ~, start := 41846, stop := 41859 ), 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 := 41868, root := ~, start := 41860, stop := 41867 ), 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 := 42279, root := ~, start := 41952, stop := 42278 ), rec( content := "\n ", count := [ 3, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 42296, root := ~, start := 41568, stop := 42295 ) ], count := [ 3, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 42310, root := ~, start := 41468, stop := 42309 ), 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 := 42376, root := ~, start := 42331, stop := 42375 ), 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 := 42420, root := ~, start := 42409, stop := 42419 ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 42430, root := ~, start := 42379, stop := 42429 ), 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 := 42468, root := ~, start := 42454, stop := 42467 ), 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 := 42522, root := ~, start := 42499, stop := 42521 ), 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 := 42588, root := ~, start := 42574, stop := 42587 ), 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 := 42634, root := ~, start := 42595, stop := 42633 ), 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 := 42675, root := ~, start := 42651, stop := 42674 ), 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 := 42689, root := ~, start := 42675, stop := 42688 ), 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 := 42697, root := ~, start := 42689, stop := 42696 ), 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 := 42717, root := ~, start := 42706, stop := 42716 ), 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 := 42724, root := ~, start := 42720, stop := 42723 ), 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 := 42747, root := ~, start := 42733, stop := 42746 ), 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 := 42785, root := ~, start := 42777, stop := 42784 ), 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 := 42829, root := ~, start := 42805, stop := 42828 ), 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 := 42843, root := ~, start := 42829, stop := 42842 ), 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 := 42851, root := ~, start := 42843, stop := 42850 ), 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 := 42895, root := ~, start := 42887, stop := 42894 ), 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 := 42922, root := ~, start := 42914, stop := 42921 ), 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 := 42966, root := ~, start := 42938, stop := 42965 ), 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 := 42980, root := ~, start := 42968, stop := 42979 ), 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 := 43040, root := ~, start := 43032, stop := 43039 ), 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 := 43058, root := ~, start := 43044, stop := 43057 ), 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 := 43524, root := ~, start := 43065, stop := 43523 ), rec( content := "\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 43541, root := ~, start := 42433, stop := 43540 ) ], count := [ 3, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 43555, root := ~, start := 42316, stop := 43554 ), rec( content := "\n\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 43569, root := ~, start := 37371, stop := 43568 ), 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 := 43622, root := ~, start := 43582, stop := 43621 ), 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 := 43691, root := ~, start := 43642, stop := 43690 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opy" ), content := 0, count := [ 3, 3, 1, 4 ], mathmode := "Text", name := "Oper", next := 43741, root := ~, start := 43694, stop := 43740 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopySame\ Mutability" ), content := 0, count := [ 3, 3, 1, 6 ], mathmode := "Text", name := "Oper", next := 43798, root := ~, start := 43744, stop := 43797 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopy" ), content := 0, count := [ 3, 3, 1, 8 ], mathmode := "Text", name := "Oper", next := 43841, root := ~, start := 43801, stop := 43840 ), 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 := 43873, root := ~, start := 43844, stop := 43872 ), 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 := 43955, root := ~, start := 43941, stop := 43954 ), 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 := 44060, root := ~, start := 44046, stop := 44059 ), 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 := 44066, root := ~, start := 44062, stop := 44065 ), 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 := 44090, root := ~, start := 44072, stop := 44089 ), 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 := 44140, root := ~, start := 44108, stop := 44139 ), 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 := 44394, root := ~, start := 44142, stop := 44393 ), rec( content := "\n ", count := [ 3, 3, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "Description", next := 44411, root := ~, start := 43876, stop := 44410 ) ], count := [ 3, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 44425, root := ~, start := 43627, stop := 44424 ), 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 := 44506, root := ~, start := 44446, stop := 44505 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopyIfMutable" ), content := 0, count := [ 3, 3, 2, 4 ], mathmode := "Text", name := "Oper", next := 44567, root := ~, start := 44509, stop := 44566 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfMutable" ), content := 0, count := [ 3, 3, 2, 6 ], mathmode := "Text", name := "Oper", next := 44626, root := ~, start := 44570, stop := 44625 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfImmutable" ), content := 0, count := [ 3, 3, 2, 8 ], mathmode := "Text", name := "Oper", next := 44687, root := ~, start := 44629, stop := 44686 ), 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 := 44719, root := ~, start := 44690, stop := 44718 ), 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 := 44820, root := ~, start := 44806, stop := 44819 ), 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 := 44853, root := ~, start := 44839, stop := 44852 ), 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 := 44934, root := ~, start := 44920, stop := 44933 ), 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 := 46196, root := ~, start := 44936, stop := 46195 ), rec( content := "\n ", count := [ 3, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "Description", next := 46213, root := ~, start := 44722, stop := 46212 ) ], count := [ 3, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 46227, root := ~, start := 44431, stop := 46226 ), 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 := 46301, root := ~, start := 46248, stop := 46300 ), 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 := 46333, root := ~, start := 46304, stop := 46332 ), 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 := 46371, root := ~, start := 46357, stop := 46370 ), 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 := 46402, root := ~, start := 46390, stop := 46401 ), 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 := 46452, root := ~, start := 46438, stop := 46451 ), 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 := 46527, root := ~, start := 46513, stop := 46526 ), 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 := 46585, root := ~, start := 46573, stop := 46584 ), 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 := 46673, root := ~, start := 46661, stop := 46672 ), 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 := 46679, root := ~, start := 46675, stop := 46678 ), 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 := 46757, root := ~, start := 46732, stop := 46756 ), 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 := 46846, root := ~, start := 46813, stop := 46845 ), 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 := 46852, root := ~, start := 46848, stop := 46851 ), 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 := 46875, root := ~, start := 46861, stop := 46874 ), 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 := 46917, root := ~, start := 46887, stop := 46916 ), 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 := 46942, root := ~, start := 46928, stop := 46941 ), 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 := 46982, root := ~, start := 46968, stop := 46981 ), 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 := 47032, root := ~, start := 46994, stop := 47031 ), 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 := 47128, root := ~, start := 47124, stop := 47127 ), 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 := 47606, root := ~, start := 47134, stop := 47605 ), rec( content := "\n ", count := [ 3, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 47623, root := ~, start := 46336, stop := 47622 ) ], count := [ 3, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 47637, root := ~, start := 46233, stop := 47636 ), 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 := 47701, root := ~, start := 47658, stop := 47700 ), rec( attributes := rec( Arg := "digraph", Name := "ReducedDigraphA\ ttr" ), content := 0, count := [ 3, 3, 4, 4 ], mathmode := "Text", name := "Attr", next := 47751, root := ~, start := 47704, stop := 47750 ), 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 := 47783, root := ~, start := 47754, stop := 47782 ), 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 := 47886, root := ~, start := 47872, stop := 47885 ), 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 := 47984, root := ~, start := 47970, stop := 47983 ), 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 := 48049, root := ~, start := 48035, stop := 48048 ), 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 := 48090, root := ~, start := 48059, stop := 48089 ), 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 := 48100, root := ~, start := 48096, stop := 48099 ), 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 := 48162, root := ~, start := 48148, stop := 48161 ), 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 := 48277, root := ~, start := 48244, stop := 48276 ), 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 := 48313, root := ~, start := 48282, stop := 48312 ), 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 := 48428, root := ~, start := 48414, stop := 48427 ), 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 := 48438, root := ~, start := 48434, stop := 48437 ), 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 := 48461, root := ~, start := 48447, stop := 48460 ), 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 := 48592, root := ~, start := 48578, stop := 48591 ), 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 := 49285, root := ~, start := 48623, stop := 49284 ), rec( content := "\n ", count := [ 3, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Description", next := 49302, root := ~, start := 47786, stop := 49301 ) ], count := [ 3, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 49316, root := ~, start := 47643, stop := 49315 ), 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 := 49392, root := ~, start := 49337, stop := 49391 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphAttr" ), content := 0, count := [ 3, 3, 5, 4 ], mathmode := "Text", name := "Attr", next := 49454, root := ~, start := 49395, stop := 49453 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoops" ), content := 0, count := [ 3, 3, 5, 6 ], mathmode := "Text", name := "Oper", next := 49524, root := ~, start := 49457, stop := 49523 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoopsAttr" ), content := 0, count := [ 3, 3, 5, 8 ], mathmode := "Text", name := "Attr", next := 49598, root := ~, start := 49527, stop := 49597 ), 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 := 49630, root := ~, start := 49601, stop := 49629 ), 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 := 49668, root := ~, start := 49654, stop := 49667 ), 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 := 49721, root := ~, start := 49688, stop := 49720 ), 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 := 49828, root := ~, start := 49814, stop := 49827 ), 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 := 49883, root := ~, start := 49869, stop := 49882 ), 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 := 49961, root := ~, start := 49949, stop := 49960 ), 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 := 50013, root := ~, start := 50001, stop := 50012 ), 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 := 50023, root := ~, start := 50019, stop := 50022 ), 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 := 50098, root := ~, start := 50053, stop := 50097 ), 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 := 50150, root := ~, start := 50146, stop := 50149 ), 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 := 50177, root := ~, start := 50163, stop := 50176 ), 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 := 50283, root := ~, start := 50269, stop := 50282 ), 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 := 50348, root := ~, start := 50344, stop := 50347 ), 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 := 50390, root := ~, start := 50358, stop := 50389 ), 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 := 50420, root := ~, start := 50392, stop := 50419 ), 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 := 50459, root := ~, start := 50430, stop := 50458 ), 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 := 51161, root := ~, start := 50486, stop := 51160 ), rec( content := "\n ", count := [ 3, 3, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "Description", next := 51178, root := ~, start := 49633, stop := 51177 ) ], count := [ 3, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 51192, root := ~, start := 49322, stop := 51191 ), 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 := 51272, root := ~, start := 51213, stop := 51271 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalAntiSymm\ etricSubdigraphAttr" ), content := 0, count := [ 3, 3, 6, 4 ], mathmode := "Text", name := "Attr", next := 51338, root := ~, start := 51275, stop := 51337 ), 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 := 51370, root := ~, start := 51341, stop := 51369 ), 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 := 51408, root := ~, start := 51394, stop := 51407 ), 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 := 51465, root := ~, start := 51428, stop := 51464 ), 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 := 51524, root := ~, start := 51510, stop := 51523 ), 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 := 51579, root := ~, start := 51565, stop := 51578 ), 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 := 51653, root := ~, start := 51641, stop := 51652 ), 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 := 51671, root := ~, start := 51657, stop := 51670 ), 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 := 51690, root := ~, start := 51678, stop := 51689 ), 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 := 51728, root := ~, start := 51716, stop := 51727 ), 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 := 51757, root := ~, start := 51743, stop := 51756 ), 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 := 51829, root := ~, start := 51817, stop := 51828 ), 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 := 51846, root := ~, start := 51834, stop := 51845 ), 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 := 51864, root := ~, start := 51850, stop := 51863 ), 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 := 51885, root := ~, start := 51877, stop := 51884 ), 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 := 51898, root := ~, start := 51890, stop := 51897 ), 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 := 51970, root := ~, start := 51942, stop := 51969 ), 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 := 51980, root := ~, start := 51976, stop := 51979 ), 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 := 52003, root := ~, start := 51989, stop := 52002 ), 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 := 52109, root := ~, start := 52095, stop := 52108 ), 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 := 52135, root := ~, start := 52131, stop := 52134 ), 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 := 52181, root := ~, start := 52145, stop := 52180 ), 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 := 52911, root := ~, start := 52208, stop := 52910 ), rec( content := "\n ", count := [ 3, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "Description", next := 52928, root := ~, start := 51373, stop := 52927 ) ], count := [ 3, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 52942, root := ~, start := 51198, stop := 52941 ), 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 := 53016, root := ~, start := 52963, stop := 53015 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingForestAttr" ), content := 0, count := [ 3, 3, 7, 4 ], mathmode := "Text", name := "Attr", next := 53076, root := ~, start := 53019, stop := 53075 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTree" ), content := 0, count := [ 3, 3, 7, 6 ], mathmode := "Text", name := "Oper", next := 53130, root := ~, start := 53079, stop := 53129 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTreeAttr" ), content := 0, count := [ 3, 3, 7, 8 ], mathmode := "Text", name := "Attr", next := 53188, root := ~, start := 53133, stop := 53187 ), 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 := 53225, root := ~, start := 53214, stop := 53224 ), rec( content := ".", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Returns", next := 53236, root := ~, start := 53191, stop := 53235 ), 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 := 53274, root := ~, start := 53260, stop := 53273 ), 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 := 53354, root := ~, start := 53323, stop := 53353 ), 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 := 53414, root := ~, start := 53400, stop := 53413 ), 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 := 53460, root := ~, start := 53449, stop := 53459 ), 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 := 53513, root := ~, start := 53466, stop := 53512 ), 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 := 53574, root := ~, start := 53570, stop := 53573 ), 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 := 53597, root := ~, start := 53583, stop := 53596 ), 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 := 53692, root := ~, start := 53646, stop := 53691 ), 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 := 53750, root := ~, start := 53711, stop := 53749 ), 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 := 53787, root := ~, start := 53758, stop := 53786 ), 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 := 53845, root := ~, start := 53831, stop := 53844 ), 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 := 53895, root := ~, start := 53884, stop := 53894 ), 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 := 53941, root := ~, start := 53902, stop := 53940 ), 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 := 54000, root := ~, start := 53996, stop := 53999 ), 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 := 54023, root := ~, start := 54009, stop := 54022 ), 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 := 54126, root := ~, start := 54112, stop := 54125 ), 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 := 54201, root := ~, start := 54187, stop := 54200 ), 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 := 54206, root := ~, start := 54202, stop := 54205 ), 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 := 54335, root := ~, start := 54306, stop := 54334 ), 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 := 54413, root := ~, start := 54382, stop := 54412 ), 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 := 55575, root := ~, start := 54420, stop := 55574 ), rec( content := "\n ", count := [ 3, 3, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Description", next := 55592, root := ~, start := 53239, stop := 55591 ) ], count := [ 3, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 55606, root := ~, start := 52948, stop := 55605 ), 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 := 55674, root := ~, start := 55627, stop := 55673 ), 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 := 55706, root := ~, start := 55677, stop := 55705 ), 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 := 55744, root := ~, start := 55730, stop := 55743 ), 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 := 55771, root := ~, start := 55763, stop := 55770 ), 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 := 55824, root := ~, start := 55810, stop := 55823 ), 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 := 55926, root := ~, start := 55912, stop := 55925 ), 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 := 55969, root := ~, start := 55961, stop := 55968 ), 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 := 55979, root := ~, start := 55975, stop := 55978 ), 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 := 56030, root := ~, start := 56016, stop := 56029 ), 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 := 56159, root := ~, start := 56145, stop := 56158 ), 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 := 56239, root := ~, start := 56235, stop := 56238 ), 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 := 56275, root := ~, start := 56261, stop := 56274 ), 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 := 56292, root := ~, start := 56284, stop := 56291 ), 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 := 56304, root := ~, start := 56296, stop := 56303 ), 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 := 56336, root := ~, start := 56328, stop := 56335 ), 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 := 56398, root := ~, start := 56384, stop := 56397 ), 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 := 56431, root := ~, start := 56423, stop := 56430 ), 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 := 56458, root := ~, start := 56450, stop := 56457 ), 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 := 56508, root := ~, start := 56500, stop := 56507 ), 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 := 56520, root := ~, start := 56512, stop := 56519 ), 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 := 56708, root := ~, start := 56704, stop := 56707 ), 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 := 56731, root := ~, start := 56717, stop := 56730 ), 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 := 56773, root := ~, start := 56743, stop := 56772 ), 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 := 56798, root := ~, start := 56784, stop := 56797 ), 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 := 56838, root := ~, start := 56824, stop := 56837 ), 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 := 56888, root := ~, start := 56850, stop := 56887 ), 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 := 56960, root := ~, start := 56956, stop := 56959 ), 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 := 57574, root := ~, start := 56966, stop := 57573 ), rec( content := "\n ", count := [ 3, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 57591, root := ~, start := 55709, stop := 57590 ) ], count := [ 3, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 57605, root := ~, start := 55612, stop := 57604 ), 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 := 57669, root := ~, start := 57626, stop := 57668 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReverseA\ ttr" ), content := 0, count := [ 3, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 57719, root := ~, start := 57672, stop := 57718 ), 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 := 57751, root := ~, start := 57722, stop := 57750 ), 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 := 57907, root := ~, start := 57894, stop := 57906 ), 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 := 57983, root := ~, start := 57970, stop := 57982 ), 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 := 57988, root := ~, start := 57984, stop := 57987 ), 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 := 58015, root := ~, start := 57994, stop := 58014 ), 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 := 58065, root := ~, start := 58051, stop := 58064 ), 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 := 58088, root := ~, start := 58074, stop := 58087 ), 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 := 58194, root := ~, start := 58180, stop := 58193 ), 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 := 59058, root := ~, start := 58237, stop := 59057 ), rec( content := "\n ", count := [ 3, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 59075, root := ~, start := 57754, stop := 59074 ) ], count := [ 3, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 59089, root := ~, start := 57611, stop := 59088 ), 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 := 59150, root := ~, start := 59110, stop := 59149 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphDualAttr\ " ), content := 0, count := [ 3, 3, 10, 4 ], mathmode := "Text", name := "Attr", next := 59197, root := ~, start := 59153, stop := 59196 ), 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 := 59229, root := ~, start := 59200, stop := 59228 ), 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 := 59265, root := ~, start := 59254, stop := 59264 ), 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 := 59283, root := ~, start := 59269, stop := 59282 ), 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 := 59323, root := ~, start := 59309, stop := 59322 ), 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 := 59375, root := ~, start := 59367, stop := 59374 ), 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 := 59387, root := ~, start := 59379, stop := 59386 ), 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 := 59431, root := ~, start := 59423, stop := 59430 ), 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 := 59443, root := ~, start := 59435, stop := 59442 ), 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 := 59461, root := ~, start := 59447, stop := 59460 ), 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 := 59482, root := ~, start := 59471, stop := 59481 ), 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 := 59524, root := ~, start := 59507, stop := 59523 ), 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 := 59529, root := ~, start := 59525, stop := 59528 ), 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 := 59553, root := ~, start := 59535, stop := 59552 ), 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 := 59600, root := ~, start := 59586, stop := 59599 ), 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 := 59623, root := ~, start := 59609, stop := 59622 ), 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 := 59740, root := ~, start := 59726, stop := 59739 ), 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 := 60206, root := ~, start := 59784, stop := 60205 ), rec( content := "\n ", count := [ 3, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "Description", next := 60223, root := ~, start := 59232, stop := 60222 ) ], count := [ 3, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 60237, root := ~, start := 59095, stop := 60236 ), 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 := 60310, root := ~, start := 60258, stop := 60309 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSymmetri\ cClosureAttr" ), content := 0, count := [ 3, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 60369, root := ~, start := 60313, stop := 60368 ), 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 := 60401, root := ~, start := 60372, stop := 60400 ), 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 := 60439, root := ~, start := 60425, stop := 60438 ), 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 := 60590, root := ~, start := 60576, stop := 60589 ), 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 := 60595, root := ~, start := 60591, stop := 60594 ), 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 := 60634, root := ~, start := 60618, stop := 60633 ), 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 := 60693, root := ~, start := 60664, stop := 60692 ), 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 := 60823, root := ~, start := 60815, stop := 60822 ), 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 := 60844, root := ~, start := 60836, stop := 60843 ), 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 := 60871, root := ~, start := 60863, stop := 60870 ), 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 := 60883, root := ~, start := 60875, stop := 60882 ), 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 := 60926, root := ~, start := 60894, stop := 60925 ), 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 := 60931, root := ~, start := 60927, stop := 60930 ), 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 := 60954, root := ~, start := 60940, stop := 60953 ), 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 := 61060, root := ~, start := 61046, stop := 61059 ), 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 := 61901, root := ~, start := 61113, stop := 61900 ), rec( content := "\n ", count := [ 3, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 61918, root := ~, start := 60404, stop := 61917 ) ], count := [ 3, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 61932, root := ~, start := 60243, stop := 61931 ), 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 := 62006, root := ~, start := 61953, stop := 62005 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veClosureAttr" ), content := 0, count := [ 3, 3, 12, 4 ], mathmode := "Text", name := "Attr", next := 62066, root := ~, start := 62009, stop := 62065 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosure" ), content := 0, count := [ 3, 3, 12, 6 ], mathmode := "Text", name := "Oper", next := 62131, root := ~, start := 62069, stop := 62130 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosureAttr" ), content := 0, count := [ 3, 3, 12, 8 ], mathmode := "Text", name := "Attr", next := 62200, root := ~, start := 62134, stop := 62199 ), 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 := 62232, root := ~, start := 62203, stop := 62231 ), 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 := 62270, root := ~, start := 62256, stop := 62269 ), 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 := 62393, root := ~, start := 62379, stop := 62392 ), 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 := 62398, root := ~, start := 62394, stop := 62397 ), 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 := 62433, root := ~, start := 62417, stop := 62432 ), 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 := 62499, root := ~, start := 62482, stop := 62498 ), 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 := 62524, root := ~, start := 62512, stop := 62523 ), 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 := 62541, root := ~, start := 62529, stop := 62540 ), 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 := 62573, root := ~, start := 62559, stop := 62572 ), 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 := 62587, root := ~, start := 62575, stop := 62586 ), 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 := 62653, root := ~, start := 62610, stop := 62652 ), 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 := 62681, root := ~, start := 62667, stop := 62680 ), 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 := 62759, root := ~, start := 62745, stop := 62758 ), 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 := 62764, root := ~, start := 62760, stop := 62763 ), 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 := 62787, root := ~, start := 62773, stop := 62786 ), 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 := 62893, root := ~, start := 62879, stop := 62892 ), 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 := 62962, root := ~, start := 62958, stop := 62961 ), 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 := 62980, root := ~, start := 62972, stop := 62979 ), 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 := 63024, root := ~, start := 63010, stop := 63023 ), 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 := 63046, root := ~, start := 63038, stop := 63045 ), 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 := 63204, root := ~, start := 63191, stop := 63203 ), 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 := 63304, root := ~, start := 63262, stop := 63303 ), 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 := 63388, root := ~, start := 63361, stop := 63387 ), 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 := 63413, root := ~, start := 63409, stop := 63412 ), 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 := 64409, root := ~, start := 63419, stop := 64408 ), rec( content := "\n ", count := [ 3, 3, 12, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "Description", next := 64426, root := ~, start := 62235, stop := 64425 ) ], count := [ 3, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 64440, root := ~, start := 61938, stop := 64439 ), 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 := 64516, root := ~, start := 64461, stop := 64515 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veReductionAttr" ), content := 0, count := [ 3, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 64578, root := ~, start := 64519, stop := 64577 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReduction" ), content := 0, count := [ 3, 3, 13, 6 ], mathmode := "Text", name := "Oper", next := 64645, root := ~, start := 64581, stop := 64644 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReductionAttr" ), content := 0, count := [ 3, 3, 13, 8 ], mathmode := "Text", name := "Attr", next := 64716, root := ~, start := 64648, stop := 64715 ), 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 := 64748, root := ~, start := 64719, stop := 64747 ), 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 := 64786, root := ~, start := 64772, stop := 64785 ), 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 := 64868, root := ~, start := 64832, stop := 64867 ), 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 := 64986, root := ~, start := 64972, stop := 64985 ), 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 := 64991, root := ~, start := 64987, stop := 64990 ), 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 := 65214, root := ~, start := 65200, stop := 65213 ), 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 := 65271, root := ~, start := 65224, stop := 65270 ), 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 := 65336, root := ~, start := 65322, stop := 65335 ), 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 := 65394, root := ~, start := 65380, stop := 65393 ), 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 := 65399, root := ~, start := 65395, stop := 65398 ), 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 := 65422, root := ~, start := 65408, stop := 65421 ), 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 := 65528, root := ~, start := 65514, stop := 65527 ), 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 := 65599, root := ~, start := 65595, stop := 65598 ), 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 := 65617, root := ~, start := 65609, stop := 65616 ), 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 := 65693, root := ~, start := 65685, stop := 65692 ), 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 := 65803, root := ~, start := 65776, stop := 65802 ), 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 := 65813, root := ~, start := 65809, stop := 65812 ), 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 := 66673, root := ~, start := 65818, stop := 66672 ), rec( content := "\n ", count := [ 3, 3, 13, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Description", next := 66690, root := ~, start := 64751, stop := 66689 ) ], count := [ 3, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 66704, root := ~, start := 64446, stop := 66703 ), 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 := 66780, root := ~, start := 66725, stop := 66779 ), 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 := 66812, root := ~, start := 66783, stop := 66811 ), 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 := 66896, root := ~, start := 66882, stop := 66895 ), 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 := 66954, root := ~, start := 66950, stop := 66953 ), 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 := 67004, root := ~, start := 66992, stop := 67003 ), 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 := 67077, root := ~, start := 67065, stop := 67076 ), 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 := 67087, root := ~, start := 67083, stop := 67086 ), 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 := 67110, root := ~, start := 67096, stop := 67109 ), 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 := 67152, root := ~, start := 67122, stop := 67151 ), 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 := 67209, root := ~, start := 67195, stop := 67208 ), 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 := 67228, root := ~, start := 67214, stop := 67227 ), 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 := 67276, root := ~, start := 67244, stop := 67275 ), 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 := 67313, root := ~, start := 67299, stop := 67312 ), 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 := 67361, root := ~, start := 67357, stop := 67360 ), 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 := 67966, root := ~, start := 67367, stop := 67965 ), rec( content := "\n ", count := [ 3, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 67983, root := ~, start := 66815, stop := 67982 ) ], count := [ 3, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 67997, root := ~, start := 66710, stop := 67996 ), 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 := 68109, root := ~, start := 68018, stop := 68108 ), 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 := 68214, root := ~, start := 68112, stop := 68213 ), 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 := 68382, root := ~, start := 68353, stop := 68381 ), 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 := 68438, root := ~, start := 68430, stop := 68437 ), 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 := 68508, root := ~, start := 68494, stop := 68507 ), 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 := 68527, root := ~, start := 68519, stop := 68526 ), 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 := 68550, root := ~, start := 68546, stop := 68549 ), 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 := 68583, root := ~, start := 68570, stop := 68582 ), 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 := 68620, root := ~, start := 68612, stop := 68619 ), 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 := 68709, root := ~, start := 68695, stop := 68708 ), 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 := 68728, root := ~, start := 68720, stop := 68727 ), 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 := 68798, root := ~, start := 68794, stop := 68797 ), 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 := 68821, root := ~, start := 68807, stop := 68820 ), 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 := 68863, root := ~, start := 68833, stop := 68862 ), 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 := 68923, root := ~, start := 68909, stop := 68922 ), 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 := 68973, root := ~, start := 68959, stop := 68972 ), 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 := 69017, root := ~, start := 68985, stop := 69016 ), rec( content := ", then\n\ ", 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 := 69042, root := ~, start := 69028, stop := 69041 ), rec( content := " itself \ is returned if no vertices are added\n (i.e. ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "m", count := [ 3, 3, 15, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := \ "Text", name := "A", next := 69108, root := ~, start := 69100, stop := 69107 ) , rec( content := "\ =0", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "C", next := 69114, root := ~, start := 69097, stop := 69113 ), rec( content := " or ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 69131, root := ~, start := 69118, stop := 69130 ), rec( content := " is empt\ y), otherwise\n the result is a new 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 := 69204, root := ~, start := 69200, stop := 69203 ), 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 := 69853, root := ~, start := 69210, stop := 69852 ), rec( content := "\n ", count := [ 3, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "Description", next := 69870, root := ~, start := 68385, stop := 69869 ) ], count := [ 3, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 69884, root := ~, start := 68003, stop := 69883 ), 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 := 69992, root := ~, start := 69905, stop := 69991 ), 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 := 70093, root := ~, start := 69995, stop := 70092 ), 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 := 70125, root := ~, start := 70096, stop := 70124 ), 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 := 70160, root := ~, start := 70149, stop := 70159 ), 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 := 70200, root := ~, start := 70186, stop := 70199 ), 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 := 70215, root := ~, start := 70205, stop := 70214 ), 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 := 70234, root := ~, start := 70224, stop := 70233 ), 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 := 70265, root := ~, start := 70251, stop := 70264 ), 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 := 70340, root := ~, start := 70326, stop := 70339 ), 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 := 70389, root := ~, start := 70378, stop := 70388 ), 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 := 70399, root := ~, start := 70389, stop := 70398 ), 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 := 70411, root := ~, start := 70401, stop := 70410 ), 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 := 70438, root := ~, start := 70427, stop := 70437 ), 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 := 70448, root := ~, start := 70438, stop := 70447 ), 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 := 70460, root := ~, start := 70450, stop := 70459 ), 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 := 70471, root := ~, start := 70467, stop := 70470 ), 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 := 70498, root := ~, start := 70484, stop := 70497 ), 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 := 70540, root := ~, start := 70510, stop := 70539 ), 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 := 70595, root := ~, start := 70581, stop := 70594 ), 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 := 70614, root := ~, start := 70600, stop := 70613 ), 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 := 70664, root := ~, start := 70626, stop := 70663 ), 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 := 70706, root := ~, start := 70692, stop := 70705 ), 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 := 70754, root := ~, start := 70750, stop := 70753 ), 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 := 71480, root := ~, start := 70760, stop := 71479 ), rec( content := "\n ", count := [ 3, 3, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "Description", next := 71497, root := ~, start := 70128, stop := 71496 ) ], count := [ 3, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 71511, root := ~, start := 69890, stop := 71510 ), 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 := 71586, root := ~, start := 71532, stop := 71585 ), 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 := 71630, root := ~, start := 71589, stop := 71629 ), 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 := 71742, root := ~, start := 71728, stop := 71741 ), 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 := 71820, root := ~, start := 71809, stop := 71819 ), 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 := 71871, root := ~, start := 71845, stop := 71870 ), 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 := 71893, root := ~, start := 71879, stop := 71892 ), 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 := 71910, root := ~, start := 71899, stop := 71909 ), 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 := 71947, root := ~, start := 71933, stop := 71946 ), 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 := 71972, root := ~, start := 71958, stop := 71971 ), 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 := 72023, root := ~, start := 72009, stop := 72022 ), 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 := 72066, root := ~, start := 72062, stop := 72065 ), 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 := 72126, root := ~, start := 72112, stop := 72125 ), 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 := 73102, root := ~, start := 72133, stop := 73101 ), rec( content := "\n ", count := [ 3, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 73119, root := ~, start := 71633, stop := 73118 ) ], count := [ 3, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 73133, root := ~, start := 71517, stop := 73132 ), 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 := 73205, root := ~, start := 73154, stop := 73204 ), 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 := 73237, root := ~, start := 73208, stop := 73236 ), 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 := 73273, root := ~, start := 73261, stop := 73272 ), 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 := 73343, root := ~, start := 73329, stop := 73342 ), 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 := 73418, root := ~, start := 73404, stop := 73417 ), 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 := 73464, root := ~, start := 73452, stop := 73463 ), 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 := 73508, root := ~, start := 73497, stop := 73507 ), 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 := 73524, root := ~, start := 73512, stop := 73523 ), 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 := 73581, root := ~, start := 73567, stop := 73580 ), 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 := 73607, root := ~, start := 73592, stop := 73606 ), 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 := 73617, root := ~, start := 73613, stop := 73616 ), 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 := 73661, root := ~, start := 73649, stop := 73660 ), 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 := 73688, root := ~, start := 73680, stop := 73687 ), 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 := 73724, root := ~, start := 73716, stop := 73723 ), 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 := 73754, root := ~, start := 73740, stop := 73753 ), 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 := 73796, root := ~, start := 73766, stop := 73795 ), 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 := 73853, root := ~, start := 73839, stop := 73852 ), 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 := 73872, root := ~, start := 73858, stop := 73871 ), 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 := 73922, root := ~, start := 73884, stop := 73921 ), 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 := 73989, root := ~, start := 73985, stop := 73988 ), 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 := 74726, root := ~, start := 73995, stop := 74725 ), rec( content := "\n ", count := [ 3, 3, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 74743, root := ~, start := 73240, stop := 74742 ) ], count := [ 3, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 74757, root := ~, start := 73139, stop := 74756 ), 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 := 74829, root := ~, start := 74778, stop := 74828 ), 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 := 74861, root := ~, start := 74832, stop := 74860 ), 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 := 74893, root := ~, start := 74885, stop := 74892 ), 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 := 74923, root := ~, start := 74909, stop := 74922 ), 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 := 74998, root := ~, start := 74984, stop := 74997 ), 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 := 75026, root := ~, start := 75018, stop := 75025 ), 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 := 75092, root := ~, start := 75084, stop := 75091 ), 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 := 75097, root := ~, start := 75093, stop := 75096 ), 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 := 75120, root := ~, start := 75106, stop := 75119 ), 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 := 75133, root := ~, start := 75125, stop := 75132 ), 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 := 75209, root := ~, start := 75194, stop := 75208 ), 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 := 75292, root := ~, start := 75259, stop := 75291 ), 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 := 75298, root := ~, start := 75294, stop := 75297 ), 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 := 75321, root := ~, start := 75307, stop := 75320 ), 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 := 75363, root := ~, start := 75333, stop := 75362 ), 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 := 75424, root := ~, start := 75410, stop := 75423 ), 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 := 75443, root := ~, start := 75429, stop := 75442 ), 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 := 75491, root := ~, start := 75459, stop := 75490 ), 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 := 75528, root := ~, start := 75514, stop := 75527 ), 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 := 75568, root := ~, start := 75564, stop := 75567 ), 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 := 76262, root := ~, start := 75574, stop := 76261 ), rec( content := "\n ", count := [ 3, 3, 19, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 76279, root := ~, start := 74864, stop := 76278 ) ], count := [ 3, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 76293, root := ~, start := 74763, stop := 76292 ), 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 := 76371, root := ~, start := 76314, stop := 76370 ), 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 := 76403, root := ~, start := 76374, stop := 76402 ), 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 := 76439, root := ~, start := 76427, stop := 76438 ), 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 := 76515, root := ~, start := 76501, stop := 76514 ), 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 := 76590, root := ~, start := 76576, stop := 76589 ), 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 := 76631, root := ~, start := 76619, stop := 76630 ), 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 := 76704, root := ~, start := 76692, stop := 76703 ), 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 := 76709, root := ~, start := 76705, stop := 76708 ), 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 := 76732, root := ~, start := 76718, stop := 76731 ), 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 := 76745, root := ~, start := 76737, stop := 76744 ), 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 := 76831, root := ~, start := 76819, stop := 76830 ) , rec( content := "\ )]", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "C", next := 76837, root := ~, start := 76801, stop := 76836 ), 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 := 76920, root := ~, start := 76887, stop := 76919 ), 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 := 76926, root := ~, start := 76922, stop := 76925 ), 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 := 76949, root := ~, start := 76935, stop := 76948 ), 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 := 76991, root := ~, start := 76961, stop := 76990 ), 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 := 77055, root := ~, start := 77041, stop := 77054 ), 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 := 77074, root := ~, start := 77060, stop := 77073 ), 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 := 77122, root := ~, start := 77090, stop := 77121 ), 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 := 77159, root := ~, start := 77145, stop := 77158 ), 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 := 77201, root := ~, start := 77197, stop := 77200 ), 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 := 77784, root := ~, start := 77207, stop := 77783 ), rec( content := "\n ", count := [ 3, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 77801, root := ~, start := 76406, stop := 77800 ) ], count := [ 3, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 77815, root := ~, start := 76299, stop := 77814 ), 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 := 77926, root := ~, start := 77836, stop := 77925 ), 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 := 78030, root := ~, start := 77929, stop := 78029 ), 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 := 78062, root := ~, start := 78033, stop := 78061 ), 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 := 78100, root := ~, start := 78086, stop := 78099 ), 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 := 78152, root := ~, start := 78141, stop := 78151 ), 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 := 78196, root := ~, start := 78182, stop := 78195 ), 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 := 78211, root := ~, start := 78201, stop := 78210 ), 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 := 78226, root := ~, start := 78216, stop := 78225 ), 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 := 78261, root := ~, start := 78247, stop := 78260 ), 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 := 78336, root := ~, start := 78322, stop := 78335 ), 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 := 78386, root := ~, start := 78375, stop := 78385 ), 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 := 78407, root := ~, start := 78390, stop := 78406 ), 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 := 78413, root := ~, start := 78409, stop := 78412 ), 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 := 78438, root := ~, start := 78424, stop := 78437 ), 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 := 78486, root := ~, start := 78450, stop := 78485 ), 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 := 78545, root := ~, start := 78531, stop := 78544 ), 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 := 78564, root := ~, start := 78550, stop := 78563 ), 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 := 78614, root := ~, start := 78576, stop := 78613 ), 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 := 78651, root := ~, start := 78637, stop := 78650 ), 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 := 78694, root := ~, start := 78690, stop := 78693 ), 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 := 78727, root := ~, start := 78713, stop := 78726 ), 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 := 78771, root := ~, start := 78739, stop := 78770 ), 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 := 78810, root := ~, start := 78796, stop := 78809 ), 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 := 78847, root := ~, start := 78836, stop := 78846 ), 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 := 78873, root := ~, start := 78856, stop := 78872 ), 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 := 78915, root := ~, start := 78901, stop := 78914 ), 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 := 78920, root := ~, start := 78916, stop := 78919 ), 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 := 79350, root := ~, start := 78926, stop := 79349 ), rec( content := "\n ", count := [ 3, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "Description", next := 79367, root := ~, start := 78065, stop := 79366 ) ], count := [ 3, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 79381, root := ~, start := 77821, stop := 79380 ), 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 := 79459, root := ~, start := 79402, stop := 79458 ), 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 := 79503, root := ~, start := 79462, stop := 79502 ), 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 := 79605, root := ~, start := 79591, stop := 79604 ), 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 := 79648, root := ~, start := 79637, stop := 79647 ), 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 := 79704, root := ~, start := 79678, stop := 79703 ), 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 := 79722, root := ~, start := 79708, stop := 79721 ), 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 := 79751, root := ~, start := 79740, stop := 79750 ), 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 := 79784, root := ~, start := 79770, stop := 79783 ), 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 := 79805, root := ~, start := 79791, stop := 79804 ), 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 := 79860, root := ~, start := 79846, stop := 79859 ), 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 := 79903, root := ~, start := 79899, stop := 79902 ), 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 := 79963, root := ~, start := 79949, stop := 79962 ), 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 := 80938, root := ~, start := 79969, stop := 80937 ), rec( content := "\n ", count := [ 3, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 80955, root := ~, start := 79506, stop := 80954 ) ], count := [ 3, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 80969, root := ~, start := 79387, stop := 80968 ), 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 := 81044, root := ~, start := 80990, stop := 81043 ), 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 := 81076, root := ~, start := 81047, stop := 81075 ), 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 := 81174, root := ~, start := 81160, stop := 81173 ), 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 := 81236, root := ~, start := 81224, stop :=\ 81235 ), 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 := 81285, root := ~, start := 81271, stop :=\ 81284 ), rec( content := ", or\n ", count := [ 3, 3, 23, 5 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], name := "Ite\ m", next := 81303, root := ~, start := 81145, stop := 81302 ), 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 := 81331, root := ~, start := 81317, stop := 81330 ), 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\ := 81361, root := ~, start := 81349, stop := 81360 ), rec( content := " is an\ empty list ", count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 3, 23, 7 ], name := "Item", next := 81386, root := ~, start := 813\ 10, stop := 81385 ) ], count := [ 3, 3, 23, 4 ], mathmode := "Text", name := "List", next := 81398, root := ~, start := 81132, stop := 81397 ), 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 := 81473, root := ~, start := 81459, stop := 81472 ), 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 := 81532, root := ~, start := 81520, stop := 81531 ), 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 := 81607, root := ~, start := 81542, stop := 81606 ), 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 := 81618, root := ~, start := 81614, stop := 81617 ), 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 := 81641, root := ~, start := 81627, stop := 81640 ), 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 := 81683, root := ~, start := 81653, stop := 81682 ), 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 := 81742, root := ~, start := 81728, stop := 81741 ), 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 := 81761, root := ~, start := 81747, stop := 81760 ), 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 := 81811, root := ~, start := 81773, stop := 81810 ), 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 := 81877, root := ~, start := 81863, stop := 81876 ), 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 := 81915, root := ~, start := 81911, stop := 81914 ), 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 := 81946, root := ~, start := 81934, stop := 81945 ), 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 := 82014, root := ~, start := 82000, stop := 82013 ), 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 := 82075, root := ~, start := 82063, stop := 82074 ), 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 := 82134, root := ~, start := 82120, stop := 82133 ), 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 := 82622, root := ~, start := 82187, stop := 82621 ), rec( content := "\n ", count := [ 3, 3, 23, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 82639, root := ~, start := 81079, stop := 82638 ) ], count := [ 3, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 82653, root := ~, start := 80975, stop := 82652 ), 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 := 82721, root := ~, start := 82674, stop := 82720 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveLo\ opsAttr" ), content := 0, count := [ 3, 3, 24, 4 ], mathmode := "Text", name := "Attr", next := 82775, root := ~, start := 82724, stop := 82774 ), 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 := 82807, root := ~, start := 82778, stop := 82806 ), 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 := 82845, root := ~, start := 82831, stop := 82844 ), 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 := 82934, root := ~, start := 82920, stop := 82933 ), 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 := 83013, root := ~, start := 83009, stop := 83012 ), 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 := 83036, root := ~, start := 83022, stop := 83035 ), 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 := 83169, root := ~, start := 83155, stop := 83168 ), 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 := 83623, root := ~, start := 83188, stop := 83622 ), rec( content := "\n ", count := [ 3, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "Description", next := 83640, root := ~, start := 82810, stop := 83639 ) ], count := [ 3, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 83654, root := ~, start := 82659, stop := 83653 ), 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 := 83733, root := ~, start := 83675, stop := 83732 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveAl\ lMultipleEdgesAttr" ), content := 0, count := [ 3, 3, 25, 4 ], mathmode := "Text", name := "Attr", next := 83798, root := ~, start := 83736, stop := 83797 ), 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 := 83830, root := ~, start := 83801, stop := 83829 ), 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 := 83868, root := ~, start := 83854, stop := 83867 ), 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 := 83957, root := ~, start := 83943, stop := 83956 ), 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 := 84048, root := ~, start := 84034, stop := 84047 ), 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 := 84095, root := ~, start := 84091, stop := 84094 ), 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 := 84118, root := ~, start := 84104, stop := 84117 ), 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 := 84246, root := ~, start := 84232, stop := 84245 ), 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 := 84799, root := ~, start := 84277, stop := 84798 ), rec( content := "\n ", count := [ 3, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "Description", next := 84816, root := ~, start := 83833, stop := 84815 ) ], count := [ 3, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 84830, root := ~, start := 83660, stop := 84829 ), 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 := 84952, root := ~, start := 84851, stop := 84951 ), 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 := 85046, root := ~, start := 84955, stop := 85045 ), 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 := 85151, root := ~, start := 85049, stop := 85150 ), 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 := 85183, root := ~, start := 85154, stop := 85182 ), 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 := 85221, root := ~, start := 85207, stop := 85220 ), 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 := 85275, root := ~, start := 85263, stop := 85274 ), 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 := 85328, root := ~, start := 85314, stop := 85327 ), 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 := 85463, root := ~, start := 85437, stop := 85462 ), 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 := 85517, root := ~, start := 85503, stop := 85516 ), 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 := 85590, root := ~, start := 85578, stop := 85589 ), 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 := 85662, root := ~, start := 85637, stop := 85661 ), 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 := 85857, root := ~, start := 85853, stop := 85856 ), 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 := 85899, root := ~, start := 85885, stop := 85898 ), 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 := 85972, root := ~, start := 85968, stop := 85971 ), 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 := 85995, root := ~, start := 85981, stop := 85994 ), 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 := 86037, root := ~, start := 86007, stop := 86036 ), 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 := 86088, root := ~, start := 86074, stop := 86087 ), 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 := 86107, root := ~, start := 86093, stop := 86106 ), 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 := 86157, root := ~, start := 86119, stop := 86156 ), 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 := 86194, root := ~, start := 86180, stop := 86193 ), 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 := 86249, root := ~, start := 86245, stop := 86248 ), 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 := 86832, root := ~, start := 86254, stop := 86831 ), rec( content := "\n ", count := [ 3, 3, 26, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "Description", next := 86849, root := ~, start := 85186, stop := 86848 ) ], count := [ 3, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 86863, root := ~, start := 84836, stop := 86862 ), 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 := 86985, root := ~, start := 86884, stop := 86984 ), 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 := 87065, root := ~, start := 86988, stop := 87064 ), 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 := 87097, root := ~, start := 87068, stop := 87096 ), 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 := 87149, root := ~, start := 87140, stop := 87148 ), 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 := 87160, root := ~, start := 87151, stop := 87159 ), 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 := 87217, root := ~, start := 87190, stop := 87216 ), 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 := 87287, root := ~, start := 87276, stop := 87286 ), 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 := 87357, root := ~, start := 87330, stop := 87356 ), 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 := 87433, root := ~, start := 87429, stop := 87432 ), 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 := 87601, root := ~, start := 87597, stop := 87600 ), 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 := 87664, root := ~, start := 87655, stop := 87663 ), 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 := 87675, root := ~, start := 87666, stop := 87674 ), 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 := 87691, root := ~, start := 87681, stop := 87690 ), 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 := 87751, root := ~, start := 87726, stop := 87750 ), 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 := 87760, root := ~, start := 87751, stop := 87759 ), 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 := 87768, root := ~, start := 87760, stop := 87767 ), 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 := 87777, root := ~, start := 87769, stop := 87776 ), 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 := 87807, root := ~, start := 87782, stop := 87806 ), 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 := 87816, root := ~, start := 87807, stop := 87815 ), 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 := 87824, root := ~, start := 87816, stop := 87823 ), 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 := 87833, root := ~, start := 87825, stop := 87832 ), 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 := 87844, root := ~, start := 87834, stop := 87843 ), 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 := 87881, root := ~, start := 87872, stop := 87880 ), 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 := 87941, root := ~, start := 87933, stop := 87940 ), 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 := 87961, root := ~, start := 87953, stop := 87960 ), 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 := 87971, root := ~, start := 87961, stop := 87970 ), 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 := 88080, root := ~, start := 88072, stop := 88079 ), 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 := 88090, root := ~, start := 88080, stop := 88089 ), 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 := 88129, root := ~, start := 88121, stop := 88128 ), 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 := 88139, root := ~, start := 88129, stop := 88138 ), 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 := 88219, root := ~, start := 88194, stop := 88218 ), 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 := 88228, root := ~, start := 88219, stop := 88227 ), 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 := 88236, root := ~, start := 88228, stop := 88235 ), 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 := 88245, root := ~, start := 88237, stop := 88244 ), 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 := 88256, root := ~, start := 88246, stop := 88255 ), 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 := 88265, root := ~, start := 88257, stop := 88264 ), 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 := 88295, root := ~, start := 88270, stop := 88294 ), 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 := 88303, root := ~, start := 88295, stop := 88302 ), 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 := 88316, root := ~, start := 88303, stop := 88315 ), 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 := 88327, root := ~, start := 88323, stop := 88326 ), 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 := 88391, root := ~, start := 88358, stop := 88390 ), 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 := 88409, root := ~, start := 88405, stop := 88408 ), 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 := 88445, root := ~, start := 88436, stop := 88444 ), 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 := 88461, root := ~, start := 88447, stop := 88460 ), 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 := 88508, root := ~, start := 88474, stop := 88507 ), 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 := 88524, root := ~, start := 88515, stop := 88523 ), 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 := 88540, root := ~, start := 88526, stop := 88539 ), 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 := 88631, root := ~, start := 88617, stop := 88630 ), 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 := 88679, root := ~, start := 88647, stop := 88678 ), 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 := 89221, root := ~, start := 88768, stop := 89220 ), rec( content := "\n ", count := [ 3, 3, 27, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "Description", next := 89238, root := ~, start := 87100, stop := 89237 ) ], count := [ 3, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 89252, root := ~, start := 86869, stop := 89251 ), 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 := 89368, root := ~, start := 89273, stop := 89367 ), 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 := 89444, root := ~, start := 89371, stop := 89443 ), 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 := 89476, root := ~, start := 89447, stop := 89475 ), 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 := 89528, root := ~, start := 89519, stop := 89527 ), 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 := 89539, root := ~, start := 89530, stop := 89538 ), 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 := 89592, root := ~, start := 89569, stop := 89591 ), 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 := 89658, root := ~, start := 89647, stop := 89657 ), 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 := 89724, root := ~, start := 89701, stop := 89723 ), 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 := 89796, root := ~, start := 89792, stop := 89795 ), 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 := 89886, root := ~, start := 89874, stop := 89885 ), 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 := 89977, root := ~, start := 89957, stop := 89976 ), 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 := 90073, root := ~, start := 90059, stop := 90072 ), 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 := 90195, root := ~, start := 90185, stop := 90194 ), 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 := 90240, root := ~, start := 90236, stop := 90239 ), 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 := 90304, root := ~, start := 90271, stop := 90303 ), 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 := 90327, root := ~, start := 90323, stop := 90326 ), 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 := 90363, root := ~, start := 90354, stop := 90362 ), 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 := 90379, root := ~, start := 90365, stop := 90378 ), 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 := 90426, root := ~, start := 90392, stop := 90425 ), 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 := 90442, root := ~, start := 90433, stop := 90441 ), 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 := 90458, root := ~, start := 90444, stop := 90457 ), 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 := 90549, root := ~, start := 90535, stop := 90548 ), 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 := 90597, root := ~, start := 90565, stop := 90596 ), 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 := 91270, root := ~, start := 90686, stop := 91269 ), rec( content := "\n ", count := [ 3, 3, 28, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "Description", next := 91287, root := ~, start := 89479, stop := 91286 ) ], count := [ 3, 3, 28, 1 ], mathmode := "Text", name := "ManSection", next := 91301, root := ~, start := 89258, stop := 91300 ), 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 := 91412, root := ~, start := 91322, stop := 91411 ), 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 := 91483, root := ~, start := 91415, stop := 91482 ), 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 := 91515, root := ~, start := 91486, stop := 91514 ), 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 := 91567, root := ~, start := 91558, stop := 91566 ), 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 := 91578, root := ~, start := 91569, stop := 91577 ), 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 := 91626, root := ~, start := 91608, stop := 91625 ), 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 := 91686, root := ~, start := 91675, stop := 91685 ), 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 := 91747, root := ~, start := 91729, stop := 91746 ), 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 := 91812, root := ~, start := 91808, stop := 91811 ), 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 := 91864, root := ~, start := 91855, stop := 91863 ), 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 := 91875, root := ~, start := 91866, stop := 91874 ), 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 := 91887, root := ~, start := 91877, stop := 91886 ), 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 := 91993, root := ~, start := 91922, stop := 91992 ), 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 := 92059, root := ~, start := 92044, stop := 92058 ), 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 := 92139, root := ~, start := 92131, stop := 92138 ), 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 := 92149, root := ~, start := 92139, stop := 92148 ), 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 := 92166, root := ~, start := 92158, stop := 92165 ), 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 := 92176, root := ~, start := 92166, stop := 92175 ), 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 := 92252, root := ~, start := 92248, stop := 92251 ), 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 := 92342, root := ~, start := 92338, stop := 92341 ), 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 := 92406, root := ~, start := 92373, stop := 92405 ), 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 := 92429, root := ~, start := 92425, stop := 92428 ), 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 := 92465, root := ~, start := 92456, stop := 92464 ), 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 := 92481, root := ~, start := 92467, stop := 92480 ), 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 := 92528, root := ~, start := 92494, stop := 92527 ), 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 := 92544, root := ~, start := 92535, stop := 92543 ), 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 := 92560, root := ~, start := 92546, stop := 92559 ), 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 := 92651, root := ~, start := 92637, stop := 92650 ), 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 := 92699, root := ~, start := 92667, stop := 92698 ), 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 := 93084, root := ~, start := 92788, stop := 93083 ), rec( content := "\n ", count := [ 3, 3, 29, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "Description", next := 93101, root := ~, start := 91518, stop := 93100 ) ], count := [ 3, 3, 29, 1 ], mathmode := "Text", name := "ManSection", next := 93115, root := ~, start := 91307, stop := 93114 ), 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 := 93240, root := ~, start := 93136, stop := 93239 ), 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 := 93327, root := ~, start := 93243, stop := 93326 ), 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 := 93359, root := ~, start := 93330, stop := 93358 ), 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 := 93412, root := ~, start := 93402, stop := 93411 ), 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 := 93424, root := ~, start := 93414, stop := 93423 ), 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 := 93484, root := ~, start := 93454, stop := 93483 ), 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 := 93554, root := ~, start := 93550, stop := 93553 ), 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 := 93594, root := ~, start := 93583, stop := 93593 ), 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 := 93667, root := ~, start := 93637, stop := 93666 ), 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 := 93774, root := ~, start := 93770, stop := 93773 ), 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 := 93843, root := ~, start := 93835, stop := 93842 ), 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 := 93853, root := ~, start := 93845, stop := 93852 ), 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 := 93947, root := ~, start := 93887, stop := 93946 ), 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 := 93994, root := ~, start := 93980, stop := 93993 ), 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 := 94016, root := ~, start := 94002, stop := 94015 ), 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 := 94035, root := ~, start := 94021, stop := 94034 ), 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 := 94071, root := ~, start := 94062, stop := 94070 ), 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 := 94088, root := ~, start := 94079, stop := 94087 ), 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 := 94100, root := ~, start := 94092, stop := 94099 ), 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 := 94119, root := ~, start := 94104, stop := 94118 ), 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 := 94158, root := ~, start := 94150, stop := 94157 ), 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 := 94170, root := ~, start := 94162, stop := 94169 ), 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 := 94182, root := ~, start := 94174, stop := 94181 ), 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 := 94192, root := ~, start := 94188, stop := 94191 ), 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 := 94244, root := ~, start := 94236, stop := 94243 ), 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 := 94291, root := ~, start := 94261, stop := 94290 ), 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 := 94367, root := ~, start := 94311, stop := 94366 ), 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 := 94595, root := ~, start := 94547, stop := 94594 ), 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 := 94647, root := ~, start := 94614, stop := 94646 ), 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 := 94695, root := ~, start := 94682, stop := 94694 ), 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 := 94719, root := ~, start := 94711, stop := 94718 ), 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 := 94753, root := ~, start := 94745, stop := 94752 ), 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 := 94803, root := ~, start := 94795, stop := 94802 ), 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 := 94817, root := ~, start := 94808, stop := 94816 ), 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 := 94917, root := ~, start := 94827, stop := 94916 ), 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 := 94927, root := ~, start := 94923, stop := 94926 ), 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 := 95039, root := ~, start := 95029, stop := 95038 ), 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 := 95051, root := ~, start := 95041, stop := 95050 ), 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 := 95063, root := ~, start := 95053, stop := 95062 ), 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 := 95114, root := ~, start := 95110, stop := 95113 ), 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 := 95548, root := ~, start := 95116, stop := 95547 ), rec( content := "\n ", count := [ 3, 3, 30, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "Description", next := 95565, root := ~, start := 93362, stop := 95564 ) ], count := [ 3, 3, 30, 1 ], mathmode := "Text", name := "ManSection", next := 95579, root := ~, start := 93121, stop := 95578 ), 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 := 95701, root := ~, start := 95600, stop := 95700 ), 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 := 95785, root := ~, start := 95704, stop := 95784 ), 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 := 95817, root := ~, start := 95788, stop := 95816 ), 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 := 95870, root := ~, start := 95860, stop := 95869 ), 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 := 95882, root := ~, start := 95872, stop := 95881 ), 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 := 95939, root := ~, start := 95912, stop := 95938 ), 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 := 96006, root := ~, start := 96002, stop := 96005 ), 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 := 96046, root := ~, start := 96035, stop := 96045 ), 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 := 96116, root := ~, start := 96089, stop := 96115 ), 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 := 96219, root := ~, start := 96215, stop := 96218 ), 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 := 96284, root := ~, start := 96276, stop := 96283 ), 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 := 96294, root := ~, start := 96286, stop := 96293 ), 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 := 96388, root := ~, start := 96328, stop := 96387 ), 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 := 96435, root := ~, start := 96421, stop := 96434 ), 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 := 96457, root := ~, start := 96443, stop := 96456 ), 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 := 96492, root := ~, start := 96484, stop := 96491 ), 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 := 96504, root := ~, start := 96496, stop := 96503 ), 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 := 96520, root := ~, start := 96512, stop := 96519 ), 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 := 96547, root := ~, start := 96538, stop := 96546 ), 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 := 96560, root := ~, start := 96551, stop := 96559 ), 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 := 96572, root := ~, start := 96564, stop := 96571 ), 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 := 96583, root := ~, start := 96579, stop := 96582 ), 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 := 96636, root := ~, start := 96628, stop := 96635 ), 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 := 96680, root := ~, start := 96653, stop := 96679 ), 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 := 96756, root := ~, start := 96700, stop := 96755 ), 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 := 96976, root := ~, start := 96931, stop := 96975 ), 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 := 97027, root := ~, start := 96994, stop := 97026 ), 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 := 97075, root := ~, start := 97062, stop := 97074 ), 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 := 97099, root := ~, start := 97091, stop := 97098 ), 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 := 97133, root := ~, start := 97125, stop := 97132 ), 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 := 97183, root := ~, start := 97175, stop := 97182 ), 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 := 97197, root := ~, start := 97188, stop := 97196 ), 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 := 97295, root := ~, start := 97207, stop := 97294 ), 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 := 97305, root := ~, start := 97301, stop := 97304 ), 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 := 97411, root := ~, start := 97401, stop := 97410 ), 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 := 97423, root := ~, start := 97413, stop := 97422 ), 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 := 97435, root := ~, start := 97425, stop := 97434 ), 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 := 97486, root := ~, start := 97482, stop := 97485 ), 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 := 97883, root := ~, start := 97488, stop := 97882 ), rec( content := "\n ", count := [ 3, 3, 31, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "Description", next := 97900, root := ~, start := 95820, stop := 97899 ) ], count := [ 3, 3, 31, 1 ], mathmode := "Text", name := "ManSection", next := 97914, root := ~, start := 95585, stop := 97913 ), 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 := 97998, root := ~, start := 97935, stop := 97997 ), 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 := 98046, root := ~, start := 98001, stop := 98045 ), 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 := 98084, root := ~, start := 98070, stop := 98083 ), 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 := 98180, root := ~, start := 98122, stop := 98179 ), 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 := 98232, root := ~, start := 98191, stop := 98231 ), 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 := 98259, root := ~, start := 98248, stop := 98258 ), 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 := 98288, root := ~, start := 98274, stop := 98287 ), 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 := 98324, root := ~, start := 98316, stop := 98323 ), 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 := 98367, root := ~, start := 98363, stop := 98366 ), 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 := 98433, root := ~, start := 98419, stop := 98432 ), 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 := 98462, root := ~, start := 98442, stop := 98461 ), 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 := 98540, root := ~, start := 98526, stop := 98539 ), 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 := 98571, root := ~, start := 98557, stop := 98570 ), 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 := 98605, root := ~, start := 98594, stop := 98604 ), 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 := 98615, root := ~, start := 98611, stop := 98614 ), 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 := 98732, root := ~, start := 98702, stop := 98731 ), 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 := 98804, root := ~, start := 98800, stop := 98803 ), 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 := 98907, root := ~, start := 98839, stop := 98906 ), 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 := 99350, root := ~, start := 98914, stop := 99349 ), rec( content := "\n ", count := [ 3, 3, 32, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "Description", next := 99367, root := ~, start := 98049, stop := 99366 ) ], count := [ 3, 3, 32, 1 ], mathmode := "Text", name := "ManSection", next := 99381, root := ~, start := 97920, stop := 99380 ), 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 := 99462, root := ~, start := 99402, stop := 99461 ), 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 := 99510, root := ~, start := 99465, stop := 99509 ), 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 := 99548, root := ~, start := 99534, stop := 99547 ), 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 := 99638, root := ~, start := 99583, stop := 99637 ), 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 := 99687, root := ~, start := 99649, stop := 99686 ), 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 := 99714, root := ~, start := 99703, stop := 99713 ), 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 := 99743, root := ~, start := 99729, stop := 99742 ), 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 := 99779, root := ~, start := 99771, stop := 99778 ), 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 := 99822, root := ~, start := 99818, stop := 99821 ), 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 := 99888, root := ~, start := 99874, stop := 99887 ), 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 := 99917, root := ~, start := 99897, stop := 99916 ), 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 := 99982, root := ~, start := 99968, stop := 99981 ), 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 := 100003, root := ~, start := 99989, stop := 100002 ), 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 := 100032, root := ~, start := 100021, stop := 100031 ), 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 := 100042, root := ~, start := 100038, stop := 100041 ), 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 := 100155, root := ~, start := 100128, stop := 100154 ), 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 := 100227, root := ~, start := 100223, stop := 100226 ), 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 := 100328, root := ~, start := 100263, stop := 100327 ), 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 := 100780, root := ~, start := 100335, stop := 100779 ), rec( content := "\n ", count := [ 3, 3, 33, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "Description", next := 100797, root := ~, start := 99513, stop := 100796 ) ], count := [ 3, 3, 33, 1 ], mathmode := "Text", name := "ManSection", next := 100811, root := ~, start := 99387, stop := 100810 ), 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 := 100872, root := ~, start := 100832, stop := 100871 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeDigraph" ), content := 0, count := [ 3, 3, 34, 4 ], mathmode := "Text", name := "Oper", next := 100915, root := ~, start := 100875, stop := 100914 ), 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 := 100947, root := ~, start := 100918, stop := 100946 ), 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 := 100998, root := ~, start := 100984, stop := 100997 ), 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 := 101103, root := ~, start := 101089, stop := 101102 ), 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 := 101152, root := ~, start := 101144, stop := 101151 ), 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 := 101173, root := ~, start := 101165, stop := 101172 ), 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 := 101249, root := ~, start := 101241, stop := 101248 ), 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 := 101310, root := ~, start := 101302, stop := 101309 ), 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 := 101315, root := ~, start := 101311, stop := 101314 ), 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 := 101422, root := ~, start := 101408, stop := 101421 ), 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 := 101641, root := ~, start := 101446, stop := 101640 ), rec( content := "\n ", count := [ 3, 3, 34, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "Description", next := 101658, root := ~, start := 100950, stop := 101657 ) ], count := [ 3, 3, 34, 1 ], mathmode := "Text", name := "ManSection", next := 101672, root := ~, start := 100817, stop := 101671 ), 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 := 101743, root := ~, start := 101693, stop := 101742 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeUndirectedD\ igraph" ), content := 0, count := [ 3, 3, 35, 4 ], mathmode := "Text", name := "Oper", next := 101796, root := ~, start := 101746, stop := 101795 ), 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 := 101828, root := ~, start := 101799, stop := 101827 ), 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 := 101889, root := ~, start := 101875, stop := 101888 ), 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 := 102008, root := ~, start := 101994, stop := 102007 ), 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 := 102164, root := ~, start := 102160, stop := 102163 ), 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 := 102271, root := ~, start := 102257, stop := 102270 ), 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 := 102535, root := ~, start := 102295, stop := 102534 ), rec( content := "\n ", count := [ 3, 3, 35, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "Description", next := 102552, root := ~, start := 101831, stop := 102551 ) ], count := [ 3, 3, 35, 1 ], mathmode := "Text", name := "ManSection", next := 102566, root := ~, start := 101678, stop := 102565 ), 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 := 102629, root := ~, start := 102587, stop := 102628 ), 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 := 102661, root := ~, start := 102632, stop := 102660 ), 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 := 102700, root := ~, start := 102686, stop := 102699 ), 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 := 102738, root := ~, start := 102730, stop := 102737 ), 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 := 102802, root := ~, start := 102788, stop := 102801 ), 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 := 102933, root := ~, start := 102916, stop := 102932 ), 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 := 102955, root := ~, start := 102938, stop := 102954 ), 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 := 102984, root := ~, start := 102971, stop := 102983 ), 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 := 103017, root := ~, start := 103003, stop := 103016 ), 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 := 103081, root := ~, start := 103077, stop := 103080 ), 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 := 103104, root := ~, start := 103090, stop := 103103 ), 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 := 103136, root := ~, start := 103122, stop := 103135 ), 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 := 103180, root := ~, start := 103166, stop := 103179 ), 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 := 103466, root := ~, start := 103276, stop := 103465 ), rec( content := "\n ", count := [ 3, 3, 36, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "Description", next := 103483, root := ~, start := 102664, stop := 103482 ) ], count := [ 3, 3, 36, 1 ], mathmode := "Text", name := "ManSection", next := 103497, root := ~, start := 102572, stop := 103496 ), 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 := 103569, root := ~, start := 103518, stop := 103568 ), 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 := 103601, root := ~, start := 103572, stop := 103600 ), 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 := 103640, root := ~, start := 103626, stop := 103639 ), 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 := 103678, root := ~, start := 103670, stop := 103677 ), 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 := 103752, root := ~, start := 103738, stop := 103751 ), 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 := 103884, root := ~, start := 103867, stop := 103883 ), 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 := 103906, root := ~, start := 103889, stop := 103905 ), 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 := 103941, root := ~, start := 103922, stop := 103940 ), 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 := 103970, root := ~, start := 103956, stop := 103969 ), 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 := 104080, root := ~, start := 104076, stop := 104079 ), 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 := 104103, root := ~, start := 104089, stop := 104102 ), 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 := 104135, root := ~, start := 104121, stop := 104134 ), 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 := 104179, root := ~, start := 104165, stop := 104178 ), 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 := 104473, root := ~, start := 104275, stop := 104472 ), rec( content := "\n ", count := [ 3, 3, 37, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "Description", next := 104490, root := ~, start := 103604, stop := 104489 ) ], count := [ 3, 3, 37, 1 ], mathmode := "Text", name := "ManSection", next := 104504, root := ~, start := 103503, stop := 104503 ), 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 := 104572, root := ~, start := 104525, stop := 104571 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphAddAllLo\ opsAttr" ), content := 0, count := [ 3, 3, 38, 4 ], mathmode := "Text", name := "Attr", next := 104626, root := ~, start := 104575, stop := 104625 ), 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 := 104658, root := ~, start := 104629, stop := 104657 ), 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 := 104707, root := ~, start := 104693, stop := 104706 ), 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 := 104780, root := ~, start := 104766, stop := 104779 ), 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 := 104872, root := ~, start := 104858, stop := 104871 ), 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 := 104877, root := ~, start := 104873, stop := 104876 ), 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 := 104900, root := ~, start := 104886, stop := 104899 ), 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 := 105058, root := ~, start := 105044, stop := 105057 ), 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 := 105840, root := ~, start := 105073, stop := 105839 ), rec( content := "\n ", count := [ 3, 3, 38, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "Description", next := 105857, root := ~, start := 104661, stop := 105856 ) ], count := [ 3, 3, 38, 1 ], mathmode := "Text", name := "ManSection", next := 105871, root := ~, start := 104510, stop := 105870 ), 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 := 105975, root := ~, start := 105892, stop := 105974 ), 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 := 106065, root := ~, start := 105978, stop := 106064 ), 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 := 106097, root := ~, start := 106068, stop := 106096 ), 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 := 106315, root := ~, start := 106301, stop := 106314 ), 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 := 106412, root := ~, start := 106398, stop := 106411 ), 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 := 106428, root := ~, start := 106420, stop := 106427 ), 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 := 106462, root := ~, start := 106451, stop := 106461 ), 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 := 106549, root := ~, start := 106469, stop := 106548 ), 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 := 106555, root := ~, start := 106551, stop := 106554 ), 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 := 106578, root := ~, start := 106564, stop := 106577 ), 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 := 106610, root := ~, start := 106596, stop := 106609 ), 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 := 106654, root := ~, start := 106640, stop := 106653 ), 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 := 107094, root := ~, start := 106750, stop := 107093 ), rec( content := "\n ", count := [ 3, 3, 39, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Description", next := 107111, root := ~, start := 106100, stop := 107110 ) ], count := [ 3, 3, 39, 1 ], mathmode := "Text", name := "ManSection", next := 107125, root := ~, start := 105877, stop := 107124 ), 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 := 107192, root := ~, start := 107146, stop := 107191 ), 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 := 107224, root := ~, start := 107195, stop := 107223 ), 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 := 107317, root := ~, start := 107303, stop := 107316 ), 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 := 107338, root := ~, start := 107330, stop := 10733\ 7 ), 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 := 1073\ 64, root := ~, start := 107350, stop := 107363 ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "E", next := 107368, root := ~, start := 107327, stop := 107367 ), 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 := 107445, root := ~, start := 107437, stop := 107444 ), 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 := 107506, root := ~, start := 107492, stop := 107505 ), 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 := 107552, root := ~, start := 107538, stop := 107551 ), 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 := 107663, root := ~, start := 107655, stop := 107662 ), 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 := 107685, root := ~, start := 107677, stop := 107684 ), 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 := 107727, root := ~, start := 107691, stop := 107726 ), 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 := 107758, root := ~, start := 107729, stop := 107757 ), 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 := 107792, root := ~, start := 107760, stop := 107791 ), 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 := 107829, root := ~, start := 107798, stop := 107828 ), 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 := 107835, root := ~, start := 107831, stop := 107834 ), 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 := 107876, root := ~, start := 107855, stop := 107875 ), 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 := 107897, root := ~, start := 107889, stop := 107896 ), 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 := 107927, root := ~, start := 107913, stop := 107926 ), 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 := 108259, root := ~, start := 107933, stop := 108258 ), rec( content := "\n ", count := [ 3, 3, 40, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Description", next := 108276, root := ~, start := 107227, stop := 108275 ) ], count := [ 3, 3, 40, 1 ], mathmode := "Text", name := "ManSection", next := 108290, root := ~, start := 107131, stop := 108289 ), 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 := 108358, root := ~, start := 108311, stop := 108357 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMycielsk\ ianAttr" ), content := 0, count := [ 3, 3, 41, 4 ], mathmode := "Text", name := "Attr", next := 108412, root := ~, start := 108361, stop := 108411 ), 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 := 108444, root := ~, start := 108415, stop := 108443 ), 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 := 108482, root := ~, start := 108468, stop := 108481 ), 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 := 108541, root := ~, start := 108516, stop := 108540 ), 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 := 108583, root := ~, start := 108569, stop := 108582 ), 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 := 108588, root := ~, start := 108584, stop := 108587 ), 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 := 108814, root := ~, start := 108762, stop := 108813 ), 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 := 108819, root := ~, start := 108815, stop := 108818 ), 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 := 108842, root := ~, start := 108828, stop := 108841 ), 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 := 108948, root := ~, start := 108934, stop := 108947 ), 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 := 108998, root := ~, start := 108994, stop := 108997 ), 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 := 109568, root := ~, start := 109004, stop := 109567 ), rec( content := "\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "Description", next := 109585, root := ~, start := 108447, stop := 109584 ) ], count := [ 3, 3, 41, 1 ], mathmode := "Text", name := "ManSection", next := 109599, root := ~, start := 108296, stop := 109598 ), rec( content := "\n\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 109613, root := ~, start := 43573, stop := 109612 ), 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 := 109660, root := ~, start := 109626, stop := 109659 ), 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 := 109729, root := ~, start := 109680, stop := 109728 ), 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 := 109761, root := ~, start := 109732, stop := 109760 ), 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 := 109824, root := ~, start := 109813, stop := 109823 ), 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 := 109970, root := ~, start := 109959, stop := 109969 ), 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 := 110004, root := ~, start := 109974, stop := 110003 ), 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 := 110072, root := ~, start := 110061, stop := 110071 ), 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 := 110112, root := ~, start := 110076, stop := 110111 ), 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 := 110197, root := ~, start := 110186, stop := 110196 ), 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 := 110257, root := ~, start := 110219, stop := 110256 ), 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 := 110281, root := ~, start := 110277, stop := 110280 ), 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 := 110298, root := ~, start := 110290, stop := 110297 ), 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 := 110383, root := ~, start := 110375, stop := 110382 ), 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 := 110466, root := ~, start := 110462, stop := 110465 ), 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 := 110512, root := ~, start := 110504, stop := 110511 ), 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 := 110553, root := ~, start := 110539, stop := 110552 ), 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 := 110564, root := ~, start := 110554, stop := 110563 ), 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 := 110579, root := ~, start := 110565, stop := 110578 ), 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 := 110678, root := ~, start := 110670, stop := 110677 ), 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 := 110695, root := ~, start := 110687, stop := 110694 ), 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 := 111093, root := ~, start := 110799, stop := 111092 ), rec( content := "\n ", count := [ 3, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 111110, root := ~, start := 109764, stop := 111109 ) ], count := [ 3, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 111124, root := ~, start := 109665, stop := 111123 ), 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 := 111191, root := ~, start := 111145, stop := 111190 ), 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 := 111223, root := ~, start := 111194, stop := 111222 ), 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 := 111255, root := ~, start := 111247, stop := 111254 ), 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 := 111340, root := ~, start := 111332, stop := 111339 ), 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 := 111395, root := ~, start := 111387, stop := 111394 ), 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 := 111454, root := ~, start := 111446, stop := 111453 ), 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 := 111477, root := ~, start := 111469, stop := 111476 ), 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 := 111608, root := ~, start := 111595, stop := 111607 ), 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 := 111628, root := ~, start := 111608, stop := 111627 ), 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 := 111636, root := ~, start := 111628, stop := 111635 ), 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 := 111646, root := ~, start := 111642, stop := 111645 ), 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 := 111950, root := ~, start := 111946, stop := 111949 ), 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 := 112159, root := ~, start := 111955, stop := 112158 ), rec( content := "\n ", count := [ 3, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 112176, root := ~, start := 111226, stop := 112175 ) ], count := [ 3, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 112190, root := ~, start := 111130, stop := 112189 ), 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 := 112258, root := ~, start := 112211, stop := 112257 ), 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 := 112290, root := ~, start := 112261, stop := 112289 ), 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 := 112353, root := ~, start := 112342, stop := 112352 ), 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 := 112499, root := ~, start := 112488, stop := 112498 ), 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 := 112533, root := ~, start := 112503, stop := 112532 ), 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 := 112601, root := ~, start := 112590, stop := 112600 ), 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 := 112641, root := ~, start := 112605, stop := 112640 ), 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 := 112726, root := ~, start := 112715, stop := 112725 ), 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 := 112786, root := ~, start := 112748, stop := 112785 ), 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 := 112810, root := ~, start := 112806, stop := 112809 ), 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 := 112827, root := ~, start := 112819, stop := 112826 ), 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 := 112914, root := ~, start := 112906, stop := 112913 ), 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 := 112955, root := ~, start := 112929, stop := 112954 ), 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 := 112961, root := ~, start := 112957, stop := 112960 ), 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 := 113153, root := ~, start := 112966, stop := 113152 ), rec( content := "\n ", count := [ 3, 4, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Description", next := 113170, root := ~, start := 112293, stop := 113169 ) ], count := [ 3, 4, 3, 1 ], mathmode := "Text", name := "ManSection", next := 113184, root := ~, start := 112196, stop := 113183 ), 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 := 113241, root := ~, start := 113205, stop := 113240 ), 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 := 113273, root := ~, start := 113244, stop := 113272 ), 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 := 113336, root := ~, start := 113325, stop := 113335 ), 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 := 113482, root := ~, start := 113471, stop := 113481 ), 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 := 113516, root := ~, start := 113486, stop := 113515 ), 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 := 113584, root := ~, start := 113573, stop := 113583 ), 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 := 113624, root := ~, start := 113588, stop := 113623 ), 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 := 113709, root := ~, start := 113698, stop := 113708 ), 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 := 113769, root := ~, start := 113731, stop := 113768 ), 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 := 113793, root := ~, start := 113789, stop := 113792 ), 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 := 113810, root := ~, start := 113802, stop := 113809 ), 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 := 113889, root := ~, start := 113881, stop := 113888 ), 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 := 113936, root := ~, start := 113928, stop := 113935 ), 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 := 113960, root := ~, start := 113952, stop := 113959 ), 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 := 113980, root := ~, start := 113972, stop := 11397\ 9 ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 113984, root := ~, start := 113965, stop := 113983 ), 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 := 114020, root := ~, start := 113990, stop := 114019 ), 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 := 114026, root := ~, start := 114022, stop := 114025 ), 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 := 114221, root := ~, start := 114031, stop := 114220 ), rec( content := "\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Description", next := 114238, root := ~, start := 113276, stop := 114237 ) ], count := [ 3, 4, 4, 1 ], mathmode := "Text", name := "ManSection", next := 114252, root := ~, start := 113190, stop := 114251 ), rec( content := "\n\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 114266, root := ~, start := 109617, stop := 114265 ) , 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 := 114315, root := ~, start := 114279, stop := 114314 ), 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 := 114378, root := ~, start := 114335, stop := 114377 ), 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 := 114410, root := ~, start := 114381, stop := 114409 ), 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 := 114442, root := ~, start := 114434, stop := 114441 ), 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 := 114513, root := ~, start := 114505, stop := 114512 ), 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 := 114538, root := ~, start := 114530, stop := 11453\ 7 ), rec( content := " - 1", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114546, root := ~, start := 114527, stop := 114545 ), 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 := 114600, root := ~, start := 114592, stop := 114599 ), 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 := 114615, root := ~, start := 114607, stop := 114614 ), 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 := 114629, root := ~, start := 114621, stop := 114628 ), 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 := 114681, root := ~, start := 114673, stop := 114680 ), 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 := 114704, root := ~, start := 114692, stop := 114703 ), 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 := 114710, root := ~, start := 114706, stop := 114709 ), 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 := 114758, root := ~, start := 114747, stop := 114757 ), 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 := 114904, root := ~, start := 114893, stop := 114903 ), 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 := 114938, root := ~, start := 114908, stop := 114937 ), 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 := 115006, root := ~, start := 114995, stop := 115005 ), 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 := 115046, root := ~, start := 115010, stop := 115045 ), 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 := 115131, root := ~, start := 115120, stop := 115130 ), 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 := 115191, root := ~, start := 115153, stop := 115190 ), 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 := 115215, root := ~, start := 115211, stop := 115214 ), 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 := 115272, root := ~, start := 115225, stop := 115271 ), 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 := 115501, root := ~, start := 115318, stop := 115500 ), rec( content := "\n ", count := [ 3, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 115518, root := ~, start := 114413, stop := 115517 ) ], count := [ 3, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 115532, root := ~, start := 114320, stop := 115531 ), 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 := 115599, root := ~, start := 115553, stop := 115598 ), 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 := 115631, root := ~, start := 115602, stop := 115630 ), 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 := 115663, root := ~, start := 115655, stop := 115662 ), 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 := 115751, root := ~, start := 115743, stop := 115750 ), 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 := 115797, root := ~, start := 115766, stop := 115796 ), 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 := 115803, root := ~, start := 115799, stop := 115802 ), 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 := 115851, root := ~, start := 115840, stop := 115850 ), 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 := 115997, root := ~, start := 115986, stop := 115996 ), 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 := 116031, root := ~, start := 116001, stop := 116030 ), 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 := 116099, root := ~, start := 116088, stop := 116098 ), 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 := 116139, root := ~, start := 116103, stop := 116138 ), 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 := 116224, root := ~, start := 116213, stop := 116223 ), 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 := 116284, root := ~, start := 116246, stop := 116283 ), 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 := 116502, root := ~, start := 116309, stop := 116501 ), rec( content := "\n ", count := [ 3, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 116519, root := ~, start := 115634, stop := 116518 ) ], count := [ 3, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 116533, root := ~, start := 115538, stop := 116532 ), 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 := 116612, root := ~, start := 116554, stop := 116611 ), 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 := 116644, root := ~, start := 116615, stop := 116643 ), 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 := 116833, root := ~, start := 116825, stop := 116832 ), 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 := 116852, root := ~, start := 116844, stop := 116851 ), 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 := 116880, root := ~, start := 116872, stop := 116879 ), 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 := 116893, root := ~, start := 116885, stop := 116892 ), 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 := 116928, root := ~, start := 116924, stop := 116927 ), 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 := 116945, root := ~, start := 116937, stop := 116944 ), 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 := 116958, root := ~, start := 116950, stop := 116957 ), 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 := 117077, root := ~, start := 117069, stop := 117076 ), 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 := 117118, root := ~, start := 117103, stop := 117117 ), 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 := 117136, root := ~, start := 117128, stop := 117135 ), 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 := 117185, root := ~, start := 117162, stop := 117184 ), 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 := 117196, root := ~, start := 117192, stop := 117195 ), 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 := 117244, root := ~, start := 117233, stop := 117243 ), 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 := 117390, root := ~, start := 117379, stop := 117389 ), 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 := 117424, root := ~, start := 117394, stop := 117423 ), 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 := 117492, root := ~, start := 117481, stop := 117491 ), 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 := 117532, root := ~, start := 117496, stop := 117531 ), 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 := 117617, root := ~, start := 117606, stop := 117616 ), 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 := 117677, root := ~, start := 117639, stop := 117676 ), 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 := 117941, root := ~, start := 117703, stop := 117940 ), rec( content := "\n ", count := [ 3, 5, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 117958, root := ~, start := 116647, stop := 117957 ) ], count := [ 3, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 117972, root := ~, start := 116539, stop := 117971 ), 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 := 118062, root := ~, start := 117997, stop := 118061 ), 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 := 118096, root := ~, start := 118067, stop := 118095 ), 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 := 118145, root := ~, start := 118132, stop := 118144 ), 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 := 118157, root := ~, start := 118149, stop := 118156 ), 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 := 118236, root := ~, start := 118228, stop := 118235 ), 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 := 118294, root := ~, start := 118286, stop := 11829\ 3 ), 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 := 118309, root := ~, start := 118301, stop := 11830\ 8 ), rec( content := "[n]]", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 118317, root := ~, start := 118282, stop := 118316 ), 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 := 118433, root := ~, start := 118429, stop := 118432 ), 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 := 118483, root := ~, start := 118472, stop := 118482 ), 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 := 118633, root := ~, start := 118622, stop := 118632 ), 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 := 118667, root := ~, start := 118637, stop := 118666 ), 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 := 118737, root := ~, start := 118726, stop := 118736 ), 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 := 118779, root := ~, start := 118741, stop := 118778 ), 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 := 118866, root := ~, start := 118855, stop := 118865 ), 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 := 118928, root := ~, start := 118888, stop := 118927 ), 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 := 119209, root := ~, start := 118955, stop := 119208 ), rec( content := "\n ", count := [ 3, 5, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "Description", next := 119226, root := ~, start := 118101, stop := 119225 ) ], count := [ 3, 5, 4, 1 ], mathmode := "Text", name := "ManSection", next := 119240, root := ~, start := 117980, stop := 119239 ), 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 := 119304, root := ~, start := 119261, stop := 119303 ), 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 := 119336, root := ~, start := 119307, stop := 119335 ), 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 := 119368, root := ~, start := 119360, stop := 119367 ), 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 := 119428, root := ~, start := 119416, stop := 119427 ), 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 := 119454, root := ~, start := 119446, stop := 119453 ), 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 := 119476, root := ~, start := 119468, stop := 119475 ), 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 := 119526, root := ~, start := 119518, stop := 119525 ), 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 := 119541, root := ~, start := 119533, stop := 119540 ), 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 := 119555, root := ~, start := 119547, stop := 119554 ), 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 := 119607, root := ~, start := 119599, stop := 119606 ), 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 := 119630, root := ~, start := 119618, stop := 119629 ), 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 := 119686, root := ~, start := 119678, stop := 119685 ), 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 := 119705, root := ~, start := 119697, stop := 119704 ), 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 := 119711, root := ~, start := 119707, stop := 119710 ), 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 := 119759, root := ~, start := 119748, stop := 119758 ), 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 := 119905, root := ~, start := 119894, stop := 119904 ), 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 := 119939, root := ~, start := 119909, stop := 119938 ), 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 := 120007, root := ~, start := 119996, stop := 120006 ), 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 := 120047, root := ~, start := 120011, stop := 120046 ), 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 := 120132, root := ~, start := 120121, stop := 120131 ), 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 := 120190, root := ~, start := 120154, stop := 120189 ), 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 := 120463, root := ~, start := 120213, stop := 120462 ), rec( content := "\n ", count := [ 3, 5, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "Description", next := 120480, root := ~, start := 119339, stop := 120479 ) ], count := [ 3, 5, 5, 1 ], mathmode := "Text", name := "ManSection", next := 120494, root := ~, start := 119246, stop := 120493 ), 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 := 120558, root := ~, start := 120515, stop := 120557 ), rec( attributes := rec( Arg := "[filt, ]n", Name := "NullDigraph" ), content := 0, count := [ 3, 5, 6, 4 ], mathmode := "Text", name := "Oper", next := 120603, root := ~, start := 120561, stop := 120602 ), 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 := 120635, root := ~, start := 120606, stop := 120634 ), 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 := 120667, root := ~, start := 120659, stop := 120666 ), 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 := 120737, root := ~, start := 120725, stop := 120736 ), 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 := 120752, root := ~, start := 120741, stop := 120751 ), 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 := 120774, root := ~, start := 120766, stop := 120773 ), 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 := 120836, root := ~, start := 120832, stop := 120835 ), 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 := 120890, root := ~, start := 120879, stop := 120889 ), 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 := 121036, root := ~, start := 121025, stop := 121035 ), 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 := 121070, root := ~, start := 121040, stop := 121069 ), 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 := 121138, root := ~, start := 121127, stop := 121137 ), 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 := 121178, root := ~, start := 121142, stop := 121177 ), 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 := 121263, root := ~, start := 121252, stop := 121262 ), 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 := 121323, root := ~, start := 121285, stop := 121322 ), 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 := 121347, root := ~, start := 121343, stop := 121346 ), 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 := 121371, root := ~, start := 121353, stop := 121370 ), 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 := 121408, root := ~, start := 121389, stop := 121407 ), 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 := 121659, root := ~, start := 121415, stop := 121658 ), rec( content := "\n ", count := [ 3, 5, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "Description", next := 121676, root := ~, start := 120638, stop := 121675 ) ], count := [ 3, 5, 6, 1 ], mathmode := "Text", name := "ManSection", next := 121690, root := ~, start := 120500, stop := 121689 ), 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 := 121759, root := ~, start := 121711, stop := 121758 ), 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 := 121791, root := ~, start := 121762, stop := 121790 ), 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 := 121823, root := ~, start := 121815, stop := 121822 ), 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 := 121836, root := ~, start := 121828, stop := 121835 ), 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 := 121976, root := ~, start := 121952, stop := 121975 ), 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 := 121991, root := ~, start := 121977, stop := 121990 ), 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 := 121997, root := ~, start := 121993, stop := 121996 ), 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 := 122045, root := ~, start := 122034, stop := 122044 ), 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 := 122191, root := ~, start := 122180, stop := 122190 ), 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 := 122225, root := ~, start := 122195, stop := 122224 ), 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 := 122293, root := ~, start := 122282, stop := 122292 ), 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 := 122333, root := ~, start := 122297, stop := 122332 ), 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 := 122418, root := ~, start := 122407, stop := 122417 ), 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 := 122478, root := ~, start := 122440, stop := 122477 ), 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 := 122502, root := ~, start := 122498, stop := 122501 ), 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 := 122532, root := ~, start := 122512, stop := 122531 ), 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 := 122547, root := ~, start := 122533, stop := 122546 ), 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 := 122590, root := ~, start := 122582, stop := 122589 ), 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 := 122629, root := ~, start := 122621, stop := 12262\ 8 ), rec( content := "]", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "C", next := 122634, root := ~, start := 122612, stop := 122633 ), 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 := 122727, root := ~, start := 122719, stop := 12272\ 6 ), rec( content := " - 1", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "M", next := 122735, root := ~, start := 122716, stop := 122734 ), 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 := 123284, root := ~, start := 122742, stop := 123283 ), rec( content := "\n ", count := [ 3, 5, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "Description", next := 123301, root := ~, start := 121794, stop := 123300 ) ], count := [ 3, 5, 7, 1 ], mathmode := "Text", name := "ManSection", next := 123315, root := ~, start := 121696, stop := 123314 ), 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 := 123377, root := ~, start := 123336, stop := 123376 ), 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 := 123409, root := ~, start := 123380, stop := 123408 ), 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 := 123491, root := ~, start := 123436, stop := 123490 ), 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 := 123496, root := ~, start := 123492, stop := 123495 ), 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 := 123852, root := ~, start := 123502, stop := 123851 ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "P", next := 123856, root := ~, start := 123852, stop := 123855 ), 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 := 123908, root := ~, start := 123897, stop := 123907 ), 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 := 124054, root := ~, start := 124043, stop := 124053 ), 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 := 124088, root := ~, start := 124058, stop := 124087 ), 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 := 124156, root := ~, start := 124145, stop := 124155 ), 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 := 124196, root := ~, start := 124160, stop := 124195 ), 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 := 124281, root := ~, start := 124270, stop := 124280 ), 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 := 124341, root := ~, start := 124303, stop := 124340 ), 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 := 124365, root := ~, start := 124361, stop := 124364 ), 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 := 124418, root := ~, start := 124380, stop := 124417 ), 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 := 124577, root := ~, start := 124421, stop := 124576 ), rec( content := "\n ", count := [ 3, 5, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "Description", next := 124594, root := ~, start := 123412, stop := 124593 ) ], count := [ 3, 5, 8, 1 ], mathmode := "Text", name := "ManSection", next := 124608, root := ~, start := 123321, stop := 124607 ), 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 := 124687, root := ~, start := 124629, stop := 124686 ), 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 := 124719, root := ~, start := 124690, stop := 124718 ), 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 := 124751, root := ~, start := 124743, stop := 124750 ), 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 := 124786, root := ~, start := 124778, stop := 124785 ), 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 := 124838, root := ~, start := 124830, stop := 12483\ 7 ), rec( content := " / 2", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "C", next := 124846, root := ~, start := 124827, stop := 124845 ), 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 := 124917, root := ~, start := 124884, stop := 124916 ), 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 := 124933, root := ~, start := 124925, stop := 12493\ 2 ), rec( content := ", k)", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "M", next := 124941, root := ~, start := 124918, stop := 124940 ), 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 := 124947, root := ~, start := 124943, stop := 124946 ), 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 := 125029, root := ~, start := 124962, stop := 125028 ), 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 := 125039, root := ~, start := 125035, stop := 125038 ), 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 := 125464, root := ~, start := 125045, stop := 125463 ), 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 := 125473, root := ~, start := 125469, stop := 125472 ), 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 := 125521, root := ~, start := 125510, stop := 125520 ), 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 := 125667, root := ~, start := 125656, stop := 125666 ), 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 := 125701, root := ~, start := 125671, stop := 125700 ), 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 := 125769, root := ~, start := 125758, stop := 125768 ), 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 := 125809, root := ~, start := 125773, stop := 125808 ), 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 := 125894, root := ~, start := 125883, stop := 125893 ), 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 := 125954, root := ~, start := 125916, stop := 125953 ), 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 := 125978, root := ~, start := 125974, stop := 125977 ), 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 := 126020, root := ~, start := 125993, stop := 126019 ), 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 := 126495, root := ~, start := 126023, stop := 126494 ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "Description", next := 126512, root := ~, start := 124722, stop := 126511 ) ], count := [ 3, 5, 9, 1 ], mathmode := "Text", name := "ManSection", next := 126526, root := ~, start := 124614, stop := 126525 ), rec( content := "\n\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 126540, root := ~, start := 114270, stop := 126539 ) , rec( content := "\n\n", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 126552, root := ~, start := 19173, stop := 126551 ), 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 := 126629, root := ~, start := 126601, stop := 126628 ) , 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 := 126683, root := ~, start := 126642, stop := 126682 ), 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 := 126724, root := ~, start := 126709, stop := 12672\ 3 ), 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 := 1267\ 42, root := ~, start := 126727, stop := 126741 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "C", next := 126746, root := ~, start := 126706, stop := 126745 ) ], count := [ 4, 1, 0, 4 ], name := "Mark", next := 126753, root := ~, start := 126700, stop := 126752 ), 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 := 126796, root := ~, start := 126788, stop := 12679\ 5 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 5 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "Index", next := 126819, root := ~, start := 126773, stop := 126818 ), 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 := 126846, root := ~, start := 126835, stop := 126845 ), 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 := 126865, root := ~, start := 126850, stop := 126864 ), 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 := 126885, root := ~, start := 126870, stop := 126884 ), 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 := 126952, root := ~, start := 126937, stop := 12695\ 1 ), 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 := 1269\ 93, root := ~, start := 126978, stop := 126992 ), rec( content := ")", count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "C", next := 126998, root := ~, start := 126921, stop := 126997 ), 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 := 127053, root := ~, start := 127049, stop := 127052 ), 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 := 127146, root := ~, start := 127131, stop := 127145 ), 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 := 127166, root := ~, start := 127151, stop := 127165 ), rec( content := ".\n \ ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], name := "Item", next := 127180, root := ~, start := 126759, stop := 127179 ), 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 := 127211, root := ~, start := 127196, stop := 12721\ 0 ), 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 := 1272\ 32, root := ~, start := 127217, stop := 127231 ) ], count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "C", next := 127236, root := ~, start := 127193, stop := 127235 ) ], count := [ 4, 1, 0, 7 ], name := "Mark", next := 127243, root := ~, start := 127187, stop := 127242 ), 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 := 127292, root := ~, start := 127281, stop := 12729\ 1 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 8 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "Index", next := 127315, root := ~, start := 127263, stop := 127314 ), 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 := 127357, root := ~, start := 127346, stop := 127356 ), 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 := 127452, root := ~, start := 127442, stop \ := 127451 ), 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 := 127483, root := ~, start := 127468, stop := 127482 ), 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 := 127529, root := ~, start := 127519, stop := 127528 ), 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\ := 127560, root := ~, start := 127545, stop := 127559 ), rec( content := ";\n\ ", count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 0, 10 ], name := "Item", next := 127578, root := ~, start := 127\ 413, stop := 127577 ), 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 := \ 127622, root := ~, start := 127606, stop := 127621 ), 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 := 127649, root :=\ ~, start := 127639, stop := 127648 ), 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 := 127688, root :=\ ~, start := 127673, stop := 127687 ), 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 := 127723, root := ~, star\ t := 127713, stop := 127722 ), 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 := 127762, root := ~, start := \ 127747, stop := 127761 ), rec( content := ";\n ", count := [ 4, 1, 0, \ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 12 ], name := "Ite\ m", next := 127780, root := ~, start := 127588, stop := 127779 ), 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 := 127824, root := ~, start := 127808, stop \ := 127823 ), 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 := 127842, root := ~, start := 127826, stop := 127841 ), 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 \ := 127890, root := ~, start := 127875, stop := 127889 ), rec( content := ")", \ count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 14 ], mathmode := "Text", name := "C", next := 127895, root :=\ ~, start := 127859, stop := 127894 ), 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 := 127951, root := ~,\ start := 127936, stop := 127950 ), rec( content := ")", count := [ 4, 1, 0, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, \ 14 ], mathmode := "Text", name := "C", next := 127956, root := ~, start := 127\ 920, stop := 127955 ), 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 := 128076, root := ~, start := 127790, stop := 128075 ) ], count := [ 4, 1, 0, 9 ], mathmode := "Text", name := "List", next := 128091, root := ~, start := 127397, stop := 128090 ), rec( content := "\n " , count := [ 4, 1, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], name := "Item", next := 128104, root := ~, start := 127249, stop := 128103 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "List", next := 128115, root := ~, start := 126688, stop := 128114 ), 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 := 128179, root := ~, start := 128135, stop := 128178 ), 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 := 128202, root := ~, start := 128191, stop := 128201 ), 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 := 128218, root := ~, start := 128206, stop := 128217 ), rec( content := ".", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 128229, root := ~, start := 128182, stop := 128228 ), 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 := 128265, root := ~, start := 128253, stop := 128264 ), 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 := 128280, root := ~, start := 128270, stop := 128279 ), 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 := 128338, root := ~, start := 128327, stop := 128337 ), 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 := 128352, root := ~, start := 128342, stop := 128351 ), 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 := 128384, root := ~, start := 128372, stop := 128383 ), 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 := 128402, root := ~, start := 128390, stop := 128401 ), 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 := 128425, root := ~, start := 128421, stop := 128424 ), 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 := 128451, root := ~, start := 128441, stop := 128450 ), 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 := 128474, root := ~, start := 128457, stop := 128473 ), 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 := 128500, root := ~, start := 128488, stop := 128499 ), 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 := 128518, root := ~, start := 128508, stop := 128517 ), 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 := 128535, root := ~, start := 128523, stop := 128534 ), 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 := 128618, root := ~, start := 128606, stop := 128617 ), 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 := 128689, root := ~, start := 128679, stop := 128688 ), 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 := 128716, root := ~, start := 128712, stop := 128715 ), 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 := 128748, root := ~, start := 128738, stop := 128747 ), 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 := 128780, root := ~, start := 128768, stop := 128779 ), 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 := 128831, root := ~, start := 128821, stop := 12883\ 0 ), 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 := 1288\ 65, root := ~, start := 128853, stop := 128864 ), rec( content := ")", count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 128870, root := ~, start := 128800, stop := 128869 ), 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 := 128914, root := ~, start := 128906, stop := 128913 ), 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 := 128927, root := ~, start := 128919, stop := 128926 ), 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 := 128991, root := ~, start := 128978, stop := 128990 ), 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 := 129007, root := ~, start := 128995, stop := 129006 ), 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 := 129034, root := ~, start := 129024, stop := 129033 ), 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 := 129044, root := ~, start := 129040, stop := 129043 ), 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 := 129652, root := ~, start := 129050, stop := 129651 ), rec( content := "\n ", count := [ 4, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 129669, root := ~, start := 128232, stop := 129668 ) ], count := [ 4, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 129683, root := ~, start := 128120, stop := 129682 ), 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 := 129759, root := ~, start := 129703, stop := 129758 ), rec( attributes := rec( Arg := "super, sub", Name := "IsUndirectedSpa\ nningForest" ), content := 0, count := [ 4, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 129820, root := ~, start := 129762, stop := 129819 ), 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 := 129843, root := ~, start := 129832, stop := 129842 ), 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 := 129859, root := ~, start := 129847, stop := 129858 ), rec( content := ".", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 129870, root := ~, start := 129823, stop := 129869 ), 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 := 129936, root := ~, start := 129905, stop := 129935 ), 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 := 129956, root := ~, start := 129945, stop := 129955 ), 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 := 129986, root := ~, start := 129976, stop := 129985 ), 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 := 130049, root := ~, start := 130037, stop := 130048 ), 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 := 130102, root := ~, start := 130069, stop := 130101 ), 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 := 130126, root := ~, start := 130115, stop := 130125 ), 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 := 130152, root := ~, start := 130142, stop := 130151 ), 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 := 130217, root := ~, start := 130205, stop := 130216 ), 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 := 130223, root := ~, start := 130219, stop := 130222 ), 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 := 130263, root := ~, start := 130232, stop := 130262 ), 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 := 130289, root := ~, start := 130277, stop := 130288 ), 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 := 130325, root := ~, start := 130313, stop := 130324 ), 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 := 130385, root := ~, start := 130358, stop := 130384 ), 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 := 130425, root := ~, start := 130394, stop := 130424 ), 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 := 130500, root := ~, start := 130454, stop := 130499 ), 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 := 130592, root := ~, start := 130560, stop := 130591 ), 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 := 130599, root := ~, start := 130595, stop := 130598 ), 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 := 130641, root := ~, start := 130608, stop := 130640 ), 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 := 130667, root := ~, start := 130655, stop := 130666 ), 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 := 130703, root := ~, start := 130691, stop := 130702 ), 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 := 130770, root := ~, start := 130738, stop := 130769 ), 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 := 130807, root := ~, start := 130775, stop := 130806 ), 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 := 130880, root := ~, start := 130868, stop := 130879 ), 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 := 130962, root := ~, start := 130950, stop := 130961 ), 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 := 131067, root := ~, start := 131021, stop := 131066 ), 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 := 131083, root := ~, start := 131071, stop := 131082 ), 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 := 131135, root := ~, start := 131089, stop := 131134 ), 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 := 131142, root := ~, start := 131138, stop := 131141 ), 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 := 131994, root := ~, start := 131247, stop := 131993 ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 132011, root := ~, start := 129873, stop := 132010 ) ], count := [ 4, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 132025, root := ~, start := 129688, stop := 132024 ), rec( content := "\n\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 132039, root := ~, start := 126633, stop := 132038 ) , rec( content := "\n", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 132050, root := ~, start := 126574, stop := 132049 ), 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 := 132166, root := ~, start := 132122, stop := 132165 ), 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 := 132215, root := ~, start := 132178, stop := 132214 ), 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 := 132279, root := ~, start := 132235, stop := 132278 ), 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 := 132320, root := ~, start := 132282, stop := 132319 ), 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 := 132391, root := ~, start := 132377, stop := 132390 ), 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 := 132397, root := ~, start := 132393, stop := 132396 ), 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 := 132496, root := ~, start := 132488, stop := 132495 ), 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 := 133137, root := ~, start := 132541, stop := 133136 ), rec( content := "\n ", count := [ 5, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 133154, root := ~, start := 132323, stop := 133153 ) ], count := [ 5, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 133168, root := ~, start := 132220, stop := 133167 ), 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 := 133235, root := ~, start := 133189, stop := 133234 ), 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 := 133268, root := ~, start := 133238, stop := 133267 ), 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 := 133349, root := ~, start := 133335, stop := 133348 ), 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 := 133355, root := ~, start := 133351, stop := 133354 ), 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 := 133929, root := ~, start := 133361, stop := 133928 ), rec( content := "\n ", count := [ 5, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 133946, root := ~, start := 133271, stop := 133945 ) ], count := [ 5, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 133960, root := ~, start := 133174, stop := 133959 ), 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 := 134022, root := ~, start := 133981, stop := 134021 ), 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 := 134060, root := ~, start := 134025, stop := 134059 ), 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 := 134100, root := ~, start := 134081, stop := 134099 ), 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 := 134154, root := ~, start := 134140, stop := 134153 ), 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 := 134230, root := ~, start := 134201, stop := 134229 ), 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 := 134268, root := ~, start := 134247, stop := 134267 ), 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 := 134278, root := ~, start := 134274, stop := 134277 ), 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 := 134319, root := ~, start := 134299, stop := 134318 ), 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 := 134333, root := ~, start := 134319, stop := 134332 ), 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 := 134341, root := ~, start := 134333, stop := 134340 ), 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 := 134410, root := ~, start := 134396, stop := 134409 ), 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 := 134443, root := ~, start := 134423, stop := 134442 ), 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 := 134457, root := ~, start := 134443, stop := 134456 ), 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 := 134465, root := ~, start := 134457, stop := 134464 ), 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 := 134517, root := ~, start := 134503, stop := 134516 ), 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 := 134550, root := ~, start := 134546, stop := 134549 ), 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 := 134590, root := ~, start := 134571, stop := 134589 ), 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 := 135280, root := ~, start := 134757, stop := 135279 ), rec( content := "\n ", count := [ 5, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 135297, root := ~, start := 134063, stop := 135296 ) ], count := [ 5, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 135311, root := ~, start := 133966, stop := 135310 ), 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 := 135375, root := ~, start := 135332, stop := 135374 ), 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 := 135408, root := ~, start := 135378, stop := 135407 ), 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 := 135500, root := ~, start := 135486, stop := 135499 ), 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 := 135947, root := ~, start := 135506, stop := 135946 ), rec( content := "\n ", count := [ 5, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 135964, root := ~, start := 135411, stop := 135963 ) ], count := [ 5, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 135978, root := ~, start := 135317, stop := 135977 ), 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 := 136040, root := ~, start := 135999, stop := 136039 ), 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 := 136081, root := ~, start := 136043, stop := 136080 ), 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 := 136177, root := ~, start := 136163, stop := 136176 ), 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 := 136276, root := ~, start := 136245, stop := 136275 ), 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 := 136574, root := ~, start := 136282, stop := 136573 ), rec( content := "\n ", count := [ 5, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 136591, root := ~, start := 136084, stop := 136590 ) ], count := [ 5, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 136605, root := ~, start := 135984, stop := 136604 ), 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 := 136669, root := ~, start := 136626, stop := 136668 ), 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 := 136710, root := ~, start := 136672, stop := 136709 ), 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 := 136819, root := ~, start := 136805, stop := 136818 ), 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 := 136918, root := ~, start := 136888, stop := 136917 ), 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 := 137220, root := ~, start := 136924, stop := 137219 ), rec( content := "\n ", count := [ 5, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 137237, root := ~, start := 136713, stop := 137236 ) ], count := [ 5, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 137251, root := ~, start := 136611, stop := 137250 ), 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 := 137323, root := ~, start := 137272, stop := 137322 ), 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 := 137378, root := ~, start := 137367, stop := 137377 ), rec( content := ".", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Returns", next := 137389, root := ~, start := 137326, stop := 137388 ), 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 := 137427, root := ~, start := 137413, stop := 137426 ), 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 := 137517, root := ~, start := 137488, stop := 137516 ), 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 := 137556, root := ~, start := 137542, stop := 137555 ), 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 := 137677, root := ~, start := 137663, stop := 137676 ), 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 := 137738, root := ~, start := 137730, stop := 137737 ), 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 := 137779, root := ~, start := 137768, stop := 137778 ), 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 := 137789, root := ~, start := 137785, stop := 137788 ), 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 := 137851, root := ~, start := 137807, stop := 137850 ), 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 := 137935, root := ~, start := 137931, stop := 137934 ), 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 := 138004, root := ~, start := 137991, stop := 138003 ), 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 := 138023, root := ~, start := 138015, stop := 138022 ), 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 := 138096, root := ~, start := 138088, stop := 138095 ), 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 := 138143, root := ~, start := 138139, stop := 138142 ), 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 := 138514, root := ~, start := 138148, stop := 138513 ), rec( content := "\n ", count := [ 5, 1, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Description", next := 138531, root := ~, start := 137392, stop := 138530 ) ], count := [ 5, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 138545, root := ~, start := 137257, stop := 138544 ), 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 := 138618, root := ~, start := 138566, stop := 138617 ), rec( attributes := rec( Arg := "digraph, i, obj" , Name := "SetDigraphVerte\ xLabel" ), content := 0, count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "Oper", next := 138680, root := ~, start := 138621, stop := 138679 ), 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 := 138718, root := ~, start := 138704, stop := 138717 ), 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 := 138802, root := ~, start := 138794, stop := 138801 ), 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 := 138880, root := ~, start := 138872, stop := 138879 ), 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 := 138898, root := ~, start := 138884, stop := 138897 ), 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 := 138943, root := ~, start := 138933, stop := 138942 ), 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 := 138950, root := ~, start := 138946, stop := 138949 ), 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 := 139075, root := ~, start := 139067, stop := 139074 ), 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 := 139115, root := ~, start := 139107, stop := 139114 ), 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 := 139121, root := ~, start := 139117, stop := 139120 ), 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 := 139144, root := ~, start := 139130, stop := 139143 ), 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 := 139216, root := ~, start := 139201, stop := 139215 ), 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 := 139297, root := ~, start := 139293, stop := 139296 ), 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 := 140233, root := ~, start := 139441, stop := 140232 ), rec( content := "\n ", count := [ 5, 1, 8, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "Description", next := 140250, root := ~, start := 138683, stop := 140249 ) ], count := [ 5, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 140264, root := ~, start := 138551, stop := 140263 ), 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 := 140333, root := ~, start := 140285, stop := 140332 ), rec( attributes := rec( Arg := "digraph, list", Name := "SetDigraphVerte\ xLabels" ), content := 0, count := [ 5, 1, 9, 4 ], mathmode := "Text", name := "Oper", next := 140393, root := ~, start := 140336, stop := 140392 ), 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 := 140431, root := ~, start := 140417, stop := 140430 ), 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 := 140477, root := ~, start := 140451, stop := 140476 ), 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 := 140544, root := ~, start := 140530, stop := 140543 ), 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 := 140579, root := ~, start := 140550, stop := 140578 ), 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 := 140647, root := ~, start := 140633, stop := 140646 ), 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 := 140702, root := ~, start := 140691, stop := 140701 ), 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 := 140708, root := ~, start := 140704, stop := 140707 ), 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 := 140833, root := ~, start := 140825, stop := 140832 ), 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 := 140873, root := ~, start := 140865, stop := 140872 ), 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 := 140879, root := ~, start := 140875, stop := 140878 ), 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 := 140902, root := ~, start := 140888, stop := 140901 ), 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 := 140974, root := ~, start := 140959, stop := 140973 ), 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 := 141055, root := ~, start := 141051, stop := 141054 ), 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 := 142018, root := ~, start := 141194, stop := 142017 ), rec( content := "\n ", count := [ 5, 1, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "Description", next := 142035, root := ~, start := 140396, stop := 142034 ) ], count := [ 5, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 142049, root := ~, start := 140270, stop := 142048 ), 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 := 142123, root := ~, start := 142070, stop := 142122 ), rec( attributes := rec( Arg := "digraph, i, j, o\ bj", Name := "SetDigraphEdgeLabel" ), content := 0, count := [ 5, 1, 10, 4 ], mathmode := "Text", name := "Oper", next := 142187, root := ~, start := 142126, stop := 142186 ), 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 := 142225, root := ~, start := 142211, stop := 142224 ), 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 := 142342, root := ~, start := 142334, stop := 142341 ), 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 := 142365, root := ~, start := 142357, stop := 142364 ), 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 := 142457, root := ~, start := 142449, stop := 142456 ), 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 := 142477, root := ~, start := 142469, stop := 142476 ), 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 := 142522, root := ~, start := 142512, stop := 142521 ), 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 := 142528, root := ~, start := 142524, stop := 142527 ), 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 := 142677, root := ~, start := 142669, stop := 142676 ), 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 := 142683, root := ~, start := 142679, stop := 142682 ), 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 := 142872, root := ~, start := 142841, stop := 142871 ), 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 := 143567, root := ~, start := 142879, stop := 143566 ), rec( content := "\n ", count := [ 5, 1, 10, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "Description", next := 143584, root := ~, start := 142190, stop := 143583 ) ], count := [ 5, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 143598, root := ~, start := 142055, stop := 143597 ), 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 := 143665, root := ~, start := 143619, stop := 143664 ), 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 := 143771, root := ~, start := 143668, stop := 143770 ), 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 := 143870, root := ~, start := 143774, stop := 143869 ), 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 := 143908, root := ~, start := 143894, stop := 143907 ), 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 := 143979, root := ~, start := 143955, stop := 143978 ), 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 := 144043, root := ~, start := 144029, stop := 144042 ), 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 := 144076, root := ~, start := 144063, stop := 144075 ), 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 := 144110, root := ~, start := 144091, stop := 144109 ), 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 := 144156, root := ~, start := 144148, stop := 144155 ), 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 := 144206, root := ~, start := 144171, stop := 144205 ), 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 := 144240, root := ~, start := 144213, stop := 144239 ), 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 := 144305, root := ~, start := 144291, stop := 144304 ), 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 := 144354, root := ~, start := 144341, stop := 144353 ), 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 := 144422, root := ~, start := 144405, stop := 144421 ), 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 := 144472, root := ~, start := 144464, stop := 144471 ), 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 := 144522, root := ~, start := 144487, stop := 144521 ), 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 := 144570, root := ~, start := 144543, stop := 144569 ), 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 := 144621, root := ~, start := 144610, stop := 144620 ), 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 := 144688, root := ~, start := 144680, stop := 144687 ), 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 := 144705, root := ~, start := 144697, stop := 144704 ), 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 := 144760, root := ~, start := 144752, stop := 144759 ), 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 := 144784, root := ~, start := 144776, stop := 144783 ), 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 := 144790, root := ~, start := 144786, stop := 144789 ), 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 := 144938, root := ~, start := 144930, stop := 144937 ), 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 := 144944, root := ~, start := 144940, stop := 144943 ), 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 := 146155, root := ~, start := 145088, stop := 146154 ), rec( content := "\n ", count := [ 5, 1, 11, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "Description", next := 146172, root := ~, start := 143873, stop := 146171 ) ], count := [ 5, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 146186, root := ~, start := 143604, stop := 146185 ), 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 := 146258, root := ~, start := 146207, stop := 146257 ), 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 := 146296, root := ~, start := 146261, stop := 146295 ), 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 := 146338, root := ~, start := 146317, stop := 146337 ), 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 := 146386, root := ~, start := 146372, stop := 146385 ), 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 := 146415, root := ~, start := 146402, stop := 146414 ), 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 := 146621, root := ~, start := 146436, stop := 146620 ), rec( content := "\n ", count := [ 5, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 146638, root := ~, start := 146299, stop := 146637 ) ], count := [ 5, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 146652, root := ~, start := 146192, stop := 146651 ), 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 := 146725, root := ~, start := 146673, stop := 146724 ), 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 := 146763, root := ~, start := 146728, stop := 146762 ), 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 := 146806, root := ~, start := 146784, stop := 146805 ), 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 := 146854, root := ~, start := 146840, stop := 146853 ), 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 := 146883, root := ~, start := 146870, stop := 146882 ), 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 := 147091, root := ~, start := 146905, stop := 147090 ), rec( content := "\n ", count := [ 5, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 147108, root := ~, start := 146766, stop := 147107 ) ], count := [ 5, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 147122, root := ~, start := 146658, stop := 147121 ), 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 := 147228, root := ~, start := 147143, stop := 147227 ), 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 := 147324, root := ~, start := 147231, stop := 147323 ), 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 := 147347, root := ~, start := 147336, stop := 147346 ), 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 := 147363, root := ~, start := 147351, stop := 147362 ), rec( content := ".", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "Returns", next := 147374, root := ~, start := 147327, stop := 147373 ), 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 := 147447, root := ~, start := 147436, stop := 147446 ), 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 := 147487, root := ~, start := 147476, stop := 147486 ), 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 := 147535, root := ~, start := 147521, stop := 147534 ), 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 := 147590, root := ~, start := 147579, stop := 147589 ), 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 := 147605, root := ~, start := 147594, stop := 147604 ), 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 := 147658, root := ~, start := 147647, stop := 147657 ), 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 := 147668, root := ~, start := 147658, stop := 147667 ), 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 := 147698, root := ~, start := 147687, stop := 147697 ), 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 := 147708, root := ~, start := 147698, stop := 147707 ), 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 := 147754, root := ~, start := 147740, stop := 147753 ), 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 := 147772, root := ~, start := 147760, stop := 147771 ), 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 := 147788, root := ~, start := 147784, stop := 147787 ), 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 := 147836, root := ~, start := 147825, stop := 147835 ), 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 := 147852, root := ~, start := 147844, stop := 14785\ 1 ), 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 := 147862, root := ~, start := 147854, stop := 14786\ 1 ), rec( content := "]", count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 14, 6 ], mathmode := "Text", name := "C", next := 147867, root := ~, start := 147840, stop := 147866 ), 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 := 147900, root := ~, start := 147886, stop := 147899 ), 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 := 147918, root := ~, start := 147906, stop := 147917 ), 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 := 148175, root := ~, start := 147931, stop := 148174 ), rec( content := "\n ", count := [ 5, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "Description", next := 148192, root := ~, start := 147377, stop := 148191 ) ], count := [ 5, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 148206, root := ~, start := 147128, stop := 148205 ), 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 := 148272, root := ~, start := 148227, stop := 148271 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsMaximalMatchi\ ng" ), content := 0, count := [ 5, 1, 15, 4 ], mathmode := "Text", name := "Oper", next := 148327, root := ~, start := 148275, stop := 148326 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsMaximumMatchi\ ng" ), content := 0, count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "Oper", next := 148382, root := ~, start := 148330, stop := 148381 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsPerfectMatchi\ ng" ), content := 0, count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "Oper", next := 148437, root := ~, start := 148385, stop := 148436 ), 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 := 148460, root := ~, start := 148449, stop := 148459 ), 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 := 148476, root := ~, start := 148464, stop := 148475 ), rec( content := ".", count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "Returns", next := 148487, root := ~, start := 148440, stop := 148486 ), 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 := 148525, root := ~, start := 148511, stop := 148524 ), 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 := 148554, root := ~, start := 148543, stop := 148553 ), 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 := 148607, root := ~, start := 148593, stop := 148606 ), 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 := 148631, root := ~, start := 148614, stop := 148630 ), 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 := 148651, root := ~, start := 148640, stop := 148650 ), 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 := 148670, root := ~, start := 148659, stop := 148669 ), 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 := 148702, root := ~, start := 148688, stop := 148701 ), 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 := 148747, root := ~, start := 148723, stop := 148746 ), 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 := 148767, root := ~, start := 148756, stop := 148766 ), 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 := 148782, root := ~, start := 148771, stop := 148781 ), 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 := 148835, root := ~, start := 148811, stop := 148834 ), 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 := 148855, root := ~, start := 148844, stop := 148854 ), 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 := 148870, root := ~, start := 148859, stop := 148869 ), 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 := 148925, root := ~, start := 148901, stop := 148924 ), 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 := 148945, root := ~, start := 148934, stop := 148944 ), 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 := 148965, root := ~, start := 148954, stop := 148964 ), 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 := 149006, root := ~, start := 148992, stop := 149005 ), 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 := 149083, root := ~, start := 149071, stop := 149082 ), 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 := 149093, root := ~, start := 149089, stop := 149092 ), 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 := 149116, root := ~, start := 149101, stop := 149115 ), 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 := 149125, root := ~, start := 149117, stop := 149124 ), 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 := 149153, root := ~, start := 149139, stop := 149152 ), 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 := 149200, root := ~, start := 149186, stop := 149199 ), 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 := 149237, root := ~, start := 149223, stop := 14923\ 6 ), rec( content := ")", count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 15, 10 ], mathmode := "Text", name := "C", next := 149242, root := ~, start := 149207, stop := 149241 ), 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 := 149295, root := ~, start := 149287, stop := 149294 ), 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 := 149349, root := ~, start := 149335, stop := 149348 ), 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 := 149453, root := ~, start := 149423, stop := 149452 ), 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 := 149477, root := ~, start := 149469, stop := 149476 ), 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 := 149499, root := ~, start := 149485, stop := 149498 ), 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 := 149578, root := ~, start := 149564, stop := 149577 ), 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 := 149659, root := ~, start := 149645, stop := 149658 ), 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 := 150473, root := ~, start := 149873, stop := 150472 ), rec( content := "\n ", count := [ 5, 1, 15, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 9 ], mathmode := "Text", name := "Description", next := 150490, root := ~, start := 148490, stop := 150489 ) ], count := [ 5, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 150504, root := ~, start := 148212, stop := 150503 ), 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 := 150576, root := ~, start := 150525, stop := 150575 ), 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 := 150626, root := ~, start := 150579, stop := 150625 ), 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 := 150717, root := ~, start := 150703, stop := 150716 ), 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 := 150727, root := ~, start := 150723, stop := 150726 ), 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 := 150810, root := ~, start := 150779, stop := 150809 ), 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 := 151278, root := ~, start := 150818, stop := 151277 ), rec( content := "\n ", count := [ 5, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 3 ], mathmode := "Text", name := "Description", next := 151295, root := ~, start := 150629, stop := 151294 ) ], count := [ 5, 1, 16, 1 ], mathmode := "Text", name := "ManSection", next := 151309, root := ~, start := 150510, stop := 151308 ), 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 := 151381, root := ~, start := 151330, stop := 151380 ), 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 := 151431, root := ~, start := 151384, stop := 151430 ), 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 := 151522, root := ~, start := 151508, stop := 151521 ), 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 := 151532, root := ~, start := 151528, stop := 151531 ), 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 := 151615, root := ~, start := 151584, stop := 151614 ), 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 := 151638, root := ~, start := 151624, stop := 151637 ), 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 := 151689, root := ~, start := 151657, stop := 151688 ), 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 := 151755, root := ~, start := 151736, stop := 151754 ), 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 := 151829, root := ~, start := 151809, stop := 151828 ), 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 := 151844, root := ~, start := 151836, stop := 151843 ), 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 := 151888, root := ~, start := 151880, stop := 151887 ), 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 := 152515, root := ~, start := 151918, stop := 152514 ), rec( content := "\n ", count := [ 5, 1, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "Description", next := 152532, root := ~, start := 151434, stop := 152531 ) ], count := [ 5, 1, 17, 1 ], mathmode := "Text", name := "ManSection", next := 152546, root := ~, start := 151315, stop := 152545 ), rec( content := "\n\n ", count := [ 5, 1, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 152560, root := ~, start := 132169, stop := 152559 ) , 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 := 152613, root := ~, start := 152573, stop := 152612 ), 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 := 152677, root := ~, start := 152633, stop := 152676 ), rec( attributes := rec( Arg := "digraph", Name := "AdjacencyMatrix\ MutableCopy" ), content := 0, count := [ 5, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 152735, root := ~, start := 152680, stop := 152734 ), 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 := 152798, root := ~, start := 152738, stop := 152797 ), 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 := 152872, root := ~, start := 152862, stop := 152871 ), 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 := 152906, root := ~, start := 152892, stop := 152905 ), 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 := 152958, root := ~, start := 152942, stop := 152957 ), 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 := 153003, root := ~, start := 152989, stop := 153002 ), 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 := 153024, root := ~, start := 153016, stop := 153023 ), 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 := 153043, root := ~, start := 153035, stop := 153042 ), 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 := 153062, root := ~, start := 153048, stop := 153061 ), 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 := 153121, root := ~, start := 153117, stop := 153120 ), 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 := 153162, root := ~, start := 153140, stop := 153161 ), 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 := 153257, root := ~, start := 153224, stop := 153256 ), 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 := 153302, root := ~, start := 153280, stop := 153301 ), 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 := 153348, root := ~, start := 153344, stop := 153347 ), 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 := 154318, root := ~, start := 153354, stop := 154317 ), rec( content := "\n ", count := [ 5, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 154335, root := ~, start := 152801, stop := 154334 ) ], count := [ 5, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 154349, root := ~, start := 152618, stop := 154348 ), 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 := 154423, root := ~, start := 154370, stop := 154422 ), 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 := 154484, root := ~, start := 154426, stop := 154483 ), 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 := 154591, root := ~, start := 154577, stop := 154590 ), 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 := 154701, root := ~, start := 154687, stop := 154700 ), 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 := 155367, root := ~, start := 154706, stop := 155366 ), rec( content := "\n ", count := [ 5, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 155384, root := ~, start := 154487, stop := 155383 ) ], count := [ 5, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 155398, root := ~, start := 154355, stop := 155397 ), 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 := 155470, root := ~, start := 155419, stop := 155469 ), rec( attributes := rec( Arg := "digraph", Name := "BooleanAdjacenc\ yMatrixMutableCopy" ), content := 0, count := [ 5, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 155535, root := ~, start := 155473, stop := 155534 ), 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 := 155585, root := ~, start := 155538, stop := 155584 ), 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 := 155623, root := ~, start := 155609, stop := 155622 ), 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 := 155684, root := ~, start := 155676, stop := 155683 ), 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 := 155721, root := ~, start := 155691, stop := 155720 ), 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 := 155735, root := ~, start := 155721, stop := 155734 ), 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 := 155743, root := ~, start := 155735, stop := 155742 ), 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 := 155795, root := ~, start := 155785, stop := 155794 ), 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 := 155813, root := ~, start := 155799, stop := 155812 ), 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 := 155866, root := ~, start := 155850, stop := 155865 ), 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 := 155881, root := ~, start := 155870, stop := 155880 ), 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 := 155939, root := ~, start := 155925, stop := 155938 ), 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 := 155960, root := ~, start := 155952, stop := 155959 ), 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 := 155983, root := ~, start := 155975, stop := 155982 ), 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 := 156003, root := ~, start := 155989, stop := 156002 ), 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 := 156062, root := ~, start := 156058, stop := 156061 ), 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 := 156156, root := ~, start := 156152, stop := 156155 ), 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 := 156205, root := ~, start := 156176, stop := 156204 ), 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 := 156321, root := ~, start := 156281, stop := 156320 ), 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 := 156377, root := ~, start := 156348, stop := 156376 ), 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 := 156423, root := ~, start := 156419, stop := 156422 ), 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 := 157258, root := ~, start := 156428, stop := 157257 ), rec( content := "\n ", count := [ 5, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "Description", next := 157275, root := ~, start := 155588, stop := 157274 ) ], count := [ 5, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 157289, root := ~, start := 155404, stop := 157288 ), 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 := 157363, root := ~, start := 157310, stop := 157362 ), 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 := 157396, root := ~, start := 157366, stop := 157395 ), 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 := 157434, root := ~, start := 157420, stop := 157433 ), 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 := 157485, root := ~, start := 157454, stop := 157484 ), 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 := 157555, root := ~, start := 157544, stop := 157554 ), 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 := 157583, root := ~, start := 157572, stop := 157582 ), 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 := 157628, root := ~, start := 157620, stop := 157627 ), 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 := 157647, root := ~, start := 157639, stop := 157646 ), 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 := 157669, root := ~, start := 157655, stop := 157668 ), 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 := 157686, root := ~, start := 157674, stop := 157685 ), 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 := 158467, root := ~, start := 157700, stop := 158466 ), rec( content := "\n ", count := [ 5, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 158484, root := ~, start := 157399, stop := 158483 ) ], count := [ 5, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 158498, root := ~, start := 157295, stop := 158497 ), 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 := 158560, root := ~, start := 158519, stop := 158559 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSource" ), content := 0, count := [ 5, 2, 5, 4 ], mathmode := "Text", name := "Attr", next := 158605, root := ~, start := 158563, stop := 158604 ), 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 := 158655, root := ~, start := 158608, stop := 158654 ), 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 := 158695, root := ~, start := 158676, stop := 158694 ), 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 := 158720, root := ~, start := 158700, stop := 158719 ), 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 := 158782, root := ~, start := 158768, stop := 158781 ), 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 := 158817, root := ~, start := 158809, stop := 158816 ), 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 := 158855, root := ~, start := 158841, stop := 15885\ 4 ), rec( content := ")", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 158860, root := ~, start := 158825, stop := 158859 ), 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 := 158889, root := ~, start := 158881, stop := 158888 ), 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 := 158918, root := ~, start := 158904, stop := 158917 ), 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 := 159546, root := ~, start := 158925, stop := 159545 ), rec( content := "\n ", count := [ 5, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "Description", next := 159563, root := ~, start := 158658, stop := 159562 ) ], count := [ 5, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 159577, root := ~, start := 158504, stop := 159576 ), 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 := 159640, root := ~, start := 159598, stop := 159639 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighbors" ) , content := 0, count := [ 5, 2, 6, 4 ], mathmode := "Text", name := "Attr", next := 159684, root := ~, start := 159643, stop := 159683 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighboursMu\ tableCopy" ), content := 0, count := [ 5, 2, 6, 6 ], mathmode := "Text", name := "Oper", next := 159740, root := ~, start := 159687, stop := 159739 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighborsMut\ ableCopy" ), content := 0, count := [ 5, 2, 6, 8 ], mathmode := "Text", name := "Oper", next := 159795, root := ~, start := 159743, stop := 159794 ), 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 := 159846, root := ~, start := 159798, stop := 159845 ), 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 := 159908, root := ~, start := 159898, stop := 159907 ), 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 := 159975, root := ~, start := 159961, stop := 159974 ), 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 := 160017, root := ~, start := 160009, stop := 160016 ), 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 := 160042, root := ~, start := 160029, stop := 160041 ), 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 := 160098, root := ~, start := 160090, stop := 160097 ), 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 := 160117, root := ~, start := 160109, stop := 160116 ), 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 := 160139, root := ~, start := 160125, stop := 160138 ), 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 := 160145, root := ~, start := 160141, stop := 160144 ), 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 := 160184, root := ~, start := 160164, stop := 160183 ), 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 := 160277, root := ~, start := 160246, stop := 160276 ), 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 := 160320, root := ~, start := 160300, stop := 160319 ), 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 := 160367, root := ~, start := 160363, stop := 160366 ), 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 := 161155, root := ~, start := 160373, stop := 161154 ), rec( content := "\n ", count := [ 5, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "Description", next := 161172, root := ~, start := 159849, stop := 161171 ) ], count := [ 5, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 161186, root := ~, start := 159583, stop := 161185 ), 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 := 161248, root := ~, start := 161207, stop := 161247 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighbors" ), content := 0, count := [ 5, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 161291, root := ~, start := 161251, stop := 161290 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighboursMut\ ableCopy" ), content := 0, count := [ 5, 2, 7, 6 ], mathmode := "Text", name := "Oper", next := 161346, root := ~, start := 161294, stop := 161345 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighborsMuta\ bleCopy" ), content := 0, count := [ 5, 2, 7, 8 ], mathmode := "Text", name := "Oper", next := 161400, root := ~, start := 161349, stop := 161399 ), 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 := 161450, root := ~, start := 161403, stop := 161449 ), 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 := 161512, root := ~, start := 161502, stop := 161511 ), 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 := 161578, root := ~, start := 161564, stop := 161577 ), 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 := 161620, root := ~, start := 161612, stop := 161619 ), 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 := 161645, root := ~, start := 161632, stop := 161644 ), 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 := 161701, root := ~, start := 161693, stop := 161700 ), 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 := 161720, root := ~, start := 161712, stop := 161719 ), 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 := 161742, root := ~, start := 161728, stop := 161741 ), 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 := 161748, root := ~, start := 161744, stop := 161747 ), 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 := 161786, root := ~, start := 161767, stop := 161785 ), 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 := 161878, root := ~, start := 161848, stop := 161877 ), 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 := 161920, root := ~, start := 161901, stop := 161919 ), 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 := 161967, root := ~, start := 161963, stop := 161966 ), 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 := 162007, root := ~, start := 161997, stop := 162006 ), 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 := 162044, root := ~, start := 162040, stop := 162043 ), 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 := 162825, root := ~, start := 162050, stop := 162824 ), rec( content := "\n ", count := [ 5, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "Description", next := 162842, root := ~, start := 161453, stop := 162841 ) ], count := [ 5, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 162856, root := ~, start := 161192, stop := 162855 ), 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 := 162916, root := ~, start := 162877, stop := 162915 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSequen\ ce" ), content := 0, count := [ 5, 2, 8, 4 ], mathmode := "Text", name := "Attr", next := 162965, root := ~, start := 162919, stop := 162964 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSet" ) , content := 0, count := [ 5, 2, 8, 6 ], mathmode := "Text", name := "Attr", next := 163009, root := ~, start := 162968, stop := 163008 ), 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 := 163063, root := ~, start := 163012, stop := 163062 ), 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 := 163115, root := ~, start := 163101, stop := 163114 ), 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 := 163129, root := ~, start := 163121, stop := 163128 ), 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 := 163174, root := ~, start := 163157, stop := 163173 ), 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 := 163211, root := ~, start := 163201, stop := 163210 ), 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 := 163230, root := ~, start := 163222, stop := 163229 ), 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 := 163267, root := ~, start := 163259, stop := 163266 ), 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 := 163285, root := ~, start := 163271, stop := 163284 ), 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 := 163313, root := ~, start := 163300, stop := 163312 ), 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 := 163354, root := ~, start := 163346, stop := 163353 ), 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 := 163395, root := ~, start := 163364, stop := 163394 ), 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 := 163401, root := ~, start := 163397, stop := 163400 ), 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 := 163444, root := ~, start := 163420, stop := 163443 ), 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 := 163528, root := ~, start := 163524, stop := 163527 ), 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 := 163566, root := ~, start := 163547, stop := 163565 ), 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 := 163659, root := ~, start := 163655, stop := 163658 ), 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 := 164029, root := ~, start := 163665, stop := 164028 ), rec( content := "\n ", count := [ 5, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Description", next := 164046, root := ~, start := 163066, stop := 164045 ) ], count := [ 5, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 164060, root := ~, start := 162862, stop := 164059 ), 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 := 164119, root := ~, start := 164081, stop := 164118 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSequenc\ e" ), content := 0, count := [ 5, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 164167, root := ~, start := 164122, stop := 164166 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSet" ), content := 0, count := [ 5, 2, 9, 6 ], mathmode := "Text", name := "Attr", next := 164210, root := ~, start := 164170, stop := 164209 ), 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 := 164264, root := ~, start := 164213, stop := 164263 ), 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 := 164316, root := ~, start := 164302, stop := 164315 ), 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 := 164330, root := ~, start := 164322, stop := 164329 ), 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 := 164374, root := ~, start := 164358, stop := 164373 ), 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 := 164411, root := ~, start := 164401, stop := 164410 ), 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 := 164430, root := ~, start := 164422, stop := 164429 ), 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 := 164467, root := ~, start := 164459, stop := 164466 ), 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 := 164485, root := ~, start := 164471, stop := 164484 ), 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 := 164513, root := ~, start := 164500, stop := 164512 ), 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 := 164553, root := ~, start := 164545, stop := 164552 ), 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 := 164593, root := ~, start := 164563, stop := 164592 ), 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 := 164599, root := ~, start := 164595, stop := 164598 ), 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 := 164641, root := ~, start := 164618, stop := 164640 ), 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 := 164725, root := ~, start := 164721, stop := 164724 ), 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 := 164762, root := ~, start := 164744, stop := 164761 ), 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 := 164855, root := ~, start := 164851, stop := 164854 ), 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 := 165227, root := ~, start := 164861, stop := 165226 ), rec( content := "\n ", count := [ 5, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Description", next := 165244, root := ~, start := 164267, stop := 165243 ) ], count := [ 5, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 165258, root := ~, start := 164066, stop := 165257 ), 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 := 165333, root := ~, start := 165279, stop := 165332 ), 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 := 165380, root := ~, start := 165336, stop := 165379 ), 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 := 165466, root := ~, start := 165453, stop := 165465 ), 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 := 165500, root := ~, start := 165486, stop := 165499 ), 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 := 165537, root := ~, start := 165524, stop := 165536 ), 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 := 165578, root := ~, start := 165564, stop := 165577 ), 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 := 165612, root := ~, start := 165599, stop := 165611 ), 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 := 165622, root := ~, start := 165618, stop := 165621 ), 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 := 165918, root := ~, start := 165627, stop := 165917 ), rec( content := "\n ", count := [ 5, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 165935, root := ~, start := 165383, stop := 165934 ) ], count := [ 5, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 165949, root := ~, start := 165264, stop := 165948 ), 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 := 166028, root := ~, start := 165970, stop := 166027 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutNeighborsOfV\ ertex" ), content := 0, count := [ 5, 2, 11, 4 ], mathmode := "Text", name := "Oper", next := 166088, root := ~, start := 166031, stop := 166087 ), 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 := 166129, root := ~, start := 166091, stop := 166128 ), 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 := 166193, root := ~, start := 166183, stop := 166192 ), 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 := 166239, root := ~, start := 166225, stop := 166238 ), 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 := 166262, root := ~, start := 166254, stop := 166261 ), 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 := 166293, root := ~, start := 166283, stop := 166292 ), 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 := 166355, root := ~, start := 166342, stop := 166354 ), 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 := 166374, root := ~, start := 166366, stop := 166373 ), 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 := 166392, root := ~, start := 166378, stop := 166391 ), 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 := 166439, root := ~, start := 166429, stop := 166438 ), 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 := 166467, root := ~, start := 166463, stop := 166466 ), 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 := 166727, root := ~, start := 166472, stop := 166726 ), rec( content := "\n ", count := [ 5, 2, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "Description", next := 166744, root := ~, start := 166132, stop := 166743 ) ], count := [ 5, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 166758, root := ~, start := 165955, stop := 166757 ), 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 := 166832, root := ~, start := 166779, stop := 166831 ), 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 := 166877, root := ~, start := 166835, stop := 166876 ), 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 := 166962, root := ~, start := 166949, stop := 166961 ), 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 := 166996, root := ~, start := 166982, stop := 166995 ), 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 := 167032, root := ~, start := 167019, stop := 167031 ), 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 := 167073, root := ~, start := 167059, stop := 167072 ), 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 := 167106, root := ~, start := 167093, stop := 167105 ), 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 := 167399, root := ~, start := 167112, stop := 167398 ), rec( content := "\n ", count := [ 5, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 167416, root := ~, start := 166880, stop := 167415 ) ], count := [ 5, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 167430, root := ~, start := 166764, stop := 167429 ), 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 := 167508, root := ~, start := 167451, stop := 167507 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "InNeighborsOfVe\ rtex" ), content := 0, count := [ 5, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 167567, root := ~, start := 167511, stop := 167566 ), 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 := 167618, root := ~, start := 167570, stop := 167617 ), 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 := 167682, root := ~, start := 167672, stop := 167681 ), 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 := 167728, root := ~, start := 167714, stop := 167727 ), 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 := 167751, root := ~, start := 167743, stop := 167750 ), 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 := 167782, root := ~, start := 167772, stop := 167781 ), 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 := 167839, root := ~, start := 167831, stop := 167838 ), 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 := 167863, root := ~, start := 167850, stop := 167862 ), 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 := 167881, root := ~, start := 167867, stop := 167880 ), 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 := 167928, root := ~, start := 167918, stop := 167927 ), 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 := 167957, root := ~, start := 167953, stop := 167956 ), 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 := 168234, root := ~, start := 167963, stop := 168233 ), rec( content := "\n ", count := [ 5, 2, 13, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 168251, root := ~, start := 167621, stop := 168250 ) ], count := [ 5, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 168265, root := ~, start := 167436, stop := 168264 ), 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 := 168327, root := ~, start := 168286, stop := 168326 ), 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 := 168368, root := ~, start := 168330, stop := 168367 ), 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 := 168406, root := ~, start := 168392, stop := 168405 ), 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 := 168445, root := ~, start := 168426, stop := 168444 ), 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 := 168514, root := ~, start := 168485, stop := 168513 ), 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 := 168532, root := ~, start := 168518, stop := 168531 ), 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 := 168596, root := ~, start := 168567, stop := 168595 ), 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 := 168602, root := ~, start := 168598, stop := 168601 ), 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 := 169032, root := ~, start := 168608, stop := 169031 ), rec( content := "\n ", count := [ 5, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 169049, root := ~, start := 168371, stop := 169048 ) ], count := [ 5, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 169063, root := ~, start := 168271, stop := 169062 ), 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 := 169156, root := ~, start := 169084, stop := 169155 ), rec( attributes := rec( Arg := "digraph, u, v", Name := "PartialOrderDig\ raphJoinOfVertices" ), content := 0, count := [ 5, 2, 15, 4 ], mathmode := "Text", name := "Oper", next := 169231, root := ~, start := 169159, stop := 169230 ), 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 := 169276, root := ~, start := 169265, stop := 169275 ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Returns", next := 169286, root := ~, start := 169234, stop := 169285 ), 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 := 169397, root := ~, start := 169358, stop := 169396 ), 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 := 169541, root := ~, start := 169530, stop := 169540 ), 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 := 169677, root := ~, start := 169639, stop := 169676 ), 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 := 169725, root := ~, start := 169687, stop := 169724 ), 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 := 170478, root := ~, start := 169827, stop := 170477 ), rec( content := "\n ", count := [ 5, 2, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Description", next := 170495, root := ~, start := 169291, stop := 170494 ) ], count := [ 5, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 170509, root := ~, start := 169069, stop := 170508 ), 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 := 170571, root := ~, start := 170530, stop := 170570 ), 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 := 170634, root := ~, start := 170574, stop := 170633 ), 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 := 170708, root := ~, start := 170698, stop := 170707 ), 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 := 170742, root := ~, start := 170728, stop := 170741 ), 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 := 170769, root := ~, start := 170761, stop := 170768 ), 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 := 170837, root := ~, start := 170829, stop := 170836 ), 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 := 170855, root := ~, start := 170841, stop := 170854 ), 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 := 170899, root := ~, start := 170891, stop := 170898 ), 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 := 170922, root := ~, start := 170908, stop := 170921 ), 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 := 170977, root := ~, start := 170973, stop := 170976 ), 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 := 171011, root := ~, start := 170987, stop := 171010 ), 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 := 171698, root := ~, start := 171038, stop := 171697 ), rec( content := "\n ", count := [ 5, 2, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 171715, root := ~, start := 170637, stop := 171714 ) ], count := [ 5, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 171729, root := ~, start := 170515, stop := 171728 ), 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 := 171794, root := ~, start := 171750, stop := 171793 ), 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 := 171844, root := ~, start := 171797, stop := 171843 ), 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 := 171928, root := ~, start := 171918, stop := 171927 ), 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 := 171962, root := ~, start := 171948, stop := 171961 ), 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 := 172068, root := ~, start := 172013, stop := 172067 ), 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 := 172092, root := ~, start := 172078, stop := 172091 ), 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 := 172147, root := ~, start := 172143, stop := 172146 ), 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 := 172183, root := ~, start := 172157, stop := 172182 ), 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 := 172217, root := ~, start := 172188, stop := 172216 ), 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 := 172751, root := ~, start := 172248, stop := 172750 ), rec( content := "\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 172768, root := ~, start := 171847, stop := 172767 ) ], count := [ 5, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 172782, root := ~, start := 171735, stop := 172781 ), rec( content := "\n\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 172796, root := ~, start := 152564, stop := 172795 ) , 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 := 172857, root := ~, start := 172809, stop := 172856 ), 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 := 172921, root := ~, start := 172877, stop := 172920 ), 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 := 172958, root := ~, start := 172947, stop := 172957 ), rec( content := ".", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 172969, root := ~, start := 172924, stop := 172968 ), 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 := 173054, root := ~, start := 173040, stop := 173053 ), 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 := 173064, root := ~, start := 173060, stop := 173063 ), 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 := 173097, root := ~, start := 173083, stop := 173096 ), 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 := 173175, root := ~, start := 173160, stop := 173174 ), 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 := 173301, root := ~, start := 173287, stop := 173300 ), 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 := 173367, root := ~, start := 173356, stop := 173366 ), 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 := 173373, root := ~, start := 173369, stop := 173372 ), 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 := 173421, root := ~, start := 173383, stop := 173420 ), 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 := 173427, root := ~, start := 173423, stop := 173426 ), 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 := 173871, root := ~, start := 173432, stop := 173870 ), rec( content := "\n ", count := [ 5, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 173888, root := ~, start := 172972, stop := 173887 ) ], count := [ 5, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 173902, root := ~, start := 172862, stop := 173901 ), 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 := 174028, root := ~, start := 173923, stop := 174027 ), 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 := 174130, root := ~, start := 174031, stop := 174129 ), 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 := 174249, root := ~, start := 174133, stop := 174248 ), 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 := 174286, root := ~, start := 174275, stop := 174285 ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Returns", next := 174296, root := ~, start := 174252, stop := 174295 ), 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 := 174374, root := ~, start := 174360, stop := 174373 ), 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 := 174402, root := ~, start := 174394, stop := 174401 ), 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 := 174422, root := ~, start := 174414, stop := 174421 ), 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 := 174580, root := ~, start := 174569, stop := 174579 ), 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 := 174644, root := ~, start := 174594, stop := 174643 ), 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 := 174688, root := ~, start := 174684, stop := 174687 ), 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 := 174738, root := ~, start := 174727, stop := 174737 ), 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 := 174849, root := ~, start := 174841, stop := 174848 ), 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 := 174866, root := ~, start := 174858, stop := 174865 ), 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 := 174872, root := ~, start := 174868, stop := 174871 ), 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 := 174962, root := ~, start := 174958, stop := 174961 ), 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 := 175012, root := ~, start := 175000, stop := 175011 ), 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 := 175029, root := ~, start := 175017, stop := 175028 ), 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 := 175109, root := ~, start := 175097, stop := 175108 ), 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 := 175130, root := ~, start := 175118, stop := 175129 ), 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 := 175226, root := ~, start := 175214, stop := 175225 ), 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 := 175269, root := ~, start := 175257, stop := 175268 ), 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 := 175320, root := ~, start := 175308, stop := 175319 ), 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 := 175337, root := ~, start := 175325, stop := 175336 ), 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 := 175401, root := ~, start := 175393, stop := 175400 ), 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 := 175406, root := ~, start := 175402, stop := 175405 ), 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 := 175770, root := ~, start := 175411, stop := 175769 ), rec( content := "\n ", count := [ 5, 3, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Description", next := 175787, root := ~, start := 174299, stop := 175786 ) ], count := [ 5, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 175801, root := ~, start := 173908, stop := 175800 ), 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 := 175875, root := ~, start := 175822, stop := 175874 ), 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 := 175913, root := ~, start := 175878, stop := 175912 ), 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 := 175951, root := ~, start := 175937, stop := 175950 ), 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 := 175978, root := ~, start := 175970, stop := 175977 ), 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 := 176040, root := ~, start := 176023, stop := 176039 ), 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 := 176058, root := ~, start := 176048, stop := 176057 ), 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 := 176129, root := ~, start := 176118, stop := 176128 ), 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 := 176147, root := ~, start := 176135, stop := 176146 ), 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 := 176189, root := ~, start := 176185, stop := 176188 ), 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 := 176243, root := ~, start := 176235, stop := 176242 ), 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 := 176262, root := ~, start := 176254, stop := 176261 ), 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 := 176302, root := ~, start := 176286, stop := 176301 ), 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 := 176424, root := ~, start := 176408, stop := 176423 ), 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 := 176443, root := ~, start := 176432, stop := 176442 ), 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 := 176529, root := ~, start := 176521, stop := 176528 ), 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 := 176556, root := ~, start := 176536, stop := 176555 ), 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 := 176582, root := ~, start := 176574, stop := 176581 ), 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 := 176592, root := ~, start := 176588, stop := 176591 ), 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 := 176713, root := ~, start := 176700, stop := 176712 ), 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 := 177209, root := ~, start := 176720, stop := 177208 ), rec( content := "\n ", count := [ 5, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 177226, root := ~, start := 175916, stop := 177225 ) ], count := [ 5, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 177240, root := ~, start := 175807, stop := 177239 ), 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 := 177325, root := ~, start := 177261, stop := 177324 ), 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 := 177367, root := ~, start := 177352, stop := 177366 ), rec( content := ".", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 177378, root := ~, start := 177328, stop := 177377 ), 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 := 177416, root := ~, start := 177402, stop := 177415 ), 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 := 177442, root := ~, start := 177434, stop := 177441 ), 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 := 177472, root := ~, start := 177458, stop := 177471 ), 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 := 177567, root := ~, start := 177553, stop := 177566 ), 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 := 177599, root := ~, start := 177591, stop := 177598 ), 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 := 177664, root := ~, start := 177614, stop := 177663 ), 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 := 177741, root := ~, start := 177737, stop := 177740 ), 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 := 1778\ 34, root := ~, start := 177826, stop := 177833 ), 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\ := 177963, root := ~, start := 177955, stop := 177962 ), 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 := \ 178077, root := ~, start := 178062, stop := 178076 ), 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 := 178168, root := ~, start := 178136, stop \ := 178167 ), 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 := 178173, root := ~, star\ t := 178169, stop := 178172 ), 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 := 178252\ , root := ~, start := 178238, stop := 178251 ), 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 := 178261, root := ~, start := 178253, sto\ p := 178260 ), rec( content := "))", count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "C", next := 178267, root := ~, start := 178183, stop := 178266 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 3, 4, 6 ], name := "Item", next := 178282, root := ~, start := 177\ 760, stop := 178281 ), 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 := 178373, root := ~, start := 178365, stop \ := 178372 ), 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 := 178469, root := ~, start := 178290, stop := 178468 ) ], count := [ 5, 3, 4, 5 ], mathmode := "Text", name := "List", next := 178481, root := ~, start := 177747, stop := 178480 ), 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 := 178519, root := ~, start := 178511, stop := 178518 ), 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 := 178543, root := ~, start := 178535, stop := 178542 ), 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 := 178575, root := ~, start := 178561, stop := 178574 ), 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 := 178608, root := ~, start := 178582, stop := 178607 ), 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 := 178920, root := ~, start := 178611, stop := 178919 ), rec( content := "\n ", count := [ 5, 3, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 178937, root := ~, start := 177381, stop := 178936 ) ], count := [ 5, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 178951, root := ~, start := 177246, stop := 178950 ), 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 := 179082, root := ~, start := 178972, stop := 179081 ), 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 := 179196, root := ~, start := 179085, stop := 179195 ), 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 := 179236, root := ~, start := 179199, stop := 179235 ), 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 := 179330, root := ~, start := 179316, stop := 179329 ), 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 := 179392, root := ~, start := 179379, stop := 179391 ), 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 := 179415, root := ~, start := 179400, stop := 179414 ), 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 := 179450, root := ~, start := 179434, stop := 179449 ), 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 := 179456, root := ~, start := 179452, stop := 179455 ), 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 := 179476, root := ~, start := 179462, stop := 179475 ), 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 := 179511, root := ~, start := 179498, stop := 179510 ), 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 := 179561, root := ~, start := 179546, stop := 179560 ), 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 := 179620, root := ~, start := 179604, stop := 179619 ), 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 := 179927, root := ~, start := 179669, stop := 179926 ), rec( content := "\n ", count := [ 5, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 179944, root := ~, start := 179239, stop := 179943 ) ], count := [ 5, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 179958, root := ~, start := 178957, stop := 179957 ), 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 := 180020, root := ~, start := 179979, stop := 180019 ), 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 := 180062, root := ~, start := 180047, stop := 180061 ), rec( content := ".", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 180073, root := ~, start := 180023, stop := 180072 ), 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 := 180133, root := ~, start := 180121, stop := 180132 ), 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 := 180163, root := ~, start := 180149, stop := 180162 ), 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 := 180185, root := ~, start := 180173, stop := 180184 ), 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 := 180305, root := ~, start := 180261, stop := 180304 ), 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 := 180383, root := ~, start := 180379, stop := 180382 ), 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 := 180406, root := ~, start := 180392, stop := 180405 ), 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 := 180481, root := ~, start := 180466, stop := 180480 ), 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 := 180501, root := ~, start := 180487, stop := 180500 ), 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 := 180562, root := ~, start := 180554, stop := 180561 ), 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 := 180572, root := ~, start := 180568, stop := 180571 ), 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 := 180712, root := ~, start := 180704, stop := 18071\ 1 ), rec( content := " ^ 3)", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "C", next := 180721, root := ~, start := 180699, stop := 180720 ), 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 := 180741, root := ~, start := 180733, stop := 180740 ), 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 := 180785, root := ~, start := 180771, stop := 180784 ), 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 := 180862, root := ~, start := 180836, stop := 180861 ), 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 := 180922, root := ~, start := 180918, stop := 180921 ), 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 := 180997, root := ~, start := 180961, stop := 180996 ), 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 := 181003, root := ~, start := 180999, stop := 181002 ), 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 := 181501, root := ~, start := 181008, stop := 181500 ), rec( content := "\n ", count := [ 5, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 181518, root := ~, start := 180076, stop := 181517 ) ], count := [ 5, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 181532, root := ~, start := 179964, stop := 181531 ), 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 := 181597, root := ~, start := 181553, stop := 181596 ), 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 := 181639, root := ~, start := 181624, stop := 181638 ), rec( content := ".", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 181650, root := ~, start := 181600, stop := 181649 ), 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 := 181714, root := ~, start := 181698, stop := 181713 ), 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 := 181744, root := ~, start := 181730, stop := 181743 ), 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 := 181770, root := ~, start := 181754, stop := 181769 ), 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 := 181909, root := ~, start := 181861, stop := 181908 ), 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 := 181987, root := ~, start := 181983, stop := 181986 ), 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 := 182010, root := ~, start := 181996, stop := 182009 ), 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 := 182099, root := ~, start := 182084, stop := 182098 ), 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 := 182123, root := ~, start := 182109, stop := 182122 ), 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 := 182183, root := ~, start := 182169, stop := 182182 ), 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 := 182244, root := ~, start := 182236, stop := 182243 ), 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 := 182254, root := ~, start := 182250, stop := 182253 ), 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 := 182295, root := ~, start := 182269, stop := 182294 ), 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 := 182301, root := ~, start := 182297, stop := 182300 ), 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 := 182881, root := ~, start := 182306, stop := 182880 ), rec( content := "\n ", count := [ 5, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 182898, root := ~, start := 181653, stop := 182897 ) ], count := [ 5, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 182912, root := ~, start := 181538, stop := 182911 ), 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 := 182984, root := ~, start := 182933, stop := 182983 ), 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 := 183025, root := ~, start := 183010, stop := 183024 ), rec( content := ".", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 183036, root := ~, start := 182987, stop := 183035 ), 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 := 183074, root := ~, start := 183060, stop := 183073 ), 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 := 183157, root := ~, start := 183143, stop := 183156 ), 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 := 183236, root := ~, start := 183223, stop := 183235 ), 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 := 183254, root := ~, start := 183241, stop := 183253 ), 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 := 183303, root := ~, start := 183295, stop := 183302 ), 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 := 183316, root := ~, start := 183308, stop := 183315 ), 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 := 183323, root := ~, start := 183319, stop := 183322 ), 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 := 183345, root := ~, start := 183333, stop := 183344 ), 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 := 183553, root := ~, start := 183549, stop := 183552 ), 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 := 183576, root := ~, start := 183562, stop := 183575 ), 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 := 183642, root := ~, start := 183627, stop := 183641 ), 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 := 183648, root := ~, start := 183644, stop := 183647 ), 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 := 184200, root := ~, start := 183653, stop := 184199 ), rec( content := "\n ", count := [ 5, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 184217, root := ~, start := 183039, stop := 184216 ) ], count := [ 5, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 184231, root := ~, start := 182918, stop := 184230 ), 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 := 184307, root := ~, start := 184252, stop := 184306 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrConnec\ tedComponents" ), content := 0, count := [ 5, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 184367, root := ~, start := 184310, stop := 184366 ), 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 := 184398, root := ~, start := 184370, stop := 184397 ), 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 := 184462, root := ~, start := 184452, stop := 184461 ), 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 := 184545, root := ~, start := 184531, stop := 184544 ), 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 := 184582, root := ~, start := 184568, stop := 184581 ), 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 := 184834, root := ~, start := 184820, stop := 184833 ), 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 := 184957, root := ~, start := 184909, stop := 184956 ), 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 := 185007, root := ~, start := 184970, stop := 185006 ), 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 := 185025, root := ~, start := 185011, stop := 185024 ), 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 := 185032, root := ~, start := 185028, stop := 185031 ), 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 := 185131, root := ~, start := 185117, stop := 185130 ), 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 := 185137, root := ~, start := 185133, stop := 185136 ), 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 := 185164, root := ~, start := 185154, stop := 185163 ), 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 := 185195, root := ~, start := 185183, stop := 185194 ), 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 := 185209, root := ~, start := 185200, stop := 185208 ), 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 := 185241, root := ~, start := 185229, stop := 185240 ), 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 := 185309, root := ~, start := 185295, stop := 185308 ), 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 := 185376, root := ~, start := 185367, stop := 185375 ), 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 := 185416, root := ~, start := 185408, stop := 185415 ), 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 := 185488, root := ~, start := 185469, stop := 185487 ), 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 := 185494, root := ~, start := 185490, stop := 185493 ), 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 := 185562, root := ~, start := 185549, stop := 185561 ), 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 := 185582, root := ~, start := 185574, stop := 185581 ), 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 := 185622, root := ~, start := 185614, stop := 185621 ), 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 := 185669, root := ~, start := 185665, stop := 185668 ), 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 := 1857\ 21, root := ~, start := 185707, stop := 185720 ), rec( content := ")", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 185726, root := ~, start := 185675, stop := 185725 ), 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 := 1858\ 07, root := ~, start := 185793, stop := 185806 ), rec( content := ").comps)", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 185819, root := ~, start := 185756, stop := 185818 ), 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 := 186753, root := ~, start := 185856, stop := 186752 ), rec( content := "\n ", count := [ 5, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 186770, root := ~, start := 184401, stop := 186769 ) ], count := [ 5, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 186784, root := ~, start := 184237, stop := 186783 ), 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 := 186867, root := ~, start := 186805, stop := 186866 ), 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 := 186908, root := ~, start := 186870, stop := 186907 ), 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 := 186945, root := ~, start := 186932, stop := 186944 ), 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 := 186987, root := ~, start := 186973, stop := 186986 ), 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 := 187061, root := ~, start := 187048, stop := 187060 ), 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 := 187083, root := ~, start := 187069, stop := 187082 ), 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 := 187133, root := ~, start := 187093, stop := 187132 ), 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 := 187425, root := ~, start := 187160, stop := 187424 ), rec( content := "\n ", count := [ 5, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 187442, root := ~, start := 186911, stop := 187441 ) ], count := [ 5, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 187456, root := ~, start := 186790, stop := 187455 ), 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 := 187540, root := ~, start := 187477, stop := 187539 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrStrong\ lyConnectedComponents" ), content := 0, count := [ 5, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 187608, root := ~, start := 187543, stop := 187607 ), 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 := 187639, root := ~, start := 187611, stop := 187638 ), 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 := 187703, root := ~, start := 187693, stop := 187702 ), 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 := 187788, root := ~, start := 187774, stop := 187787 ), 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 := 187825, root := ~, start := 187811, stop := 187824 ), 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 := 188059, root := ~, start := 188045, stop := 188058 ), 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 := 188069, root := ~, start := 188065, stop := 188068 ), 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 := 188096, root := ~, start := 188086, stop := 188095 ), 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 := 188127, root := ~, start := 188115, stop := 188126 ), 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 := 188141, root := ~, start := 188132, stop := 188140 ), 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 := 188173, root := ~, start := 188161, stop := 188172 ), 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 := 188242, root := ~, start := 188228, stop := 188241 ), 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 := 188309, root := ~, start := 188300, stop := 188308 ), 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 := 188349, root := ~, start := 188341, stop := 188348 ), 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 := 188423, root := ~, start := 188404, stop := 188422 ), 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 := 188429, root := ~, start := 188425, stop := 188428 ), 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 := 188539, root := ~, start := 188520, stop := 188538 ), 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 := 188572, root := ~, start := 188559, stop := 188571 ), 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 := 188591, root := ~, start := 188583, stop := 188590 ), 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 := 188664, root := ~, start := 188656, stop := 188663 ), 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 := 188711, root := ~, start := 188707, stop := 188710 ), 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 := 1887\ 71, root := ~, start := 188757, stop := 188770 ), rec( content := ")", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 188776, root := ~, start := 188717, stop := 188775 ), 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 := 1888\ 69, root := ~, start := 188855, stop := 188868 ), rec( content := ").comps)", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 188881, root := ~, start := 188810, stop := 188880 ), 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 := 189824, root := ~, start := 188918, stop := 189823 ), rec( content := "\n ", count := [ 5, 3, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 189841, root := ~, start := 187642, stop := 189840 ) ], count := [ 5, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 189855, root := ~, start := 187462, stop := 189854 ), 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 := 189946, root := ~, start := 189876, stop := 189945 ), 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 := 189987, root := ~, start := 189949, stop := 189986 ), 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 := 190024, root := ~, start := 190011, stop := 190023 ), 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 := 190066, root := ~, start := 190052, stop := 190065 ), 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 := 190149, root := ~, start := 190136, stop := 190148 ), 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 := 190171, root := ~, start := 190157, stop := 190170 ), 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 := 190229, root := ~, start := 190181, stop := 190228 ), 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 := 190536, root := ~, start := 190256, stop := 190535 ), rec( content := "\n ", count := [ 5, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 190553, root := ~, start := 189990, stop := 190552 ) ], count := [ 5, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 190567, root := ~, start := 189861, stop := 190566 ), 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 := 190636, root := ~, start := 190588, stop := 190635 ), 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 := 190691, root := ~, start := 190680, stop := 190690 ), rec( content := ".", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Returns", next := 190702, root := ~, start := 190639, stop := 190701 ), 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 := 190740, root := ~, start := 190726, stop := 190739 ), 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 := 190824, root := ~, start := 190786, stop := 190823 ), 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 := 190857, root := ~, start := 190831, stop := 190856 ), 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 := 190910, root := ~, start := 190896, stop := 190909 ), 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 := 190953, root := ~, start := 190927, stop := 190952 ), 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 := 190973, root := ~, start := 190962, stop := 190972 ), 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 := 190978, root := ~, start := 190974, stop := 190977 ), 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 := 191196, root := ~, start := 191177, stop := 191195 ), 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 := 191214, root := ~, start := 191200, stop := 191213 ), 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 := 191274, root := ~, start := 191260, stop := 191273 ), 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 := 191342, root := ~, start := 191328, stop := 191341 ), 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 := 191348, root := ~, start := 191344, stop := 191347 ), 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 := 191543, root := ~, start := 191505, stop := 191542 ), 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 := 192136, root := ~, start := 191570, stop := 192135 ), rec( content := "\n ", count := [ 5, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Description", next := 192153, root := ~, start := 190705, stop := 192152 ) ], count := [ 5, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 192167, root := ~, start := 190573, stop := 192166 ), 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 := 192235, root := ~, start := 192188, stop := 192234 ), 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 := 192276, root := ~, start := 192238, stop := 192275 ), 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 := 192338, root := ~, start := 192320, stop := 192337 ), 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 := 192417, root := ~, start := 192385, stop := 192416 ), 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 := 192481, root := ~, start := 192467, stop := 192480 ), 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 := 192547, root := ~, start := 192539, stop := 192546 ), 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 := 192565, root := ~, start := 192551, stop := 192564 ), 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 := 192663, root := ~, start := 192638, stop := 192662 ), 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 := 192668, root := ~, start := 192664, stop := 192667 ), 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 := 192699, root := ~, start := 192674, stop := 192698 ), 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 := 192763, root := ~, start := 192749, stop := 192762 ), 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 := 192837, root := ~, start := 192823, stop := 192836 ), 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 := 192860, root := ~, start := 192856, stop := 192859 ), 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 := 192913, root := ~, start := 192909, stop := 192912 ), 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 := 192981, root := ~, start := 192968, stop := 192980 ), 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 := 193000, root := ~, start := 192992, stop := 192999 ), 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 := 193036, root := ~, start := 193028, stop := 193035 ), 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 := 193086, root := ~, start := 193082, stop := 193085 ), 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 := 193109, root := ~, start := 193101, stop := 193108 ), 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 := 193149, root := ~, start := 193128, stop := 193148 ), 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 := 193266, root := ~, start := 193258, stop := 193265 ), 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 := 193296, root := ~, start := 193288, stop := 193295 ), 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 := 193335, root := ~, start := 193327, stop := 193334 ), 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 := 193529, root := ~, start := 193525, stop := 193528 ), 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 := 193578, root := ~, start := 193544, stop := 193577 ), 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 := 193621, root := ~, start := 193588, stop := 193620 ), 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 := 194026, root := ~, start := 193623, stop := 194025 ), rec( content := "\n ", count := [ 5, 3, 14, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 194043, root := ~, start := 192279, stop := 194042 ) ], count := [ 5, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 194057, root := ~, start := 192173, stop := 194056 ), 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 := 194108, root := ~, start := 194078, stop := 194107 ), 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 := 194163, root := ~, start := 194111, stop := 194162 ), 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 := 194230, root := ~, start := 194207, stop := 194229 ), 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 := 194309, root := ~, start := 194277, stop := 194308 ), 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 := 194365, root := ~, start := 194357, stop := 194364 ), 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 := 194439, root := ~, start := 194426, stop := 194438 ), 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 := 194451, root := ~, start := 194443, stop := 194450 ), 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 := 194536, root := ~, start := 194523, stop := 194535 ), 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 := 194541, root := ~, start := 194537, stop := 194540 ), 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 := 194561, root := ~, start := 194547, stop := 194560 ), 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 := 194603, root := ~, start := 194595, stop := 194602 ), 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 := 194671, root := ~, start := 194663, stop := 194670 ), 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 := 194694, root := ~, start := 194690, stop := 194693 ), 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 := 194747, root := ~, start := 194743, stop := 194746 ), 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 := 194815, root := ~, start := 194802, stop := 194814 ), 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 := 194834, root := ~, start := 194826, stop := 194833 ), 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 := 194870, root := ~, start := 194862, stop := 194869 ), 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 := 194925, root := ~, start := 194921, stop := 194924 ), 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 := 194942, root := ~, start := 194934, stop := 194941 ), 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 := 195058, root := ~, start := 195026, stop := 195057 ), 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 := 195110, root := ~, start := 195102, stop := 195109 ), 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 := 195140, root := ~, start := 195132, stop := 195139 ), 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 := 195179, root := ~, start := 195171, stop := 195178 ), 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 := 195372, root := ~, start := 195368, stop := 195371 ), 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 := 195421, root := ~, start := 195387, stop := 195420 ), 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 := 195464, root := ~, start := 195431, stop := 195463 ), 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 := 195840, root := ~, start := 195466, stop := 195839 ), rec( content := "\n ", count := [ 5, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Description", next := 195857, root := ~, start := 194166, stop := 195856 ) ], count := [ 5, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 195871, root := ~, start := 194063, stop := 195870 ), 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 := 195932, root := ~, start := 195892, stop := 195931 ), rec( attributes := rec( Arg := "D", Name := "StrongOrientati\ onAttr" ), content := 0, count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "Attr", next := 195979, root := ~, start := 195935, stop := 195978 ), 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 := 196015, root := ~, start := 196004, stop := 196014 ), rec( content := ".", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Returns", next := 196026, root := ~, start := 195982, stop := 196025 ), 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 := 196074, root := ~, start := 196049, stop := 196073 ), 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 := 196116, root := ~, start := 196108, stop := 196115 ), 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 := 196179, root := ~, start := 196171, stop := 196178 ), 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 := 196191, root := ~, start := 196183, stop := 196190 ), 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 := 196234, root := ~, start := 196221, stop := 196233 ), 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 := 196246, root := ~, start := 196238, stop := 196245 ), 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 := 196267, root := ~, start := 196254, stop := 196266 ), 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 := 196288, root := ~, start := 196275, stop := 196287 ), 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 := 196311, root := ~, start := 196303, stop := 196310 ), 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 := 196472, root := ~, start := 196439, stop := 196471 ), 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 := 196484, root := ~, start := 196480, stop := 196483 ), 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 := 196557, root := ~, start := 196549, stop := 196556 ), 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 := 196574, root := ~, start := 196566, stop := 196573 ), 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 := 196600, root := ~, start := 196592, stop := 196599 ), 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 := 196641, root := ~, start := 196633, stop := 196640 ), 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 := 196716, root := ~, start := 196705, stop := 196715 ), 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 := 196745, root := ~, start := 196737, stop := 196744 ), 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 := 196797, root := ~, start := 196793, stop := 196796 ), 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 := 196814, root := ~, start := 196806, stop := 196813 ), 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 := 196858, root := ~, start := 196850, stop := 19685\ 7 ), rec( content := ")", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "C", next := 196863, root := ~, start := 196829, stop := 196862 ), 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 := 196913, root := ~, start := 196905, stop := 196912 ), 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 := 196964, root := ~, start := 196956, stop := 19696\ 3 ), rec( content := ")", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "C", next := 196969, root := ~, start := 196935, stop := 196968 ), 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 := 197001, root := ~, start := 196997, stop := 197000 ), 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 := 197069, root := ~, start := 197056, stop := 197068 ), 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 := 197088, root := ~, start := 197080, stop := 197087 ), 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 := 197124, root := ~, start := 197116, stop := 197123 ), 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 := 197670, root := ~, start := 197171, stop := 197669 ), rec( content := "\n ", count := [ 5, 3, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Description", next := 197687, root := ~, start := 196029, stop := 197686 ) ], count := [ 5, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 197701, root := ~, start := 195877, stop := 197700 ), 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 := 197764, root := ~, start := 197722, stop := 197763 ), 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 := 197797, root := ~, start := 197767, stop := 197796 ), 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 := 197880, root := ~, start := 197866, stop := 197879 ), 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 := 197885, root := ~, start := 197881, stop := 197884 ), 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 := 197918, root := ~, start := 197904, stop := 197917 ), 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 := 198075, root := ~, start := 198061, stop := 198074 ), 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 := 198095, root := ~, start := 198081, stop := 198094 ), 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 := 198159, root := ~, start := 198151, stop := 198158 ), 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 := 198224, root := ~, start := 198174, stop := 198223 ), 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 := 198269, root := ~, start := 198265, stop := 198268 ), 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 := 198310, root := ~, start := 198302, stop := 198309 ), 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 := 198341, root := ~, start := 198325, stop := 198340 ), 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 := 198384, root := ~, start := 198352, stop := 198383 ), 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 := 198390, root := ~, start := 198386, stop := 198389 ), 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 := 198933, root := ~, start := 198395, stop := 198932 ), rec( content := "\n ", count := [ 5, 3, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 198950, root := ~, start := 197800, stop := 198949 ) ], count := [ 5, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 198964, root := ~, start := 197707, stop := 198963 ), 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 := 199054, root := ~, start := 198985, stop := 199053 ), 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 := 199085, root := ~, start := 199057, stop := 199084 ), 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 := 199123, root := ~, start := 199109, stop := 199122 ), 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 := 199150, root := ~, start := 199142, stop := 199149 ), 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 := 199213, root := ~, start := 199196, stop := 199212 ), 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 := 199231, root := ~, start := 199221, stop := 199230 ), 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 := 199345, root := ~, start := 199331, stop := 199344 ), 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 := 199351, root := ~, start := 199347, stop := 199350 ), 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 := 199398, root := ~, start := 199371, stop := 199397 ), 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 := 199445, root := ~, start := 199434, stop := 199444 ), 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 := 199460, root := ~, start := 199447, stop := 199459 ), 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 := 199477, root := ~, start := 199466, stop := 199476 ), 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 := 199510, root := ~, start := 199497, stop := 199509 ), 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 := 199526, root := ~, start := 199515, stop := 199525 ), 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 := 199587, root := ~, start := 199576, stop := 199586 ), 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 := 199659, root := ~, start := 199649, stop := 199658 ), 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 := 199721, root := ~, start := 199710, stop := 199720 ), 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 := 199800, root := ~, start := 199796, stop := 199799 ), 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 := 199851, root := ~, start := 199841, stop := 199850 ), 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 := 199889, root := ~, start := 199873, stop := 199888 ), 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 := 199910, root := ~, start := 199899, stop := 199909 ), 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 := 199955, root := ~, start := 199947, stop := 199954 ), 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 := 199974, root := ~, start := 199966, stop := 199973 ), 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 := 200013, root := ~, start := 199997, stop := 200012 ), 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 := 200040, root := ~, start := 200027, stop := 200039 ), 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 := 200101, root := ~, start := 200074, stop := 200100 ), 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 := 200141, root := ~, start := 200130, stop := 200140 ), 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 := 200212, root := ~, start := 200198, stop := 200211 ), 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 := 200218, root := ~, start := 200214, stop := 200217 ), 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 := 200416, root := ~, start := 200220, stop := 200415 ), 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 := 200443, root := ~, start := 200433, stop := 200442 ), 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 := 200527, root := ~, start := 200500, stop := 200526 ), 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 := 200633, root := ~, start := 200629, stop := 200632 ), 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 := 201339, root := ~, start := 200639, stop := 201338 ), rec( content := "\n ", count := [ 5, 3, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 201356, root := ~, start := 199088, stop := 201355 ) ], count := [ 5, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 201370, root := ~, start := 198970, stop := 201369 ), 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 := 201437, root := ~, start := 201391, stop := 201436 ), 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 := 201460, root := ~, start := 201449, stop := 201459 ), 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 := 201476, root := ~, start := 201464, stop := 201475 ), rec( content := ".", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Returns", next := 201487, root := ~, start := 201440, stop := 201486 ), 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 := 201542, root := ~, start := 201531, stop := 201541 ), 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 := 201611, root := ~, start := 201603, stop := 201610 ), 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 := 201630, root := ~, start := 201622, stop := 201629 ), 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 := 201660, root := ~, start := 201646, stop := 201659 ), 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 := 201682, root := ~, start := 201670, stop := 201681 ), 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 := 201790, root := ~, start := 201746, stop := 201789 ), 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 := 201854, root := ~, start := 201850, stop := 201853 ), 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 := 201893, root := ~, start := 201875, stop := 201892 ), 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 := 201944, root := ~, start := 201923, stop := 201943 ), 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 := 201959, root := ~, start := 201951, stop := 201958 ), 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 := 201995, root := ~, start := 201987, stop := 201994 ), 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 := 202040, root := ~, start := 202026, stop := 202039 ), 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 := 202289, root := ~, start := 202043, stop := 202288 ), rec( content := "\n ", count := [ 5, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 202306, root := ~, start := 201490, stop := 202305 ) ], count := [ 5, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 202320, root := ~, start := 201376, stop := 202319 ), 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 := 202387, root := ~, start := 202341, stop := 202386 ), 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 := 202430, root := ~, start := 202419, stop := 202429 ), rec( content := ".", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Returns", next := 202441, root := ~, start := 202390, stop := 202440 ), 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 := 202560, root := ~, start := 202552, stop := 202559 ), 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 := 202569, root := ~, start := 202561, stop := 202568 ), 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 := 202578, root := ~, start := 202570, stop := 202577 ), 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 := 202600, root := ~, start := 202592, stop := 202599 ), 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 := 202623, root := ~, start := 202615, stop := 202622 ), 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 := 202653, root := ~, start := 202639, stop := 202652 ), 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 := 202778, root := ~, start := 202767, stop := 202777 ), 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 := 202834, root := ~, start := 202793, stop := 202833 ), 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 := 202907, root := ~, start := 202903, stop := 202906 ), 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 := 202978, root := ~, start := 202968, stop := 202977 ), 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 := 203030, root := ~, start := 202984, stop := 203029 ), 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 := 203082, root := ~, start := 203070, stop := 203081 ), 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 := 203136, root := ~, start := 203128, stop := 203135 ), 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 := 203176, root := ~, start := 203149, stop := 203175 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 20, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 20, 6 ], name := "Item", next := 203191, root := ~, start := 203\ 113, stop := 203190 ), 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 := 203221, root := ~ , star\ t := 203213, stop := 203220 ), 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 \ := 203286, root := ~, start := 203255, stop := 203285 ), 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\ := 203330, root := ~, start := 203315, stop := 203329 ), 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 := 203342, root := ~, start := 20333\ 2, stop := 203341 ), 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 := 203384, root := ~, start := 2\ 03370, stop := 203383 ), 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 := 203425, root := ~, start :\ = 203396, stop := 203424 ), 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\ := 203439, root := ~, start := 203425, stop := 203438 ), 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 := 203447, root := ~, start := 203439, stop := 203\ 446 ), 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 := 203457, root := ~ , star\ t := 203447, stop := 203456 ), 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\ := 203465, root := ~, start := 203457, stop := 203464 ), 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 := 203510, root := ~, start := 203500, stop := 203509 ), 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 := 203666, root := ~, start := 203198, stop := 203665 ) ], count := [ 5, 3, 20, 5 ], mathmode := "Text", name := "List", next := 203678, root := ~, start := 203100, stop := 203677 ), 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 := 203717, root := ~, start := 203699, stop := 203716 ), 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 := 203768, root := ~, start := 203747, stop := 203767 ), 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 := 203783, root := ~, start := 203775, stop := 203782 ), 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 := 203819, root := ~, start := 203811, stop := 203818 ), 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 := 203864, root := ~, start := 203850, stop := 203863 ), 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 := 204147, root := ~, start := 203867, stop := 204146 ), rec( content := "\n ", count := [ 5, 3, 20, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 204164, root := ~, start := 202444, stop := 204163 ) ], count := [ 5, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 204178, root := ~, start := 202326, stop := 204177 ), 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 := 204253, root := ~, start := 204199, stop := 204252 ), 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 := 204296, root := ~, start := 204285, stop := 204295 ), rec( content := ".", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Returns", next := 204307, root := ~, start := 204256, stop := 204306 ), 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 := 204414, root := ~, start := 204406, stop := 204413 ), 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 := 204437, root := ~, start := 204429, stop := 204436 ), 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 := 204476, root := ~, start := 204468, stop := 20447\ 5 ), 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 := 204491, root := ~, start := 204483, stop := 20449\ 0 ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "C", next := 204495, root := ~, start := 204465, stop := 204494 ), 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 := 204616, root := ~, start := 204605, stop := 204615 ), 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 := 204676, root := ~, start := 204635, stop := 204675 ), 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 := 204749, root := ~, start := 204745, stop := 204748 ), 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 := 204784, root := ~, start := 204759, stop := 204783 ), 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 := 204858, root := ~, start := 204832, stop := 204857 ), 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 := 204907, root := ~, start := 204892, stop := 204906 ), 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 := 204922, root := ~, start := 204914, stop := 204921 ), 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 := 204958, root := ~, start := 204950, stop := 204957 ), 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 := 205003, root := ~, start := 204989, stop := 205002 ), 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 := 205355, root := ~, start := 205006, stop := 205354 ), rec( content := "\n ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Description", next := 205372, root := ~, start := 204310, stop := 205371 ) ], count := [ 5, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 205386, root := ~, start := 204184, stop := 205385 ), 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 := 205457, root := ~, start := 205407, stop := 205456 ), 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 := 205491, root := ~, start := 205460, stop := 205490 ), 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 := 205529, root := ~, start := 205515, stop := 205528 ), 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 := 205629, root := ~, start := 205602, stop := 205628 ), 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 := 205760, root := ~, start := 205752, stop := 205759 ), 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 := 205769, root := ~, start := 205761, stop := 205768 ), 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 := 205778, root := ~, start := 205770, stop := 205777 ), 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 := 205797, root := ~, start := 205783, stop := 205796 ), 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 := 205822, root := ~, start := 205814, stop := 205821 ), 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 := 205849, root := ~, start := 205841, stop := 205848 ), 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 := 205859, root := ~, start := 205855, stop := 205858 ), 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 := 205894, root := ~, start := 205869, stop := 205893 ), 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 := 206048, root := ~, start := 206004, stop := 206047 ), 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 := 206144, root := ~, start := 206103, stop := 206143 ), 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 := 206217, root := ~, start := 206213, stop := 206216 ), 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 := 206702, root := ~, start := 206219, stop := 206701 ), rec( content := "\n ", count := [ 5, 3, 22, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 206719, root := ~, start := 205494, stop := 206718 ) ], count := [ 5, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 206733, root := ~, start := 205392, stop := 206732 ), 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 := 206807, root := ~, start := 206754, stop := 206806 ), 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 := 206857, root := ~, start := 206810, stop := 206856 ), 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 := 206895, root := ~, start := 206881, stop := 206894 ), 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 := 206946, root := ~, start := 206915, stop := 206945 ), 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 := 206995, root := ~, start := 206973, stop := 206994 ), 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 := 207013, root := ~, start := 206999, stop := 207012 ), 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 := 207019, root := ~, start := 207015, stop := 207018 ), 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 := 207081, root := ~, start := 207037, stop := 207080 ), 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 := 207193, root := ~, start := 207189, stop := 207192 ), 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 := 207412, root := ~, start := 207381, stop := 207411 ), 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 := 207575, root := ~, start := 207559, stop := 207574 ), 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 := 207601, root := ~, start := 207584, stop := 207600 ), 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 := 207630, root := ~, start := 207622, stop := 207629 ), 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 := 207644, root := ~, start := 207635, stop := 207643 ), 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 := 207710, root := ~, start := 207679, stop := 207709 ), 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 := 207750, root := ~, start := 207746, stop := 207749 ), 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 := 207813, root := ~, start := 207805, stop := 207812 ), 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 := 207867, root := ~, start := 207859, stop := 207866 ), 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 := 207966, root := ~, start := 207900, stop := 207965 ), 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 := 208027, root := ~, start := 208002, stop := 208026 ), 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 := 208110, root := ~, start := 208096, stop := 208109 ), 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 := 2081\ 58, root := ~, start := 208144, stop := 208157 ), rec( content := ")", count := [ 5, 3, 23, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "C", next := 208163, root := ~, start := 208116, stop := 208162 ), 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 := 208232, root := ~, start := 208228, stop := 208231 ), 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 := 208885, root := ~, start := 208238, stop := 208884 ), rec( content := "\n ", count := [ 5, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 208902, root := ~, start := 206860, stop := 208901 ) ], count := [ 5, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 208916, root := ~, start := 206739, stop := 208915 ), 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 := 208993, root := ~, start := 208937, stop := 208992 ), 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 := 209039, root := ~, start := 209028, stop := 209038 ), rec( content := ".", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Returns", next := 209050, root := ~, start := 208996, stop := 209049 ), 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 := 209088, root := ~, start := 209074, stop := 209087 ), 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 := 209142, root := ~, start := 209108, stop := 209141 ), 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 := 209188, root := ~, start := 209167, stop := 209187 ), 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 := 209206, root := ~, start := 209192, stop := 209205 ), 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 := 209268, root := ~, start := 209224, stop := 209267 ), 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 := 209365, root := ~, start := 209361, stop := 209364 ), 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 := 209431, root := ~, start := 209399, stop := 209430 ), 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 := 209445, root := ~, start := 209431, stop := 209444 ), 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 := 209453, root := ~, start := 209445, stop := 209452 ), 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 := 209507, root := ~, start := 209493, stop := 209506 ), 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 := 209678, root := ~, start := 209640, stop := 209677 ), 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 := 209683, root := ~, start := 209679, stop := 209682 ), 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 := 209706, root := ~, start := 209692, stop := 209705 ), 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 := 209774, root := ~, start := 209763, stop := 209773 ), 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 := 209794, root := ~, start := 209780, stop := 209793 ), 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 := 209891, root := ~, start := 209887, stop := 209890 ), 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 := 210363, root := ~, start := 209897, stop := 210362 ), rec( content := "\n ", count := [ 5, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Description", next := 210380, root := ~, start := 209053, stop := 210379 ) ], count := [ 5, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 210394, root := ~, start := 208922, stop := 210393 ), 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 := 210465, root := ~, start := 210415, stop := 210464 ), 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 := 210494, root := ~, start := 210468, stop := 210493 ), 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 := 210556, root := ~, start := 210545, stop := 210555 ), 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 := 210581, root := ~, start := 210567, stop := 210580 ), 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 := 210662, root := ~, start := 210649, stop := 210661 ), 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 := 210684, root := ~, start := 210670, stop := 210683 ), 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 := 210700, root := ~, start := 210688, stop := 210699 ), 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 := 210722, root := ~, start := 210709, stop := 210721 ), 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 := 210764, root := ~, start := 210756, stop := 210763 ), 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 := 210894, root := ~, start := 210782, stop := 210893 ), rec( content := "\n ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Description", next := 210911, root := ~, start := 210497, stop := 210910 ) ], count := [ 5, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 210925, root := ~, start := 210400, stop := 210924 ), 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 := 210992, root := ~, start := 210946, stop := 210991 ), 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 := 211042, root := ~, start := 211031, stop := 211041 ), rec( content := ".", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Returns", next := 211053, root := ~, start := 210995, stop := 211052 ), 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 := 211091, root := ~, start := 211077, stop := 211090 ), 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 := 211180, root := ~, start := 211148, stop := 211179 ), 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 := 211213, root := ~, start := 211185, stop := 211212 ), 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 := 211280, root := ~, start := 211266, stop := 211279 ), 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 := 211286, root := ~, start := 211282, stop := 211285 ), 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 := 211349, root := ~, start := 211341, stop := 211348 ), 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 := 211395, root := ~, start := 211381, stop := 211394 ), 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 := 211483, root := ~, start := 211475, stop := 211482 ), 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 := 211537, root := ~, start := 211533, stop := 211536 ), 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 := 211560, root := ~, start := 211546, stop := 211559 ), 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 := 211643, root := ~, start := 211632, stop := 211642 ), 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 := 212098, root := ~, start := 211649, stop := 212097 ), rec( content := "\n ", count := [ 5, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Description", next := 212115, root := ~, start := 211056, stop := 212114 ) ], count := [ 5, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 212129, root := ~, start := 210931, stop := 212128 ), 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 := 212204, root := ~, start := 212150, stop := 212203 ), 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 := 212250, root := ~, start := 212239, stop := 212249 ), rec( content := ".", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Returns", next := 212261, root := ~, start := 212207, stop := 212260 ), 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 := 212299, root := ~, start := 212285, stop := 212298 ), 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 := 212352, root := ~, start := 212327, stop := 212351 ), 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 := 212366, root := ~, start := 212352, stop := 212365 ), 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 := 212374, root := ~, start := 212366, stop := 212373 ), 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 := 212413, root := ~, start := 212405, stop := 212412 ), 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 := 212451, root := ~, start := 212420, stop := 212450 ), 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 := 212561, root := ~, start := 212547, stop := 212560 ), 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 := 212566, root := ~, start := 212562, stop := 212565 ), 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 := 212611, root := ~, start := 212597, stop := 212610 ), 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 := 212637, root := ~, start := 212622, stop := 212636 ), 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 := 212675, root := ~, start := 212661, stop := 212674 ), 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 := 212723, root := ~, start := 212715, stop := 212722 ), 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 := 212766, root := ~, start := 212748, stop := 212765 ), 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 := 212791, root := ~, start := 212783, stop := 212790 ), 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 := 212837, root := ~, start := 212833, stop := 212836 ), 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 := 212871, root := ~, start := 212846, stop := 212870 ), 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 := 212885, root := ~, start := 212871, stop := 212884 ), 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 := 212893, root := ~, start := 212885, stop := 212892 ), 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 := 212913, root := ~, start := 212902, stop := 212912 ), 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 := 212958, root := ~, start := 212947, stop := 212957 ), 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 := 213513, root := ~, start := 212964, stop := 213512 ), rec( content := "\n ", count := [ 5, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Description", next := 213530, root := ~, start := 212264, stop := 213529 ) ], count := [ 5, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 213544, root := ~, start := 212135, stop := 213543 ), 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 := 213609, root := ~, start := 213565, stop := 213608 ), 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 := 213642, root := ~, start := 213631, stop := 213641 ), rec( content := ".", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Returns", next := 213653, root := ~, start := 213612, stop := 213652 ), 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 := 213727, root := ~, start := 213716, stop := 213726 ), 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 := 213739, root := ~, start := 213735, stop := 213738 ), 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 := 213770, root := ~, start := 213747, stop := 213769 ), 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 := 213797, root := ~, start := 213789, stop := 213796 ), 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 := 213847, root := ~, start := 213839, stop := 213846 ), 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 := 213866, root := ~, start := 213852, stop := 213865 ), 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 := 214022, root := ~, start := 213984, stop := 214021 ), 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 := 214046, root := ~, start := 214032, stop := 214045 ), 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 := 214099, root := ~, start := 214077, stop := 214098 ), 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 := 214117, root := ~, start := 214108, stop := 214116 ), 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 := 214135, root := ~, start := 214125, stop := 214134 ), 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 := 214154, root := ~, start := 214150, stop := 214153 ), 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 := 214269, root := ~, start := 214236, stop := 214268 ), 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 := 214877, root := ~, start := 214276, stop := 214876 ), rec( content := "\n ", count := [ 5, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Description", next := 214894, root := ~, start := 213656, stop := 214893 ) ], count := [ 5, 3, 28, 1 ], mathmode := "Text", name := "ManSection", next := 214908, root := ~, start := 213550, stop := 214907 ), 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 := 214973, root := ~, start := 214929, stop := 214972 ), 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 := 215006, root := ~, start := 214976, stop := 215005 ), 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 := 215107, root := ~, start := 215093, stop := 215106 ), 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 := 215135, root := ~, start := 215113, stop := 215134 ), 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 := 215174, root := ~, start := 215160, stop := 215173 ), 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 := 215211, root := ~, start := 215207, stop := 215210 ), 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 := 215254, root := ~, start := 215222, stop := 215253 ), 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 := 215302, root := ~, start := 215263, stop := 215301 ), 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 := 215728, root := ~, start := 215331, stop := 215727 ), rec( content := "\n ", count := [ 5, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "Description", next := 215745, root := ~, start := 215009, stop := 215744 ) ], count := [ 5, 3, 29, 1 ], mathmode := "Text", name := "ManSection", next := 215759, root := ~, start := 214914, stop := 215758 ), 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 := 215872, root := ~, start := 215780, stop := 215871 ), rec( attributes := rec( Arg := "digraph, source" , Label := "for a source", Name := "DigraphDijkstra\ " ), content := 0, count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "Oper", next := 215948, root := ~, start := 215875, stop := 215947 ), 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 := 215980, root := ~, start := 215951, stop := 215979 ), 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 := 216016, root := ~, start := 216002, stop := 216015 ), 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 := 216047, root := ~, start := 216034, stop := 216046 ), 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 := 216065, root := ~, start := 216052, stop := 216064 ), 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 := 216099, root := ~, start := 216085, stop := 216098 ), 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 := 216128, root := ~, start := 216106, stop := 216127 ), 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 := 216192, root := ~, start := 216179, stop := 216191 ), 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 := 216209, root := ~, start := 216196, stop := 216208 ), 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 := 216331, root := ~, start := 216318, stop := 216330 ), 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 := 216432, root := ~, start := 216419, stop := 216431 ), 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 := 216499, root := ~, start := 216486, stop := 216498 ), 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 := 216553, root := ~, start := 216538, stop := 216552 ), 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 := 216657, root := ~, start := 216644, stop := 216656 ), 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 := 216707, root := ~, start := 216694, stop := 216706 ), 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 := 216779, root := ~, start := 216770, stop := 216778 ), 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 := 216784, root := ~, start := 216780, stop := 216783 ), 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 := 216835, root := ~, start := 216822, stop := 216834 ), 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 := 216883, root := ~, start := 216861, stop := 216882 ), 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 := 216928, root := ~, start := 216915, stop := 216927 ), 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 := 216985, root := ~, start := 216972, stop := 216984 ), 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 := 217388, root := ~, start := 216990, stop := 217387 ), rec( content := "\n", count := [ 5, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "Description", next := 217403, root := ~, start := 215983, stop := 217402 ) ], count := [ 5, 3, 30, 1 ], mathmode := "Text", name := "ManSection", next := 217417, root := ~, start := 215765, stop := 217416 ), rec( content := "\n\n ", count := [ 5, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 217431, root := ~, start := 172800, stop := 217430 ) , 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 := 217486, root := ~, start := 217444, stop := 217485 ), 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 := 217555, root := ~, start := 217506, stop := 217554 ), rec( attributes := rec( Arg := "digraph", Name := "SemigroupOfCayl\ eyDigraph" ), content := 0, count := [ 5, 4, 1, 4 ], mathmode := "Text", name := "Attr", next := 217611, root := ~, start := 217558, stop := 217610 ), 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 := 217654, root := ~, start := 217614, stop := 217653 ), 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 := 217692, root := ~, start := 217678, stop := 217691 ), 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 := 217741, root := ~, start := 217733, stop := 217740 ), 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 := 217764, root := ~, start := 217750, stop := 217763 ), 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 := 217818, root := ~, start := 217789, stop := 217817 ), 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 := 217856, root := ~, start := 217829, stop := 217855 ), 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 := 217873, root := ~, start := 217865, stop := 217872 ), 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 := 217883, root := ~, start := 217879, stop := 217882 ), 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 := 217906, root := ~, start := 217892, stop := 217905 ), 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 := 217948, root := ~, start := 217940, stop := 217947 ), 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 := 217971, root := ~, start := 217957, stop := 217970 ), 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 := 218025, root := ~, start := 217996, stop := 218024 ), 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 := 218067, root := ~, start := 218036, stop := 218066 ), 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 := 218084, root := ~, start := 218076, stop := 218083 ), 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 := 218094, root := ~, start := 218090, stop := 218093 ), 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 := 218148, root := ~, start := 218109, stop := 218147 ), 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 := 218380, root := ~, start := 218154, stop := 218379 ), rec( content := "\n ", count := [ 5, 4, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Description", next := 218397, root := ~, start := 217657, stop := 218396 ) ], count := [ 5, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 218411, root := ~, start := 217491, stop := 218410 ), 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 := 218486, root := ~, start := 218432, stop := 218485 ), 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 := 218529, root := ~, start := 218489, stop := 218528 ), 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 := 218567, root := ~, start := 218553, stop := 218566 ), 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 := 218672, root := ~, start := 218661, stop := 218671 ), 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 := 218691, root := ~, start := 218677, stop := 218690 ), 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 := 218749, root := ~, start := 218720, stop := 218748 ), 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 := 218792, root := ~, start := 218760, stop := 218791 ), 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 := 218834, root := ~, start := 218823, stop := 218833 ), 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 := 218864, root := ~, start := 218850, stop := 218863 ), 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 := 218885, root := ~, start := 218881, stop := 218884 ), 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 := 218934, root := ~, start := 218900, stop := 218933 ), 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 := 218981, root := ~, start := 218943, stop := 218980 ), 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 := 219374, root := ~, start := 218987, stop := 219373 ), rec( content := "\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 219391, root := ~, start := 218532, stop := 219390 ) ], count := [ 5, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 219405, root := ~, start := 218417, stop := 219404 ), rec( content := "\n\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 219419, root := ~, start := 217435, stop := 219418 ) , 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 := 219472, root := ~, start := 219432, stop := 219471 ), 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 := 219583, root := ~, start := 219496, stop := 219582 ), rec( attributes := rec( Arg := "filt, digraph", Name := "AsMonoid" ), content := 0, count := [ 5, 5, 1, 4 ], mathmode := "Text", name := "Oper", next := 219631, root := ~, start := 219588, stop := 219630 ), 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 := 219686, root := ~, start := 219636, stop := 219685 ), 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 := 219743, root := ~, start := 219725, stop := 219742 ), 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 := 219769, root := ~, start := 219758, stop := 219768 ), 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 := 219839, root := ~, start := 219788, stop := 219838 ), 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 := 219864, root := ~, start := 219850, stop := 219863 ), 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 := 219914, root := ~, start := 219876, stop := 219913 ), 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 := 219948, root := ~, start := 219918, stop := 219947 ), 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 := 219978, root := ~, start := 219960, stop := 219977 ), 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 := 220114, root := ~, start := 220100, stop := 220113 ), 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 := 220194, root := ~, start := 220147, stop := 220193 ), 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 := 220219, root := ~, start := 220205, stop := 220218 ), 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 := 220275, root := ~, start := 220237, stop := 220274 ), 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 := 220328, root := ~, start := 220290, stop := 220327 ), 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 := 220358, root := ~, start := 220340, stop := 220357 ), 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 := 220494, root := ~, start := 220480, stop := 220493 ), 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 := 220574, root := ~, start := 220527, stop := 220573 ), 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 := 220586, root := ~, start := 220582, stop := 220585 ), 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 := 220623, root := ~, start := 220608, stop := 220622 ), 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 := 220663, root := ~, start := 220645, stop := 220662 ), 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 := 220693, root := ~, start := 220682, stop := 220692 ), 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 := 220769, root := ~, start := 220721, stop := 220768 ), 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 := 220791, root := ~, start := 220777, stop := 220790 ), 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 := 220835, root := ~, start := 220805, stop := 220834 ), 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 := 220905, root := ~, start := 220868, stop := 220904 ), 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 := 220911, root := ~, start := 220907, stop := 220910 ), 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 := 221064, root := ~, start := 221007, stop := 221063 ), 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 := 221139, root := ~, start := 221089, stop := 221138 ), 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 := 221771, root := ~, start := 221229, stop := 221770 ), rec( content := "\n ", count := [ 5, 5, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Description", next := 221790, root := ~, start := 219691, stop := 221789 ) ], count := [ 5, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 221806, root := ~, start := 219479, stop := 221805 ), 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 := 221945, root := ~, start := 221831, stop := 221944 ), 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 := 222009, root := ~, start := 221950, stop := 222008 ), 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 := 222066, root := ~, start := 222048, stop := 222065 ), 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 := 222092, root := ~, start := 222081, stop := 222091 ), 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 := 222162, root := ~, start := 222111, stop := 222161 ), 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 := 222181, root := ~, start := 222173, stop := 222180 ), 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 := 222225, root := ~, start := 222187, stop := 222224 ), 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 := 222273, root := ~, start := 222235, stop := 222272 ), 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 := 222291, root := ~, start := 222281, stop := 222290 ), 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 := 222363, root := ~, start := 222352, stop := 222362 ), 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 := 222412, root := ~, start := 222399, stop := 222411 ), 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 := 222464, root := ~, start := 222450, stop := 222463 ), 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 := 222492, root := ~, start := 222474, stop := 222491 ), 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 := 222515, root := ~, start := 222505, stop := 222514 ), 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 := 222558, root := ~, start := 222545, stop := 222557 ), 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 := 222581, root := ~, start := 222568, stop := 222580 ), 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 := 222655, root := ~, start := 222637, stop := 222654 ), 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 := 222777, root := ~, start := 222755, stop := 222776 ), 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 := 223354, root := ~, start := 222785, stop := 223353 ), rec( content := "\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 223373, root := ~, start := 222014, stop := 223372 ) ], count := [ 5, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 223389, root := ~, start := 221814, stop := 223388 ), rec( content := "\n\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 223403, root := ~, start := 219423, stop := 223402 ) , 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 := 223444, root := ~, start := 223416, stop := 223443 ), 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 := 223519, root := ~, start := 223464, stop := 223518 ), 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 := 223552, root := ~, start := 223541, stop := 223551 ), rec( content := ".", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Returns", next := 223563, root := ~, start := 223522, stop := 223562 ), 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 := 223617, root := ~, start := 223584, stop := 223616 ), 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 := 223751, root := ~, start := 223737, stop := 223750 ), 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 := 223795, root := ~, start := 223781, stop := 223794 ), 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 := 223829, root := ~, start := 223818, stop := 223828 ), 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 := 223847, root := ~, start := 223833, stop := 223846 ), 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 := 223913, root := ~, start := 223880, stop := 223912 ), 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 := 223978, root := ~, start := 223964, stop := 223977 ), 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 := 224039, root := ~, start := 224031, stop := 224038 ), 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 := 224123, root := ~, start := 224115, stop := 224122 ), 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 := 224136, root := ~, start := 224128, stop := 224135 ), 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 := 224142, root := ~, start := 224138, stop := 224141 ), 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 := 224216, root := ~, start := 224202, stop := 224215 ), 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 := 224279, root := ~, start := 224265, stop := 224278 ), 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 := 224295, root := ~, start := 224291, stop := 224294 ), 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 := 224339, root := ~, start := 224310, stop := 224338 ), 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 := 224389, root := ~, start := 224348, stop := 224388 ), 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 := 224399, root := ~, start := 224395, stop := 224398 ), 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 := 224561, root := ~, start := 224543, stop := 224560 ), 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 := 225863, root := ~, start := 224564, stop := 225862 ), rec( content := "\n ", count := [ 5, 6, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Description", next := 225880, root := ~, start := 223566, stop := 225879 ) ], count := [ 5, 6, 1, 1 ], mathmode := "Text", name := "ManSection", next := 225894, root := ~, start := 223449, stop := 225893 ), 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 := 225975, root := ~, start := 225915, stop := 225974 ), 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 := 226008, root := ~, start := 225997, stop := 226007 ), rec( content := ".", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Returns", next := 226019, root := ~, start := 225978, stop := 226018 ), 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 := 226078, root := ~, start := 226040, stop := 226077 ), 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 := 226222, root := ~, start := 226208, stop := 226221 ), 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 := 226270, root := ~, start := 226256, stop := 226269 ), 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 := 226306, root := ~, start := 226295, stop := 226305 ), 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 := 226328, root := ~, start := 226314, stop := 226327 ), 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 := 226405, root := ~, start := 226367, stop := 226404 ), 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 := 226470, root := ~, start := 226456, stop := 226469 ), 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 := 226531, root := ~, start := 226523, stop := 226530 ), 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 := 226615, root := ~, start := 226607, stop := 226614 ), 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 := 226628, root := ~, start := 226620, stop := 226627 ), 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 := 226634, root := ~, start := 226630, stop := 226633 ), 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 := 226708, root := ~, start := 226694, stop := 226707 ), 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 := 226771, root := ~, start := 226757, stop := 226770 ), 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 := 226798, root := ~, start := 226794, stop := 226797 ), 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 := 226852, root := ~, start := 226818, stop := 226851 ), 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 := 226898, root := ~, start := 226858, stop := 226897 ), 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 := 226950, root := ~, start := 226909, stop := 226949 ), 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 := 226960, root := ~, start := 226956, stop := 226959 ), 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 := 227118, root := ~, start := 227100, stop := 227117 ), 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 := 228611, root := ~, start := 227121, stop := 228610 ), rec( content := "\n ", count := [ 5, 6, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Description", next := 228628, root := ~, start := 226022, stop := 228627 ) ], count := [ 5, 6, 2, 1 ], mathmode := "Text", name := "ManSection", next := 228642, root := ~, start := 225900, stop := 228641 ), 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 := 228707, root := ~, start := 228663, stop := 228706 ), 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 := 228740, root := ~, start := 228729, stop := 228739 ), rec( content := ".", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Returns", next := 228751, root := ~, start := 228710, stop := 228750 ), 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 := 228789, root := ~, start := 228775, stop := 228788 ), 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 := 228843, root := ~, start := 228821, stop := 228842 ), 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 := 228935, root := ~, start := 228921, stop := 228934 ), 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 := 229022, root := ~, start := 229008, stop := 229021 ), 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 := 229058, root := ~, start := 229047, stop := 229057 ), 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 := 229076, root := ~, start := 229072, stop := 229075 ), 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 := 229150, root := ~, start := 229136, stop := 229149 ), 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 := 229192, root := ~, start := 229170, stop := 229191 ), 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 := 229202, root := ~, start := 229198, stop := 229201 ), 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 := 229251, root := ~, start := 229222, stop := 229250 ), 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 := 229261, root := ~, start := 229257, stop := 229260 ), 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 := 229419, root := ~, start := 229401, stop := 229418 ), 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 := 230637, root := ~, start := 229422, stop := 230636 ), rec( content := "\n ", count := [ 5, 6, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Description", next := 230654, root := ~, start := 228754, stop := 230653 ) ], count := [ 5, 6, 3, 1 ], mathmode := "Text", name := "ManSection", next := 230668, root := ~, start := 228648, stop := 230667 ), 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 := 230738, root := ~, start := 230689, stop := 230737 ), 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 := 230771, root := ~, start := 230760, stop := 230770 ), rec( content := ".", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Returns", next := 230782, root := ~, start := 230741, stop := 230781 ), 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 := 230820, root := ~, start := 230806, stop := 230819 ), 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 := 230885, root := ~, start := 230858, stop := 230884 ), 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 := 230977, root := ~, start := 230963, stop := 230976 ), 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 := 231064, root := ~, start := 231050, stop := 231063 ), 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 := 231106, root := ~, start := 231095, stop := 231105 ), 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 := 231124, root := ~, start := 231120, stop := 231123 ), 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 := 231198, root := ~, start := 231184, stop := 231197 ), 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 := 231245, root := ~, start := 231218, stop := 231244 ), 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 := 231255, root := ~, start := 231251, stop := 231254 ), 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 := 231304, root := ~, start := 231270, stop := 231303 ), 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 := 231314, root := ~, start := 231310, stop := 231313 ), 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 := 231472, root := ~, start := 231454, stop := 231471 ), 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 := 232710, root := ~, start := 231475, stop := 232709 ), rec( content := "\n ", count := [ 5, 6, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Description", next := 232727, root := ~, start := 230785, stop := 232726 ) ], count := [ 5, 6, 4, 1 ], mathmode := "Text", name := "ManSection", next := 232741, root := ~, start := 230674, stop := 232740 ), 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 := 232818, root := ~, start := 232762, stop := 232817 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK33" ), content := 0, count := [ 5, 6, 5, 4 ], mathmode := "Text", name := "Attr", next := 232877, root := ~, start := 232821, stop := 232876 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK4" ), content := 0, count := [ 5, 6, 5, 6 ], mathmode := "Text", name := "Attr", next := 232935, root := ~, start := 232880, stop := 232934 ), 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 := 232968, root := ~, start := 232957, stop := 232967 ), rec( content := ".", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Returns", next := 232979, root := ~, start := 232938, stop := 232978 ), 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 := 233119, root := ~, start := 233105, stop := 233118 ), 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 := 233238, root := ~, start := 233230, stop := 233237 ), 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 := 233255, root := ~, start := 233247, stop := 233254 ), 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 := 233320, root := ~, start := 233312, stop := 233319 ), 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 := 233337, root := ~, start := 233329, stop := 233336 ), 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 := 233374, root := ~, start := 233366, stop := 233373 ), 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 := 233406, root := ~, start := 233392, stop := 233405 ), 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 := 233448, root := ~, start := 233437, stop := 233447 ), 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 := 233471, root := ~, start := 233467, stop := 233470 ), 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 := 233515, root := ~, start := 233486, stop := 233514 ), 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 := 233560, root := ~, start := 233520, stop := 233559 ), 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 := 233582, root := ~, start := 233578, stop := 233581 ), 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 := 233740, root := ~, start := 233722, stop := 233739 ), 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 := 236052, root := ~, start := 233743, stop := 236051 ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Description", next := 236069, root := ~, start := 232982, stop := 236068 ) ], count := [ 5, 6, 5, 1 ], mathmode := "Text", name := "ManSection", next := 236083, root := ~, start := 232747, stop := 236082 ), rec( content := "\n\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 0, 1 ], mathmode := "Text", name := "Section", next := 236097, root := ~, start := 223407, stop := 236096 ) , rec( content := "\n\n", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 236109, root := ~, start := 132079, stop := 236108 ), 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 := 236226, root := ~, start := 236185, stop := 236225 ) , 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 := 236273, root := ~, start := 236239, stop := 236272 ), 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 := 236337, root := ~, start := 236293, stop := 236336 ), 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 := 236360, root := ~, start := 236349, stop := 236359 ), 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 := 236376, root := ~, start := 236364, stop := 236375 ), rec( content := ".", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 236387, root := ~, start := 236340, stop := 236386 ), 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 := 236427, root := ~, start := 236416, stop := 236426 ), 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 := 236457, root := ~, start := 236443, stop := 236456 ), 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 := 236489, root := ~, start := 236477, stop := 236488 ), 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 := 236561, root := ~, start := 236557, stop := 236560 ), 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 := 236603, root := ~, start := 236589, stop := 236602 ), 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 := 237198, root := ~, start := 236700, stop := 237197 ), rec( content := "\n ", count := [ 6, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 237215, root := ~, start := 236390, stop := 237214 ) ], count := [ 6, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 237229, root := ~, start := 236278, stop := 237228 ), 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 := 237301, root := ~, start := 237250, stop := 237300 ), rec( attributes := rec( Arg := "digraph", Name := "IsAntisymmetric\ Digraph" ), content := 0, count := [ 6, 1, 2, 4 ], mathmode := "Text", name := "Prop", next := 237355, root := ~, start := 237304, stop := 237354 ), 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 := 237378, root := ~, start := 237367, stop := 237377 ), 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 := 237394, root := ~, start := 237382, stop := 237393 ), rec( content := ".", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 237405, root := ~, start := 237358, stop := 237404 ), 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 := 237454, root := ~, start := 237443, stop := 237453 ), 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 := 237484, root := ~, start := 237470, stop := 237483 ), 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 := 237523, root := ~, start := 237511, stop := 237522 ), 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 := 237546, root := ~, start := 237542, stop := 237545 ), 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 := 237585, root := ~, start := 237565, stop := 237584 ), 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 := 237639, root := ~, start := 237631, stop := 237638 ), 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 := 237658, root := ~, start := 237650, stop := 237657 ), 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 := 237692, root := ~, start := 237684, stop := 237691 ), 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 := 237715, root := ~, start := 237707, stop := 237714 ), 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 := 237743, root := ~, start := 237735, stop := 237742 ), 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 := 237756, root := ~, start := 237748, stop := 237755 ), 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 := 237776, root := ~, start := 237772, stop := 237775 ), 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 := 237818, root := ~, start := 237804, stop := 237817 ), 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 := 238345, root := ~, start := 237919, stop := 238344 ), rec( content := "\n ", count := [ 6, 1, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 238362, root := ~, start := 237408, stop := 238361 ) ], count := [ 6, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 238376, root := ~, start := 237235, stop := 238375 ), 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 := 238444, root := ~, start := 238397, stop := 238443 ), 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 := 238467, root := ~, start := 238456, stop := 238466 ), 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 := 238483, root := ~, start := 238471, stop := 238482 ), rec( content := ".", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 238494, root := ~, start := 238447, stop := 238493 ), 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 := 238541, root := ~, start := 238530, stop := 238540 ), 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 := 238571, root := ~, start := 238557, stop := 238570 ), 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 := 238604, root := ~, start := 238592, stop := 238603 ), 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 := 238690, root := ~, start := 238676, stop := 238689 ), 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 := 238796, root := ~, start := 238782, stop := 238795 ), 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 := 238956, root := ~, start := 238898, stop := 238955 ), 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 := 238962, root := ~, start := 238958, stop := 238961 ), 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 := 239008, root := ~, start := 238975, stop := 239007 ), 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 := 239018, root := ~, start := 239014, stop := 239017 ), 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 := 239060, root := ~, start := 239046, stop := 239059 ), 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 := 239533, root := ~, start := 239160, stop := 239532 ), rec( content := "\n ", count := [ 6, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 239550, root := ~, start := 238497, stop := 239549 ) ], count := [ 6, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 239564, root := ~, start := 238382, stop := 239563 ), 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 := 239640, root := ~, start := 239585, stop := 239639 ), 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 := 239663, root := ~, start := 239652, stop := 239662 ), 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 := 239679, root := ~, start := 239667, stop := 239678 ), rec( content := ".", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 239690, root := ~, start := 239643, stop := 239689 ), 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 := 239730, root := ~, start := 239719, stop := 239729 ), 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 := 239760, root := ~, start := 239746, stop := 239759 ), 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 := 239814, root := ~, start := 239802, stop := 239813 ), 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 := 239833, root := ~, start := 239829, stop := 239832 ), 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 := 239887, root := ~, start := 239854, stop := 239886 ), 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 := 239948, root := ~, start := 239916, stop := 239947 ), 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 := 240005, root := ~, start := 239997, stop := 240004 ), 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 := 240024, root := ~, start := 240016, stop := 240023 ), 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 := 240048, root := ~, start := 240040, stop := 240047 ), 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 := 240061, root := ~, start := 240053, stop := 240060 ), 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 := 240113, root := ~, start := 240099, stop := 240112 ), 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 := 240158, root := ~, start := 240119, stop := 240157 ), 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 := 240164, root := ~, start := 240160, stop := 240163 ), 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 := 240238, root := ~, start := 240230, stop := 240237 ), 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 := 240255, root := ~, start := 240247, stop := 240254 ), 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 := 240300, root := ~, start := 240290, stop := 240299 ), 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 := 240350, root := ~, start := 240346, stop := 240349 ), 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 := 240403, root := ~, start := 240365, stop := 240402 ), 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 := 240413, root := ~, start := 240409, stop := 240412 ), 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 := 240455, root := ~, start := 240441, stop := 240454 ), 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 := 240986, root := ~, start := 240556, stop := 240985 ), rec( content := "\n ", count := [ 6, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 241003, root := ~, start := 239693, stop := 241002 ) ], count := [ 6, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 241017, root := ~, start := 239570, stop := 241016 ), 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 := 241084, root := ~, start := 241038, stop := 241083 ), 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 := 241107, root := ~, start := 241096, stop := 241106 ), 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 := 241123, root := ~, start := 241111, stop := 241122 ), rec( content := ".", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 241134, root := ~, start := 241087, stop := 241133 ), 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 := 241174, root := ~, start := 241163, stop := 241173 ), 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 := 241204, root := ~, start := 241190, stop := 241203 ), 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 := 241238, root := ~, start := 241226, stop := 241237 ), 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 := 241257, root := ~, start := 241253, stop := 241256 ), 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 := 241291, root := ~, start := 241276, stop := 241290 ), 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 := 241343, root := ~, start := 241328, stop := 241342 ), 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 := 241361, root := ~, start := 241353, stop := 241360 ), 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 := 241374, root := ~, start := 241366, stop := 241373 ), 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 := 241426, root := ~, start := 241418, stop := 241425 ), 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 := 241445, root := ~, start := 241437, stop := 241444 ), 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 := 241489, root := ~, start := 241481, stop := 241488 ), 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 := 241552, root := ~, start := 241537, stop := 241551 ), 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 := 241601, root := ~, start := 241597, stop := 241600 ), 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 := 241643, root := ~, start := 241629, stop := 241642 ), 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 := 242151, root := ~, start := 241744, stop := 242150 ), rec( content := "\n ", count := [ 6, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 242168, root := ~, start := 241137, stop := 242167 ) ], count := [ 6, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 242182, root := ~, start := 241023, stop := 242181 ), 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 := 242261, root := ~, start := 242203, stop := 242260 ), 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 := 242284, root := ~, start := 242273, stop := 242283 ), 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 := 242300, root := ~, start := 242288, stop := 242299 ), rec( content := ".", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 242311, root := ~, start := 242264, stop := 242310 ), 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 := 242365, root := ~, start := 242354, stop := 242364 ), 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 := 242383, root := ~, start := 242369, stop := 242382 ), 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 := 242440, root := ~, start := 242428, stop := 242439 ), 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 := 242453, root := ~, start := 242449, stop := 242452 ), 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 := 242510, root := ~, start := 242474, stop := 242509 ), 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 := 242715, root := ~, start := 242711, stop := 242714 ), 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 := 242757, root := ~, start := 242743, stop := 242756 ), 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 := 243200, root := ~, start := 242858, stop := 243199 ), rec( content := "\n ", count := [ 6, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 243217, root := ~, start := 242314, stop := 243216 ) ], count := [ 6, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 243231, root := ~, start := 242188, stop := 243230 ), 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 := 243295, root := ~, start := 243252, stop := 243294 ), rec( attributes := rec( Arg := "digraph", Name := "IsNullDigraph" ), content := 0, count := [ 6, 1, 7, 4 ], mathmode := "Text", name := "Prop", next := 243340, root := ~, start := 243298, stop := 243339 ), 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 := 243363, root := ~, start := 243352, stop := 243362 ), 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 := 243379, root := ~, start := 243367, stop := 243378 ), rec( content := ".", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Returns", next := 243390, root := ~, start := 243343, stop := 243389 ), 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 := 243430, root := ~, start := 243419, stop := 243429 ), 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 := 243460, root := ~, start := 243446, stop := 243459 ), 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 := 243491, root := ~, start := 243479, stop := 243490 ), 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 := 243531, root := ~, start := 243519, stop := 243530 ), 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 := 243559, root := ~, start := 243555, stop := 243558 ), 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 := 243585, root := ~, start := 243565, stop := 243584 ), 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 := 243624, root := ~, start := 243603, stop := 243623 ), 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 := 243634, root := ~, start := 243630, stop := 243633 ), 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 := 243676, root := ~, start := 243662, stop := 243675 ), 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 := 244069, root := ~, start := 243777, stop := 244068 ), rec( content := "\n ", count := [ 6, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Description", next := 244086, root := ~, start := 243393, stop := 244085 ) ], count := [ 6, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 244100, root := ~, start := 243237, stop := 244099 ), 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 := 244170, root := ~, start := 244121, stop := 244169 ), 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 := 244193, root := ~, start := 244182, stop := 244192 ), 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 := 244209, root := ~, start := 244197, stop := 244208 ), rec( content := ".", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 244220, root := ~, start := 244173, stop := 244219 ), 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 := 244358, root := ~, start := 244326, stop := 244357 ), 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 := 244396, root := ~, start := 244364, stop := 244395 ), 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 := 244434, root := ~, start := 244401, stop := 244433 ), 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 := 244470, root := ~, start := 244456, stop := 244469 ), 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 := 244519, root := ~, start := 244515, stop := 244518 ), 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 := 244561, root := ~, start := 244547, stop := 244560 ), 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 := 244812, root := ~, start := 244658, stop := 244811 ), rec( content := "\n ", count := [ 6, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 244829, root := ~, start := 244223, stop := 244828 ) ], count := [ 6, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 244843, root := ~, start := 244106, stop := 244842 ), 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 := 244912, root := ~, start := 244864, stop := 244911 ), 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 := 244935, root := ~, start := 244924, stop := 244934 ), 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 := 244951, root := ~, start := 244939, stop := 244950 ), rec( content := ".", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Returns", next := 244962, root := ~, start := 244915, stop := 244961 ), 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 := 245011, root := ~, start := 245000, stop := 245010 ), 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 := 245041, root := ~, start := 245027, stop := 245040 ), 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 := 245065, root := ~, start := 245061, stop := 245064 ), 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 := 245101, root := ~, start := 245084, stop := 245100 ), 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 := 245162, root := ~, start := 245158, stop := 245161 ), 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 := 245204, root := ~, start := 245190, stop := 245203 ), 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 := 245715, root := ~, start := 245304, stop := 245714 ), rec( content := "\n ", count := [ 6, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 245732, root := ~, start := 244965, stop := 245731 ) ], count := [ 6, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 245746, root := ~, start := 244849, stop := 245745 ), 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 := 245810, root := ~, start := 245767, stop := 245809 ), 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 := 245833, root := ~, start := 245822, stop := 245832 ), 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 := 245849, root := ~, start := 245837, stop := 245848 ), rec( content := ".", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Returns", next := 245860, root := ~, start := 245813, stop := 245859 ), 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 := 245902, root := ~, start := 245883, stop := 245901 ), 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 := 245974, root := ~, start := 245970, stop := 245973 ), 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 := 246010, root := ~, start := 245996, stop := 246009 ), 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 := 246734, root := ~, start := 246111, stop := 246733 ), rec( content := "\n ", count := [ 6, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 246751, root := ~, start := 245863, stop := 246750 ) ], count := [ 6, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 246765, root := ~, start := 245752, stop := 246764 ), 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 := 246833, root := ~, start := 246786, stop := 246832 ), 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 := 246856, root := ~, start := 246845, stop := 246855 ), 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 := 246872, root := ~, start := 246860, stop := 246871 ), rec( content := ".", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Returns", next := 246883, root := ~, start := 246836, stop := 246882 ), 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 := 246932, root := ~, start := 246921, stop := 246931 ), 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 := 246962, root := ~, start := 246948, stop := 246961 ), 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 := 246997, root := ~, start := 246985, stop := 246996 ), 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 := 247045, root := ~, start := 247029, stop := 247044 ), 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 := 247084, root := ~, start := 247080, stop := 247083 ), 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 := 247093, root := ~, start := 247089, stop := 247092 ), 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 := 247135, root := ~, start := 247121, stop := 247134 ), 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 := 247508, root := ~, start := 247235, stop := 247507 ), rec( content := "\n ", count := [ 6, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Description", next := 247525, root := ~, start := 246886, stop := 247524 ) ], count := [ 6, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 247539, root := ~, start := 246771, stop := 247538 ), 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 := 247607, root := ~, start := 247560, stop := 247606 ), 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 := 247630, root := ~, start := 247619, stop := 247629 ), 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 := 247646, root := ~, start := 247634, stop := 247645 ), rec( content := ".", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 247657, root := ~, start := 247610, stop := 247656 ), 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 := 247706, root := ~, start := 247695, stop := 247705 ), 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 := 247736, root := ~, start := 247722, stop := 247735 ), 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 := 247771, root := ~, start := 247759, stop := 247770 ), 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 := 247789, root := ~, start := 247785, stop := 247788 ), 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 := 247821, root := ~, start := 247797, stop := 247820 ), 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 := 247885, root := ~, start := 247877, stop := 247884 ), 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 := 247904, root := ~, start := 247896, stop := 247903 ), 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 := 247960, root := ~, start := 247952, stop := 247959 ), 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 := 247979, root := ~, start := 247971, stop := 247978 ), 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 := 248003, root := ~, start := 247995, stop := 248002 ), 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 := 248034, root := ~, start := 248026, stop := 248033 ), 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 := 248053, root := ~, start := 248045, stop := 248052 ), 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 := 248092, root := ~, start := 248084, stop := 248091 ), 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 := 248123, root := ~, start := 248115, stop := 248122 ), 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 := 248142, root := ~, start := 248134, stop := 248141 ), 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 := 248247, root := ~, start := 248218, stop := 248246 ), 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 := 248257, root := ~, start := 248253, stop := 248256 ), 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 := 248299, root := ~, start := 248285, stop := 248298 ), 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 := 249019, root := ~, start := 248399, stop := 249018 ), rec( content := "\n ", count := [ 6, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 249036, root := ~, start := 247660, stop := 249035 ) ], count := [ 6, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 249050, root := ~, start := 247545, stop := 249049 ), 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 := 249112, root := ~, start := 249071, stop := 249111 ), 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 := 249135, root := ~, start := 249124, stop := 249134 ), 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 := 249151, root := ~, start := 249139, stop := 249150 ), rec( content := ".", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Returns", next := 249162, root := ~, start := 249115, stop := 249161 ), 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 := 249211, root := ~, start := 249200, stop := 249210 ), 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 := 249241, root := ~, start := 249227, stop := 249240 ), 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 := 249279, root := ~, start := 249267, stop := 249278 ), 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 := 249298, root := ~, start := 249294, stop := 249297 ), 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 := 249533, root := ~, start := 249529, stop := 249532 ), 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 := 249575, root := ~, start := 249561, stop := 249574 ), 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 := 250156, root := ~, start := 249675, stop := 250155 ), rec( content := "\n ", count := [ 6, 1, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 250173, root := ~, start := 249165, stop := 250172 ) ], count := [ 6, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 250187, root := ~, start := 249056, stop := 250186 ), 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 := 250256, root := ~, start := 250208, stop := 250255 ), 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 := 250279, root := ~, start := 250268, stop := 250278 ), 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 := 250295, root := ~, start := 250283, stop := 250294 ), rec( content := ".", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Returns", next := 250306, root := ~, start := 250259, stop := 250305 ), 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 := 250355, root := ~, start := 250344, stop := 250354 ), 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 := 250385, root := ~, start := 250371, stop := 250384 ), 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 := 250421, root := ~, start := 250409, stop := 250420 ), 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 := 250471, root := ~, start := 250454, stop := 250470 ), 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 := 250499, root := ~, start := 250484, stop := 250498 ), 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 := 250523, root := ~, start := 250508, stop := 250522 ), 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 := 250570, root := ~, start := 250555, stop := 250569 ), 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 := 250611, root := ~, start := 250607, stop := 250610 ), 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 := 250629, root := ~, start := 250621, stop := 250628 ), 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 := 250705, root := ~, start := 250697, stop := 250704 ), 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 := 250875, root := ~, start := 250862, stop := 250874 ), 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 := 250977, root := ~, start := 250941, stop := 250976 ), 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 := 251040, root := ~, start := 251013, stop := 251039 ), 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 := 251080, root := ~, start := 251076, stop := 251079 ), 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 := 251122, root := ~, start := 251108, stop := 251121 ), 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 := 251226, root := ~, start := 251222, stop := 251225 ), 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 := 251688, root := ~, start := 251231, stop := 251687 ), rec( content := "\n ", count := [ 6, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Description", next := 251705, root := ~, start := 250309, stop := 251704 ) ], count := [ 6, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 251719, root := ~, start := 250193, stop := 251718 ), 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 := 251786, root := ~, start := 251740, stop := 251785 ), rec( attributes := rec( Arg := "digraph", Name := "IsQuasiorderDig\ raph" ), content := 0, count := [ 6, 1, 15, 4 ], mathmode := "Text", name := "Prop", next := 251837, root := ~, start := 251789, stop := 251836 ), 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 := 251860, root := ~, start := 251849, stop := 251859 ), 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 := 251876, root := ~, start := 251864, stop := 251875 ), rec( content := ".", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Returns", next := 251887, root := ~, start := 251840, stop := 251886 ), 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 := 252019, root := ~, start := 251987, stop := 252018 ), 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 := 252057, root := ~, start := 252024, stop := 252056 ), 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 := 252120, root := ~, start := 252106, stop := 252119 ), 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 := 252173, root := ~, start := 252162, stop := 252172 ), 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 := 252200, root := ~, start := 252185, stop := 252199 ), 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 := 252233, root := ~, start := 252220, stop := 252232 ), 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 := 252262, root := ~, start := 252248, stop := 252261 ), 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 := 252272, root := ~, start := 252268, stop := 252271 ), 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 := 252314, root := ~, start := 252300, stop := 252313 ), 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 := 252947, root := ~, start := 252411, stop := 252946 ), rec( content := "\n ", count := [ 6, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Description", next := 252964, root := ~, start := 251890, stop := 252963 ) ], count := [ 6, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 252978, root := ~, start := 251725, stop := 252977 ), 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 := 253049, root := ~, start := 252999, stop := 253048 ), 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 := 253072, root := ~, start := 253061, stop := 253071 ), 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 := 253088, root := ~, start := 253076, stop := 253087 ), rec( content := ".", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Returns", next := 253099, root := ~, start := 253052, stop := 253098 ), 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 := 253238, root := ~, start := 253206, stop := 253237 ), 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 := 253280, root := ~, start := 253244, stop := 253279 ), 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 := 253318, root := ~, start := 253285, stop := 253317 ), 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 := 253354, root := ~, start := 253340, stop := 253353 ), 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 := 253412, root := ~, start := 253401, stop := 253411 ), 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 := 253439, root := ~, start := 253424, stop := 253438 ), 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 := 253472, root := ~, start := 253459, stop := 253471 ), 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 := 253501, root := ~, start := 253487, stop := 253500 ), 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 := 253511, root := ~, start := 253507, stop := 253510 ), 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 := 253553, root := ~, start := 253539, stop := 253552 ), 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 := 254057, root := ~, start := 253650, stop := 254056 ), rec( content := "\n ", count := [ 6, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Description", next := 254074, root := ~, start := 253102, stop := 254073 ) ], count := [ 6, 1, 16, 1 ], mathmode := "Text", name := "ManSection", next := 254088, root := ~, start := 252984, stop := 254087 ), 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 := 254162, root := ~, start := 254109, stop := 254161 ), rec( attributes := rec( Arg := "digraph", Name := "IsJoinSemilatti\ ceDigraph" ), content := 0, count := [ 6, 1, 17, 4 ], mathmode := "Text", name := "Prop", next := 254218, root := ~, start := 254165, stop := 254217 ), rec( attributes := rec( Arg := "digraph", Name := "IsLatticeDigrap\ h" ), content := 0, count := [ 6, 1, 17, 6 ], mathmode := "Text", name := "Prop", next := 254266, root := ~, start := 254221, stop := 254265 ), 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 := 254289, root := ~, start := 254278, stop := 254288 ), 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 := 254305, root := ~, start := 254293, stop := 254304 ), rec( content := ".", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Returns", next := 254316, root := ~, start := 254269, stop := 254315 ), 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 := 254368, root := ~, start := 254337, stop := 254367 ), 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 := 254388, root := ~, start := 254377, stop := 254387 ), 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 := 254422, root := ~, start := 254408, stop := 254421 ), 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 := 254477, root := ~, start := 254446, stop := 254476 ), 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 := 254501, root := ~, start := 254490, stop := 254500 ), 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 := 254531, root := ~, start := 254517, stop := 254530 ), 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 := 254587, root := ~, start := 254564, stop := 254586 ), 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 := 254607, root := ~, start := 254596, stop := 254606 ), 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 := 254641, root := ~, start := 254627, stop := 254640 ), 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 := 254689, root := ~, start := 254685, stop := 254688 ), 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 := 254758, root := ~, start := 254723, stop := 254757 ), 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 := 254822, root := ~, start := 254811, stop := 254821 ), 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 := 254854, root := ~, start := 254839, stop := 254853 ), 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 := 254883, root := ~, start := 254870, stop := 254882 ), 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 := 254938, root := ~, start := 254915, stop := 254937 ), 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 := 255103, root := ~, start := 255080, stop := 255102 ), 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 := 255229, root := ~, start := 255225, stop := 255228 ), 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 := 255271, root := ~, start := 255257, stop := 255270 ), 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 := 256259, root := ~, start := 255372, stop := 256258 ), rec( content := "\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Description", next := 256276, root := ~, start := 254319, stop := 256275 ) ], count := [ 6, 1, 17, 1 ], mathmode := "Text", name := "ManSection", next := 256290, root := ~, start := 254094, stop := 256289 ), rec( content := "\n\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 256304, root := ~, start := 236230, stop := 256303 ) , 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 := 256346, root := ~, start := 256317, stop := 256345 ), 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 := 256413, root := ~, start := 256366, stop := 256412 ), 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 := 256436, root := ~, start := 256425, stop := 256435 ), 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 := 256452, root := ~, start := 256440, stop := 256451 ), rec( content := ".", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Returns", next := 256463, root := ~, start := 256416, stop := 256462 ), 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 := 256512, root := ~, start := 256501, stop := 256511 ), 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 := 256544, root := ~, start := 256536, stop := 256543 ), 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 := 256584, root := ~, start := 256576, stop := 256583 ), 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 := 256610, root := ~, start := 256596, stop := 256609 ), 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 := 256641, root := ~, start := 256633, stop := 256640 ), 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 := 256671, root := ~, start := 256663, stop := 256670 ), 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 := 256720, root := ~, start := 256687, stop := 256719 ), 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 := 256759, root := ~, start := 256729, stop := 256758 ), 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 := 256769, root := ~, start := 256765, stop := 256768 ), 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 := 256811, root := ~, start := 256797, stop := 256810 ), 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 := 257041, root := ~, start := 256911, stop := 257040 ), rec( content := "\n ", count := [ 6, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 257058, root := ~, start := 256466, stop := 257057 ) ], count := [ 6, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 257072, root := ~, start := 256351, stop := 257071 ), 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 := 257141, root := ~, start := 257093, stop := 257140 ), 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 := 257164, root := ~, start := 257153, stop := 257163 ), 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 := 257180, root := ~, start := 257168, stop := 257179 ), rec( content := ".", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 257191, root := ~, start := 257144, stop := 257190 ), 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 := 257240, root := ~, start := 257229, stop := 257239 ), 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 := 257272, root := ~, start := 257264, stop := 257271 ), 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 := 257312, root := ~, start := 257304, stop := 257311 ), 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 := 257338, root := ~, start := 257324, stop := 257337 ), 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 := 257369, root := ~, start := 257361, stop := 257368 ), 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 := 257396, root := ~, start := 257388, stop := 257395 ), 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 := 257406, root := ~, start := 257402, stop := 257405 ), 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 := 257453, root := ~, start := 257421, stop := 257452 ), 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 := 257492, root := ~, start := 257462, stop := 257491 ), 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 := 257502, root := ~, start := 257498, stop := 257501 ), 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 := 257544, root := ~, start := 257530, stop := 257543 ), 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 := 257776, root := ~, start := 257644, stop := 257775 ), rec( content := "\n ", count := [ 6, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 257793, root := ~, start := 257194, stop := 257792 ) ], count := [ 6, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 257807, root := ~, start := 257078, stop := 257806 ), 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 := 257873, root := ~, start := 257828, stop := 257872 ), 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 := 257896, root := ~, start := 257885, stop := 257895 ), 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 := 257912, root := ~, start := 257900, stop := 257911 ), rec( content := ".", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 257923, root := ~, start := 257876, stop := 257922 ), 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 := 257972, root := ~, start := 257961, stop := 257971 ), 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 := 258004, root := ~, start := 257996, stop := 258003 ), 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 := 258044, root := ~, start := 258036, stop := 258043 ), 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 := 258070, root := ~, start := 258056, stop := 258069 ), 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 := 258101, root := ~, start := 258093, stop := 258100 ), 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 := 258144, root := ~, start := 258136, stop := 258143 ), 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 := 258193, root := ~, start := 258182, stop := 258192 ), 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 := 258211, root := ~, start := 258197, stop := 258210 ), 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 := 258302, root := ~, start := 258270, stop := 258301 ), 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 := 258344, root := ~, start := 258311, stop := 258343 ), 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 := 258354, root := ~, start := 258350, stop := 258353 ), 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 := 258396, root := ~, start := 258382, stop := 258395 ), 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 := 258622, root := ~, start := 258496, stop := 258621 ), rec( content := "\n ", count := [ 6, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 258639, root := ~, start := 257926, stop := 258638 ) ], count := [ 6, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 258653, root := ~, start := 257813, stop := 258652 ), 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 := 258727, root := ~, start := 258674, stop := 258726 ), 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 := 258750, root := ~, start := 258739, stop := 258749 ), 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 := 258766, root := ~, start := 258754, stop := 258765 ), rec( content := ".", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 258777, root := ~, start := 258730, stop := 258776 ), 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 := 258815, root := ~, start := 258801, stop := 258814 ), 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 := 258885, root := ~, start := 258874, stop := 258884 ), 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 := 258918, root := ~, start := 258910, stop := 258917 ), 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 := 258931, root := ~, start := 258923, stop := 258930 ), 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 := 258949, root := ~, start := 258935, stop := 258948 ), 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 := 258983, root := ~, start := 258975, stop := 258982 ), 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 := 258996, root := ~, start := 258988, stop := 258995 ), 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 := 259013, root := ~, start := 259005, stop := 259012 ), 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 := 259052, root := ~, start := 259038, stop := 259051 ), 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 := 259097, root := ~, start := 259089, stop := 259096 ), 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 := 259115, root := ~, start := 259107, stop := 259114 ), 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 := 259137, root := ~, start := 259129, stop := 259136 ), 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 := 259151, root := ~, start := 259143, stop := 259150 ), 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 := 259180, root := ~, start := 259172, stop := 259179 ), 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 := 259190, root := ~, start := 259182, stop := 259189 ), 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 := 259234, root := ~, start := 259226, stop := 259233 ), 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 := 259251, root := ~, start := 259243, stop := 259250 ), 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 := 259256, root := ~, start := 259252, stop := 259255 ), 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 := 259358, root := ~, start := 259348, stop := 259357 ), 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 := 259370, root := ~, start := 259360, stop := 259369 ), 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 := 259384, root := ~, start := 259376, stop := 259383 ), 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 := 259428, root := ~, start := 259420, stop := 259427 ), 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 := 259438, root := ~, start := 259430, stop := 259437 ), 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 := 259456, root := ~, start := 259442, stop := 259455 ), 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 := 259484, root := ~, start := 259476, stop := 259483 ), 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 := 259524, root := ~, start := 259514, stop := 259523 ), 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 := 259546, root := ~, start := 259538, stop := 259545 ), 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 := 259585, root := ~, start := 259573, stop := 259584 ), 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 := 259604, root := ~, start := 259596, stop := 259603 ), 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 := 259620, root := ~, start := 259610, stop := 259619 ), 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 := 259646, root := ~, start := 259638, stop := 259645 ), 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 := 259681, root := ~, start := 259669, stop := 259680 ), 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 := 259700, root := ~, start := 259692, stop := 259699 ), 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 := 259761, root := ~, start := 259747, stop := 259760 ), 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 := 259786, root := ~, start := 259782, stop := 259785 ), 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 := 259824, root := ~, start := 259810, stop := 259823 ), 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 := 259892, root := ~, start := 259880, stop := 259891 ), 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 := 259902, root := ~, start := 259898, stop := 259901 ), 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 := 259944, root := ~, start := 259930, stop := 259943 ), 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 := 260320, root := ~, start := 260045, stop := 260319 ), rec( content := "\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 260337, root := ~, start := 258780, stop := 260336 ) ], count := [ 6, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 260351, root := ~, start := 258659, stop := 260350 ), rec( content := "\n\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 260365, root := ~, start := 256308, stop := 260364 ) , 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 := 260420, root := ~, start := 260378, stop := 260419 ), 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 := 260485, root := ~, start := 260440, stop := 260484 ), 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 := 260508, root := ~, start := 260497, stop := 260507 ), 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 := 260524, root := ~, start := 260512, stop := 260523 ), rec( content := ".", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 260535, root := ~, start := 260488, stop := 260534 ), 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 := 260584, root := ~, start := 260573, stop := 260583 ), 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 := 260614, root := ~, start := 260600, stop := 260613 ), 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 := 260647, root := ~, start := 260635, stop := 260646 ), 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 := 260689, root := ~, start := 260675, stop := 260688 ), 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 := 260807, root := ~, start := 260757, stop := 260806 ), 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 := 260888, root := ~, start := 260884, stop := 260887 ), 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 := 260956, root := ~, start := 260943, stop := 260955 ), 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 := 260975, root := ~, start := 260967, stop := 260974 ), 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 := 261048, root := ~, start := 261040, stop := 261047 ), 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 := 261095, root := ~, start := 261091, stop := 261094 ), 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 := 261131, root := ~, start := 261117, stop := 261130 ), 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 := 261880, root := ~, start := 261232, stop := 261879 ), rec( content := "\n ", count := [ 6, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 261897, root := ~, start := 260538, stop := 261896 ) ], count := [ 6, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 261911, root := ~, start := 260425, stop := 261910 ), 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 := 261975, root := ~, start := 261932, stop := 261974 ), 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 := 261998, root := ~, start := 261987, stop := 261997 ), 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 := 262014, root := ~, start := 262002, stop := 262013 ), rec( content := ".", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 262025, root := ~, start := 261978, stop := 262024 ), 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 := 262067, root := ~, start := 262046, stop := 262066 ), 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 := 262087, root := ~, start := 262076, stop := 262086 ), 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 := 262117, root := ~, start := 262103, stop := 262116 ), 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 := 262211, root := ~, start := 262197, stop := 262210 ), 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 := 262229, root := ~, start := 262217, stop := 262228 ), 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 := 262280, root := ~, start := 262248, stop := 262279 ), 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 := 262285, root := ~, start := 262281, stop := 262284 ), 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 := 262318, root := ~, start := 262306, stop := 262317 ), 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 := 262443, root := ~, start := 262415, stop := 262442 ), 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 := 262476, root := ~, start := 262452, stop := 262475 ), 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 := 262486, root := ~, start := 262482, stop := 262485 ), 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 := 262528, root := ~, start := 262514, stop := 262527 ), 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 := 263187, root := ~, start := 262625, stop := 263186 ), rec( content := "\n ", count := [ 6, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 263204, root := ~, start := 262028, stop := 263203 ) ], count := [ 6, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 263218, root := ~, start := 261917, stop := 263217 ), 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 := 263286, root := ~, start := 263239, stop := 263285 ), 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 := 263309, root := ~, start := 263298, stop := 263308 ), 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 := 263325, root := ~, start := 263313, stop := 263324 ), rec( content := ".", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 263336, root := ~, start := 263289, stop := 263335 ), 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 := 263385, root := ~, start := 263374, stop := 263384 ), 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 := 263415, root := ~, start := 263401, stop := 263414 ), 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 := 263456, root := ~, start := 263444, stop := 263455 ), 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 := 263500, root := ~, start := 263486, stop := 263499 ), 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 := 263527, root := ~, start := 263504, stop := 263526 ), 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 := 263696, root := ~, start := 263692, stop := 263695 ), 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 := 263761, root := ~, start := 263750, stop := 263760 ), 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 := 263810, root := ~, start := 263783, stop := 263809 ), 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 := 263874, root := ~, start := 263861, stop := 263873 ), 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 := 263894, root := ~, start := 263886, stop := 263893 ), 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 := 263934, root := ~, start := 263926, stop := 263933 ), 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 := 263986, root := ~, start := 263982, stop := 263985 ), 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 := 264028, root := ~, start := 264014, stop := 264027 ), 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 := 264490, root := ~, start := 264129, stop := 264489 ), rec( content := "\n ", count := [ 6, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 264507, root := ~, start := 263339, stop := 264506 ) ], count := [ 6, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 264521, root := ~, start := 263224, stop := 264520 ), 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 := 264591, root := ~, start := 264542, stop := 264590 ), 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 := 264614, root := ~, start := 264603, stop := 264613 ), 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 := 264630, root := ~, start := 264618, stop := 264629 ), rec( content := ".", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 264641, root := ~, start := 264594, stop := 264640 ), 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 := 264703, root := ~, start := 264685, stop := 264702 ), 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 := 264782, root := ~, start := 264750, stop := 264781 ), 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 := 264830, root := ~, start := 264822, stop := 264829 ), 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 := 264888, root := ~, start := 264861, stop := 264887 ), 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 := 264936, root := ~, start := 264903, stop := 264935 ), 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 := 264978, root := ~, start := 264946, stop := 264977 ), 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 := 265003, root := ~, start := 264982, stop := 265002 ), 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 := 265049, root := ~, start := 265045, stop := 265048 ), 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 := 265082, root := ~, start := 265055, stop := 265081 ), 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 := 265102, root := ~, start := 265091, stop := 265101 ), 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 := 265136, root := ~, start := 265122, stop := 265135 ), 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 := 265169, root := ~, start := 265157, stop := 265168 ), 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 := 265230, root := ~, start := 265203, stop := 265229 ), 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 := 265251, root := ~, start := 265239, stop := 265250 ), 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 := 265273, root := ~, start := 265259, stop := 265272 ), 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 := 265296, root := ~, start := 265292, stop := 265295 ), 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 := 265338, root := ~, start := 265324, stop := 265337 ), 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 := 265437, root := ~, start := 265433, stop := 265436 ), 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 := 265490, root := ~, start := 265486, stop := 265489 ), 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 := 265558, root := ~, start := 265545, stop := 265557 ), 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 := 265577, root := ~, start := 265569, stop := 265576 ), 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 := 265613, root := ~, start := 265605, stop := 265612 ), 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 := 265668, root := ~, start := 265664, stop := 265667 ), 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 := 265704, root := ~, start := 265683, stop := 265703 ), 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 := 265738, root := ~, start := 265706, stop := 265737 ), 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 := 265782, root := ~, start := 265749, stop := 265781 ), 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 := 265792, root := ~, start := 265788, stop := 265791 ), 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 := 266180, root := ~, start := 265800, stop := 266179 ), rec( content := "\n ", count := [ 6, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 266197, root := ~, start := 264644, stop := 266196 ) ], count := [ 6, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 266211, root := ~, start := 264527, stop := 266210 ), 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 := 266274, root := ~, start := 266232, stop := 266273 ), 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 := 266297, root := ~, start := 266286, stop := 266296 ), 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 := 266313, root := ~, start := 266301, stop := 266312 ), rec( content := ".", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Returns", next := 266324, root := ~, start := 266277, stop := 266323 ), 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 := 266385, root := ~, start := 266368, stop := 266384 ), 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 := 266464, root := ~, start := 266432, stop := 266463 ), 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 := 266508, root := ~, start := 266500, stop := 266507 ), 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 := 266577, root := ~, start := 266539, stop := 266576 ), 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 := 266612, root := ~, start := 266586, stop := 266611 ), 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 := 266654, root := ~, start := 266622, stop := 266653 ), 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 := 266679, root := ~, start := 266658, stop := 266678 ), 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 := 266725, root := ~, start := 266721, stop := 266724 ), 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 := 266757, root := ~, start := 266731, stop := 266756 ), 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 := 266777, root := ~, start := 266766, stop := 266776 ), 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 := 266805, root := ~, start := 266797, stop := 266804 ), 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 := 266837, root := ~, start := 266825, stop := 266836 ), 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 := 266897, root := ~, start := 266871, stop := 266896 ), 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 := 266918, root := ~, start := 266906, stop := 266917 ), 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 := 266934, root := ~, start := 266926, stop := 266933 ), 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 := 266957, root := ~, start := 266953, stop := 266956 ), 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 := 266993, root := ~, start := 266985, stop := 266992 ), 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 := 267092, root := ~, start := 267088, stop := 267091 ), 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 := 267145, root := ~, start := 267141, stop := 267144 ), 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 := 267213, root := ~, start := 267200, stop := 267212 ), 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 := 267232, root := ~, start := 267224, stop := 267231 ), 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 := 267268, root := ~, start := 267260, stop := 267267 ), 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 := 267323, root := ~, start := 267319, stop := 267322 ), 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 := 267359, root := ~, start := 267338, stop := 267358 ), 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 := 267393, root := ~, start := 267361, stop := 267392 ), 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 := 267438, root := ~, start := 267404, stop := 267437 ), 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 := 268262, root := ~, start := 267447, stop := 268261 ), rec( content := "\n ", count := [ 6, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Description", next := 268279, root := ~, start := 266327, stop := 268278 ) ], count := [ 6, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 268293, root := ~, start := 266217, stop := 268292 ), 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 := 268369, root := ~, start := 268314, stop := 268368 ), 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 := 268392, root := ~, start := 268381, stop := 268391 ), 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 := 268408, root := ~, start := 268396, stop := 268407 ), rec( content := ".", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 268419, root := ~, start := 268372, stop := 268418 ), 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 := 268468, root := ~, start := 268457, stop := 268467 ), 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 := 268498, root := ~, start := 268484, stop := 268497 ), 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 := 268541, root := ~, start := 268529, stop := 268540 ), 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 := 268560, root := ~, start := 268556, stop := 268559 ), 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 := 268590, root := ~, start := 268576, stop := 268589 ), 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 := 268619, root := ~, start := 268594, stop := 268618 ), 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 := 268756, root := ~, start := 268706, stop := 268755 ), 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 := 268807, root := ~, start := 268803, stop := 268806 ), 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 := 268902, root := ~, start := 268877, stop := 268901 ), 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 := 268935, root := ~, start := 268922, stop := 268934 ), 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 := 268951, root := ~, start := 268943, stop := 268950 ), 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 := 269024, root := ~, start := 269016, stop := 269023 ), 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 := 269079, root := ~, start := 269075, stop := 269078 ), 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 := 269121, root := ~, start := 269107, stop := 269120 ), 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 := 269798, root := ~, start := 269222, stop := 269797 ), rec( content := "\n ", count := [ 6, 3, 6, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 269815, root := ~, start := 268422, stop := 269814 ) ], count := [ 6, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 269829, root := ~, start := 268299, stop := 269828 ), 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 := 269897, root := ~, start := 269850, stop := 269896 ), 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 := 269920, root := ~, start := 269909, stop := 269919 ), 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 := 269936, root := ~, start := 269924, stop := 269935 ), rec( content := ".", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 269947, root := ~, start := 269900, stop := 269946 ), 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 := 269996, root := ~, start := 269985, stop := 269995 ), 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 := 270026, root := ~, start := 270012, stop := 270025 ), 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 := 270086, root := ~, start := 270057, stop := 270085 ), 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 := 270145, root := ~, start := 270133, stop := 270144 ), 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 := 270155, root := ~, start := 270151, stop := 270154 ), 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 := 270197, root := ~, start := 270183, stop := 270196 ), 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 := 270742, root := ~, start := 270297, stop := 270741 ), rec( content := "\n ", count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 270759, root := ~, start := 269950, stop := 270758 ) ], count := [ 6, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 270773, root := ~, start := 269835, stop := 270772 ), 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 := 270837, root := ~, start := 270794, stop := 270836 ), 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 := 270860, root := ~, start := 270849, stop := 270859 ), 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 := 270876, root := ~, start := 270864, stop := 270875 ), rec( content := ".", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 270887, root := ~, start := 270840, stop := 270886 ), 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 := 270927, root := ~, start := 270916, stop := 270926 ), 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 := 270957, root := ~, start := 270943, stop := 270956 ), 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 := 270998, root := ~, start := 270986, stop := 270997 ), 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 := 271017, root := ~, start := 271013, stop := 271016 ), 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 := 271045, root := ~, start := 271025, stop := 271044 ), 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 := 271237, root := ~, start := 271233, stop := 271236 ), 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 := 271280, root := ~, start := 271252, stop := 271279 ), 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 := 271290, root := ~, start := 271286, stop := 271289 ), 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 := 271332, root := ~, start := 271318, stop := 271331 ), 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 := 271916, root := ~, start := 271433, stop := 271915 ), rec( content := "\n ", count := [ 6, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 271933, root := ~, start := 270890, stop := 271932 ) ], count := [ 6, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 271947, root := ~, start := 270779, stop := 271946 ), 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 := 272013, root := ~, start := 271968, stop := 272012 ), rec( attributes := rec( Arg := "digraph", Name := "IsUndirectedFor\ est" ), content := 0, count := [ 6, 3, 9, 4 ], mathmode := "Text", name := "Prop", next := 272063, root := ~, start := 272016, stop := 272062 ), 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 := 272086, root := ~, start := 272075, stop := 272085 ), 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 := 272102, root := ~, start := 272090, stop := 272101 ), rec( content := ".", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 272113, root := ~, start := 272066, stop := 272112 ), 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 := 272170, root := ~, start := 272147, stop := 272169 ), 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 := 272190, root := ~, start := 272179, stop := 272189 ), 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 := 272224, root := ~, start := 272210, stop := 272223 ), 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 := 272294, root := ~, start := 272269, stop := 272293 ), 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 := 272314, root := ~, start := 272303, stop := 272313 ), 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 := 272332, root := ~, start := 272318, stop := 272331 ), 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 := 272409, root := ~, start := 272397, stop := 272408 ), 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 := 272415, root := ~, start := 272411, stop := 272414 ), 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 := 272446, root := ~, start := 272424, stop := 272445 ), 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 := 272540, root := ~, start := 272532, stop := 272539 ), 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 := 272553, root := ~, start := 272545, stop := 272552 ), 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 := 272607, root := ~, start := 272599, stop := 272606 ), 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 := 272619, root := ~, start := 272611, stop := 272618 ), 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 := 272663, root := ~, start := 272625, stop := 272662 ), 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 := 272697, root := ~, start := 272668, stop := 272696 ), 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 := 272763, root := ~, start := 272719, stop := 272762 ), 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 := 272884, root := ~, start := 272880, stop := 272883 ), 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 := 272917, root := ~, start := 272893, stop := 272916 ), 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 := 273141, root := ~, start := 273094, stop := 273140 ), 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 := 273218, root := ~, start := 273146, stop := 273217 ), 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 := 273290, root := ~, start := 273286, stop := 273289 ), 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 := 273429, root := ~, start := 273425, stop := 273428 ), 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 := 273471, root := ~, start := 273457, stop := 273470 ), 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 := 274172, root := ~, start := 273572, stop := 274171 ), rec( content := "\n ", count := [ 6, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 274189, root := ~, start := 272116, stop := 274188 ) ], count := [ 6, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 274203, root := ~, start := 271953, stop := 274202 ), 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 := 274270, root := ~, start := 274224, stop := 274269 ), 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 := 274293, root := ~, start := 274282, stop := 274292 ), 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 := 274309, root := ~, start := 274297, stop := 274308 ), rec( content := ".", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Returns", next := 274320, root := ~, start := 274273, stop := 274319 ), 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 := 274397, root := ~, start := 274383, stop := 274396 ), 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 := 274419, root := ~, start := 274415, stop := 274418 ), 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 := 274470, root := ~, start := 274455, stop := 274469 ), 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 := 274626, root := ~, start := 274582, stop := 274625 ), 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 := 274767, root := ~, start := 274763, stop := 274766 ), 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 := 274809, root := ~, start := 274795, stop := 274808 ), 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 := 275372, root := ~, start := 274910, stop := 275371 ), rec( content := "\n ", count := [ 6, 3, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 275389, root := ~, start := 274323, stop := 275388 ) ], count := [ 6, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 275403, root := ~, start := 274209, stop := 275402 ), 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 := 275473, root := ~, start := 275424, stop := 275472 ), 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 := 275496, root := ~, start := 275485, stop := 275495 ), 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 := 275512, root := ~, start := 275500, stop := 275511 ), rec( content := ".", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Returns", next := 275523, root := ~, start := 275476, stop := 275522 ), 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 := 275561, root := ~, start := 275547, stop := 275560 ), 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 := 275620, root := ~, start := 275609, stop := 275619 ), 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 := 275638, root := ~, start := 275626, stop := 275637 ), 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 := 275657, root := ~, start := 275653, stop := 275656 ), 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 := 275686, root := ~, start := 275678, stop := 275685 ), 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 := 275717, root := ~, start := 275699, stop := 275716 ), 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 := 275767, root := ~, start := 275759, stop := 275766 ), 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 := 275830, root := ~, start := 275781, stop := 275829 ), 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 := 275960, root := ~, start := 275956, stop := 275959 ), 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 := 276070, root := ~, start := 276037, stop := 276069 ), 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 := 276080, root := ~, start := 276076, stop := 276079 ), 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 := 276122, root := ~, start := 276108, stop := 276121 ), 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 := 276698, root := ~, start := 276223, stop := 276697 ), rec( content := "\n ", count := [ 6, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Description", next := 276715, root := ~, start := 275526, stop := 276714 ) ], count := [ 6, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 276729, root := ~, start := 275409, stop := 276728 ), 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 := 276793, root := ~, start := 276750, stop := 276792 ), 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 := 276816, root := ~, start := 276805, stop := 276815 ), 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 := 276832, root := ~, start := 276820, stop := 276831 ), rec( content := ".", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 276843, root := ~, start := 276796, stop := 276842 ), 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 := 276885, root := ~, start := 276864, stop := 276884 ), 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 := 276905, root := ~, start := 276894, stop := 276904 ), 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 := 276935, root := ~, start := 276921, stop := 276934 ), 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 := 277029, root := ~, start := 277015, stop := 277028 ), 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 := 277047, root := ~, start := 277035, stop := 277046 ), 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 := 277098, root := ~, start := 277066, stop := 277097 ), 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 := 277103, root := ~, start := 277099, stop := 277102 ), 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 := 277136, root := ~, start := 277124, stop := 277135 ), 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 := 277236, root := ~, start := 277232, stop := 277235 ), 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 := 277278, root := ~, start := 277264, stop := 277277 ), 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 := 277763, root := ~, start := 277375, stop := 277762 ), rec( content := "\n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 277780, root := ~, start := 276846, stop := 277779 ) ], count := [ 6, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 277794, root := ~, start := 276735, stop := 277793 ), rec( content := "\n\n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 277808, root := ~, start := 260369, stop := 277807 ) , 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 := 277851, root := ~, start := 277823, stop := 277850 ), 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 := 277915, root := ~, start := 277871, stop := 277914 ), 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 := 277938, root := ~, start := 277927, stop := 277937 ), 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 := 277954, root := ~, start := 277942, stop := 277953 ), rec( content := ".", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 277965, root := ~, start := 277918, stop := 277964 ), 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 := 278001, root := ~, start := 277988, stop := 278000 ), 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 := 278242, root := ~, start := 278234, stop := 278241 ), 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 := 278322, root := ~, start := 278314, stop := 278321 ), 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 := 278335, root := ~, start := 278327, stop := 278334 ), 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 := 278345, root := ~, start := 278341, stop := 278344 ), 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 := 278377, root := ~, start := 278355, stop := 278376 ), 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 := 278397, root := ~, start := 278386, stop := 278396 ), 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 := 278427, root := ~, start := 278413, stop := 278426 ), 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 := 278458, root := ~, start := 278446, stop := 278457 ), 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 := 278541, root := ~, start := 278527, stop := 278540 ), 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 := 278579, root := ~, start := 278557, stop := 278578 ), 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 := 278589, root := ~, start := 278585, stop := 278588 ), 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 := 278639, root := ~, start := 278605, stop := 278638 ), 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 := 278649, root := ~, start := 278645, stop := 278648 ), 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 := 278807, root := ~, start := 278789, stop := 278806 ), 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 := 279346, root := ~, start := 278814, stop := 279345 ), rec( content := "\n ", count := [ 6, 4, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 279363, root := ~, start := 277968, stop := 279362 ) ], count := [ 6, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 279377, root := ~, start := 277856, stop := 279376 ), 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 := 279447, root := ~, start := 279398, stop := 279446 ), 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 := 279470, root := ~, start := 279459, stop := 279469 ), 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 := 279486, root := ~, start := 279474, stop := 279485 ), rec( content := ".", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 279497, root := ~, start := 279450, stop := 279496 ), 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 := 279540, root := ~, start := 279521, stop := 279539 ), 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 := 279856, root := ~, start := 279848, stop := 279855 ), 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 := 279936, root := ~, start := 279928, stop := 279935 ), 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 := 279949, root := ~, start := 279941, stop := 279948 ), 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 := 279959, root := ~, start := 279955, stop := 279958 ), 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 := 279996, root := ~, start := 279969, stop := 279995 ), 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 := 280016, root := ~, start := 280005, stop := 280015 ), 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 := 280050, root := ~, start := 280036, stop := 280049 ), 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 := 280083, root := ~, start := 280071, stop := 280082 ), 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 := 280166, root := ~, start := 280152, stop := 280165 ), 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 := 280208, root := ~, start := 280186, stop := 280207 ), 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 := 280215, root := ~, start := 280211, stop := 280214 ), 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 := 280259, root := ~, start := 280230, stop := 280258 ), 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 := 280418, root := ~, start := 280400, stop := 280417 ), 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 := 281218, root := ~, start := 280421, stop := 281217 ), rec( content := "\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 281235, root := ~, start := 279500, stop := 281234 ) ], count := [ 6, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 281249, root := ~, start := 279383, stop := 281248 ), rec( content := "\n\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 281263, root := ~, start := 277814, stop := 281262 ) , 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 := 281328, root := ~, start := 281276, stop := 281327 ), 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 := 281390, root := ~, start := 281348, stop := 281389 ), 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 := 281413, root := ~, start := 281402, stop := 281412 ), 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 := 281429, root := ~, start := 281417, stop := 281428 ), rec( content := ".", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 281440, root := ~, start := 281393, stop := 281439 ), 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 := 281494, root := ~, start := 281483, stop := 281493 ), 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 := 281512, root := ~, start := 281498, stop := 281511 ), 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 := 281544, root := ~, start := 281532, stop := 281543 ), 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 := 281562, root := ~, start := 281558, stop := 281561 ), 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 := 281586, root := ~, start := 281578, stop := 281585 ), 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 := 281603, root := ~, start := 281592, stop := 281602 ), 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 := 281660, root := ~, start := 281652, stop := 281659 ), 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 := 281712, root := ~, start := 281704, stop := 281711 ), 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 := 281728, root := ~, start := 281720, stop := 281727 ), 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 := 281764, root := ~, start := 281756, stop := 281763 ), 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 := 281824, root := ~, start := 281816, stop := 281823 ), 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 := 281855, root := ~, start := 281847, stop := 281854 ), 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 := 281865, root := ~, start := 281861, stop := 281864 ), 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 := 281907, root := ~, start := 281893, stop := 281906 ), 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 := 282368, root := ~, start := 282008, stop := 282367 ), rec( content := "\n ", count := [ 6, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 282385, root := ~, start := 281443, stop := 282384 ) ], count := [ 6, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 282399, root := ~, start := 281333, stop := 282398 ), 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 := 282465, root := ~, start := 282420, stop := 282464 ), 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 := 282488, root := ~, start := 282477, stop := 282487 ), 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 := 282504, root := ~, start := 282492, stop := 282503 ), rec( content := ".", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 282515, root := ~, start := 282468, stop := 282514 ), 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 := 282553, root := ~, start := 282539, stop := 282552 ), 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 := 282623, root := ~, start := 282600, stop := 282622 ), 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 := 282643, root := ~, start := 282632, stop := 282642 ), 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 := 282661, root := ~, start := 282647, stop := 282660 ), 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 := 282702, root := ~, start := 282690, stop := 282701 ), 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 := 282753, root := ~, start := 282731, stop := 282752 ), 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 := 282871, root := ~, start := 282835, stop := 282870 ), 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 := 282882, root := ~, start := 282878, stop := 282881 ), 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 := 282924, root := ~, start := 282910, stop := 282923 ), 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 := 283271, root := ~, start := 283025, stop := 283270 ), rec( content := "\n ", count := [ 6, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 283288, root := ~, start := 282518, stop := 283287 ) ], count := [ 6, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 283302, root := ~, start := 282405, stop := 283301 ), 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 := 283370, root := ~, start := 283323, stop := 283369 ), 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 := 283393, root := ~, start := 283382, stop := 283392 ), 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 := 283409, root := ~, start := 283397, stop := 283408 ), rec( content := ".", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 283420, root := ~, start := 283373, stop := 283419 ), 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 := 283458, root := ~, start := 283444, stop := 283457 ), 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 := 283503, root := ~, start := 283478, stop := 283502 ), 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 := 283527, root := ~, start := 283516, stop := 283526 ), 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 := 283545, root := ~, start := 283531, stop := 283544 ), 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 := 283584, root := ~, start := 283572, stop := 283583 ), 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 := 283637, root := ~, start := 283613, stop := 283636 ), 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 := 283711, root := ~, start := 283707, stop := 283710 ), 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 := 283753, root := ~, start := 283739, stop := 283752 ), 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 := 283984, root := ~, start := 283854, stop := 283983 ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 284001, root := ~, start := 283423, stop := 284000 ) ], count := [ 6, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 284015, root := ~, start := 283308, stop := 284014 ), rec( content := "\n\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 284029, root := ~, start := 281267, stop := 284028 ) , 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 := 284044, root := ~, start := 236145, stop := 284043 ), 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 := 284131, root := ~, start := 284099, stop := 284130 ) , 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 := 284775, root := ~, start := 284738, stop := 284774 ), 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 := 284877, root := ~, start := 284795, stop := 284876 ), 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 := 284973, root := ~, start := 284880, stop := 284972 ), 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 := 285005, root := ~, start := 284976, stop := 285004 ), 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 := 285043, root := ~, start := 285029, stop := 285042 ), 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 := 285093, root := ~, start := 285082, stop := 285092 ), 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 := 285121, root := ~, start := 285103, stop := 285120 ), 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 := 285155, root := ~, start := 285141, stop := 285154 ), 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 := 285260, root := ~, start := 285246, stop := 285259 ), 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 := 285285, root := ~, start := 285274, stop := 285284 ), 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 := 285330, root := ~, start := 285322, stop := 285329 ), 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 := 285381, root := ~, start := 285367, stop := 28538\ 0 ), 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 := 285396, root := ~, start := 285353, stop := 285395 ), 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 := 285410, root := ~, start := 285396, stop := 285409 ), 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 := 285420, root := ~, start := 285416, stop := 285419 ), 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 := 285475, root := ~, start := 285454, stop := 285474 ), 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 := 285488, root := ~, start := 285476, stop := 285487 ), 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 := 285526, root := ~, start := 285512, stop := 285525 ), 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 := 285661, root := ~, start := 285649, stop := 285660 ), 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 := 285731, root := ~, start := 285719, stop := 285730 ), 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 := 285853, root := ~, start := 285839, stop := 285852 ), 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 := 285883, root := ~, start := 285871, stop := 285882 ), 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 := 285980, root := ~, start := 285966, stop := 28597\ 9 ), 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 := 2860\ 08, root := ~, start := 285996, stop := 286007 ), rec( content := "))", count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "C", next := 286014, root := ~, start := 285952, stop := 286013 ), 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 := 286024, root := ~, start := 286020, stop := 286023 ), 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 := 286067, root := ~, start := 286034, stop := 286066 ), 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 := 286085, root := ~, start := 286071, stop := 286084 ), 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 := 286140, root := ~, start := 286136, stop := 286139 ), 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 := 286163, root := ~, start := 286149, stop := 286162 ), 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 := 286205, root := ~, start := 286175, stop := 286204 ), 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 := 286283, root := ~, start := 286269, stop := 286282 ), 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 := 286306, root := ~, start := 286292, stop := 286305 ), 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 := 286350, root := ~, start := 286318, stop := 286349 ), 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 := 286391, root := ~, start := 286377, stop := 286390 ), 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 := 286437, root := ~, start := 286433, stop := 286436 ), 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 := 287080, root := ~, start := 286443, stop := 287079 ), rec( content := "\n ", count := [ 7, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 287097, root := ~, start := 285008, stop := 287096 ) ], count := [ 7, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 287111, root := ~, start := 284780, stop := 287110 ), 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 := 287182, root := ~, start := 287132, stop := 287181 ), 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 := 287285, root := ~, start := 287185, stop := 287284 ), 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 := 287317, root := ~, start := 287288, stop := 287316 ), 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 := 287355, root := ~, start := 287341, stop := 287354 ), 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 := 287385, root := ~, start := 287374, stop := 287384 ), 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 := 287496, root := ~, start := 287482, stop := 287495 ), 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 := 287607, root := ~, start := 287593, stop := 287606 ), 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 := 287639, root := ~, start := 287625, stop := 287638 ), 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 := 287658, root := ~, start := 287644, stop := 287657 ), 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 := 287678, root := ~, start := 287674, stop := 287677 ), 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 := 287726, root := ~, start := 287704, stop := 287725 ), 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 := 287797, root := ~, start := 287785, stop := 287796 ), 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 := 287814, root := ~, start := 287802, stop := 287813 ), 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 := 287834, root := ~, start := 287830, stop := 287833 ), 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 := 287878, root := ~, start := 287864, stop := 28787\ 7 ), 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 := 2879\ 22, root := ~, start := 287908, stop := 287921 ), rec( content := ", [perm, ()\ ])", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 287939, root := ~, start := 287850, stop := 287938 ), 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 := 287957, root := ~, start := 287946, stop := 287956 ), 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 := 288012, root := ~, start := 287998, stop := 288011 ), 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 := 288144, root := ~, start := 288127, stop := 288143 ), 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 := 288178, root := ~, start := 288156, stop := 288177 ), 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 := 288202, root := ~, start := 288188, stop := 288201 ), 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 := 288244, root := ~, start := 288214, stop := 288243 ), 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 := 288322, root := ~, start := 288308, stop := 288321 ), 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 := 288345, root := ~, start := 288331, stop := 288344 ), 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 := 288389, root := ~, start := 288357, stop := 288388 ), 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 := 288430, root := ~, start := 288416, stop := 288429 ), 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 := 288476, root := ~, start := 288472, stop := 288475 ), 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 := 288921, root := ~, start := 288482, stop := 288920 ), rec( content := "\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 288938, root := ~, start := 287320, stop := 288937 ) ], count := [ 7, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 288952, root := ~, start := 287117, stop := 288951 ), rec( content := "\n\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 288966, root := ~, start := 284729, stop := 288965 ) , 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 := 289086, root := ~, start := 289030, stop := 289085 ), 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 := 289303, root := ~, start := 289285, stop := 289302 ), 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 := 289332, root := ~, start := 289308, stop := 289331 ), 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 := 289454, root := ~, start := 289416, stop := 289453 ), rec( attributes := rec( Arg := "", Name := "DigraphsUseBlis\ s" ), content := 0, count := [ 7, 2, 1, 4 ], mathmode := "Text", name := "Func", next := 289495, root := ~, start := 289457, stop := 289494 ), 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 := 289525, root := ~, start := 289498, stop := 289524 ), 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 := 289787, root := ~, start := 289764, stop := 289786 ), 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 := 289891, root := ~, start := 289887, stop := 289890 ), 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 := 290182, root := ~, start := 290128, stop := 290181 ), 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 := 290270, root := ~, start := 290188, stop := 290269 ), 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 := 290330, root := ~, start := 290276, stop := 290329 ), 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 := 290422, root := ~, start := 290340, stop := 290421 ), 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 := 290442, root := ~, start := 290438, stop := 290441 ), 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 := 290601, root := ~, start := 290565, stop := 290600 ), 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 := 290668, root := ~, start := 290610, stop := 290667 ), 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 := 290700, root := ~, start := 289528, stop := 290699 ) ], count := [ 7, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 290714, root := ~, start := 289401, stop := 290713 ), 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 := 290803, root := ~, start := 290735, stop := 290802 ), 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 := 290845, root := ~, start := 290806, stop := 290844 ), 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 := 290883, root := ~, start := 290869, stop := 290882 ), 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 := 290975, root := ~, start := 290961, stop := 290974 ), 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 := 291000, root := ~, start := 290981, stop := 290999 ), 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 := 291018, root := ~, start := 291004, stop := 291017 ), 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 := 291060, root := ~, start := 291046, stop := 291059 ), 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 := 291137, root := ~, start := 291076, stop := 291136 ), 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 := 291199, root := ~, start := 291195, stop := 291198 ), 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 := 291222, root := ~, start := 291208, stop := 291221 ), 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 := 291360, root := ~, start := 291346, stop := 291359 ), 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 := 291366, root := ~, start := 291362, stop := 291365 ), 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 := 291389, root := ~, start := 291375, stop := 291388 ), 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 := 291545, root := ~, start := 291531, stop := 291544 ), 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 := 291616, root := ~, start := 291602, stop := 291615 ), 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 := 291742, root := ~, start := 291653, stop := 291741 ), 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 := 291773, root := ~, start := 291769, stop := 291772 ), 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 := 292058, root := ~, start := 292000, stop := 292057 ), 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 := 292101, root := ~, start := 292065, stop := 292100 ), 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 := 292133, root := ~, start := 292103, stop := 292132 ), 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 := 292173, root := ~, start := 292143, stop := 292172 ), 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 := 292183, root := ~, start := 292179, stop := 292182 ), 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 := 292219, root := ~, start := 292205, stop := 292218 ), 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 := 292705, root := ~, start := 292321, stop := 292704 ), rec( content := "\n ", count := [ 7, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 292724, root := ~, start := 290848, stop := 292723 ) ], count := [ 7, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 292740, root := ~, start := 290720, stop := 292739 ), 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 := 292834, root := ~, start := 292761, stop := 292833 ), 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 := 292951, root := ~, start := 292837, stop := 292950 ), 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 := 293089, root := ~, start := 292954, stop := 293088 ), 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 := 293131, root := ~, start := 293092, stop := 293130 ), 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 := 293169, root := ~, start := 293155, stop := 293168 ), 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 := 293261, root := ~, start := 293247, stop := 293260 ), 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 := 293335, root := ~, start := 293331, stop := 293334 ), 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 := 293408, root := ~, start := 293355, stop := 293407 ), 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 := 293510, root := ~, start := 293427, stop := 293509 ), 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 := 293572, root := ~, start := 293543, stop := 293571 ), 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 := 293618, root := ~, start := 293576, stop := 293617 ), 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 := 293701, root := ~, start := 293697, stop := 293700 ), 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 := 293808, root := ~, start := 293720, stop := 293807 ), 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 := 293863, root := ~, start := 293834, stop := 293862 ), 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 := 293971, root := ~, start := 293950, stop := 293970 ), 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 := 294000, root := ~, start := 293981, stop := 293999 ), 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 := 294107, root := ~, start := 294019, stop := 294106 ), 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 := 294153, root := ~, start := 294123, stop := 294152 ), 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 := 294189, root := ~, start := 294159, stop := 294188 ), 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 := 294199, root := ~, start := 294195, stop := 294198 ), 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 := 294235, root := ~, start := 294221, stop := 294234 ), 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 := 294474, root := ~, start := 294337, stop := 294473 ), rec( content := "\n ", count := [ 7, 2, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Description", next := 294493, root := ~, start := 293134, stop := 294492 ) ], count := [ 7, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 294509, root := ~, start := 292746, stop := 294508 ), 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 := 294597, root := ~, start := 294530, stop := 294596 ), 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 := 294639, root := ~, start := 294600, stop := 294638 ), 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 := 294677, root := ~, start := 294663, stop := 294676 ), 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 := 294769, root := ~, start := 294755, stop := 294768 ), 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 := 295004, root := ~, start := 294951, stop := 295003 ), 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 := 295106, root := ~, start := 295023, stop := 295105 ), 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 := 295168, root := ~, start := 295139, stop := 295167 ), 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 := 295237, root := ~, start := 295173, stop := 295236 ), 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 := 295319, root := ~, start := 295315, stop := 295318 ), 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 := 295364, root := ~, start := 295334, stop := 295363 ), 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 := 295400, root := ~, start := 295370, stop := 295399 ), 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 := 295410, root := ~, start := 295406, stop := 295409 ), 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 := 295446, root := ~, start := 295432, stop := 295445 ), 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 := 295702, root := ~, start := 295548, stop := 295701 ), rec( content := "\n ", count := [ 7, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 295721, root := ~, start := 294642, stop := 295720 ) ], count := [ 7, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 295737, root := ~, start := 294515, stop := 295736 ), 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 := 295867, root := ~, start := 295758, stop := 295866 ), 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 := 295909, root := ~, start := 295870, stop := 295908 ), 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 := 296099, root := ~, start := 296085, stop := 296098 ), 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 := 296137, root := ~, start := 296118, stop := 296136 ), 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 := 296151, root := ~, start := 296143, stop := 296150 ), 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 := 296199, root := ~, start := 296185, stop := 296198 ), 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 := 296234, root := ~, start := 296215, stop := 296233 ), 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 := 296331, root := ~, start := 296323, stop := 2\ 96330 ), 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 := 296367, root := ~, start := 296348, stop := 296366 ), 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 := 296377, root := ~, start := 296367, stop \ := 296376 ), 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 := 296418, root := ~, start := 296410, stop \ := 296417 ), 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 := 296453, root := ~, start := 296438, stop := 296452 ), 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\ := 296487, root := ~, start := 296471, stop := 296486 ), rec( content := "; o\ r\n ", count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 5, 5 ], name := "Item", next := 296505, root := ~, start := 296\ 298, stop := 296504 ), 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 := 296618, root := ~, start := 296604\ , stop := 296617 ), rec( content := ")", count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "C", next := 296623, root := ~, start := 296585, stop := 29662\ 2 ), 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 := 296662, root := ~, start := 296643, stop := 296661 ), 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 := 296672, root := ~, start := 296662, stop \ := 296671 ), 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 := 296729, root := ~, start := 296721, stop \ := 296728 ), rec( content := ".\n ", count := [ 7, 2, 5, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], name := "Ite\ m", next := 296744, root := ~, start := 296512, stop := 296743 ) ], count := [ 7, 2, 5, 4 ], mathmode := "Text", name := "List", next := 296756, root := ~, start := 296285, stop := 296755 ), 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 := 296791, root := ~, start := 296766, stop := 296790 ), 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 := 296828, root := ~, start := 296814, stop := 296827 ), 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 := 296867, root := ~, start := 296848, stop := 296866 ), 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 := 296940, root := ~, start := 296921, stop := 296939 ), 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 := 296958, root := ~, start := 296944, stop := 296957 ), 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 := 297021, root := ~, start := 297007, stop := 297020 ), 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 := 2971\ 12, root := ~, start := 297098, stop := 297111 ), rec( content := ")", count := [ 7, 2, 5, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "C", next := 297117, root := ~, start := 297077, stop := 297116 ), 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 := 297216, root := ~, start := 297197, stop := 297215 ), 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 := 297280, root := ~, start := 297226, stop := 297279 ), 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 := 297373, root := ~, start := 297290, stop := 297372 ), 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 := 297444, root := ~, start := 297440, stop := 297443 ), 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 := 297467, root := ~, start := 297453, stop := 297466 ), 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 := 297605, root := ~, start := 297591, stop := 297604 ), 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 := 297611, root := ~, start := 297607, stop := 297610 ), 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 := 297634, root := ~, start := 297620, stop := 297633 ), 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 := 297790, root := ~, start := 297776, stop := 297789 ), 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 := 297861, root := ~, start := 297847, stop := 297860 ), 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 := 297987, root := ~, start := 297898, stop := 297986 ), 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 := 298018, root := ~, start := 298014, stop := 298017 ), 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 := 298329, root := ~, start := 298245, stop := 298328 ), 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 := 298371, root := ~, start := 298335, stop := 298370 ), 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 := 298403, root := ~, start := 298373, stop := 298402 ), 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 := 298443, root := ~, start := 298413, stop := 298442 ), 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 := 298965, root := ~, start := 298448, stop := 298964 ), rec( content := "\n ", count := [ 7, 2, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 298982, root := ~, start := 295912, stop := 298981 ) ], count := [ 7, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 298996, root := ~, start := 295743, stop := 298995 ), 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 := 299149, root := ~, start := 299017, stop := 299148 ), 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 := 299191, root := ~, start := 299152, stop := 299190 ), 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 := 299391, root := ~, start := 299377, stop := 299390 ), 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 := 299426, root := ~, start := 299407, stop := 299425 ), 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 := 299447, root := ~, start := 299428, stop := 299446 ), 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 := 299465, root := ~, start := 299457, stop := 299464 ), 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 := 299509, root := ~, start := 299495, stop := 299508 ), 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 := 299613, root := ~, start := 299594, stop := 29961\ 2 ), 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 :\ = 299633, root := ~, start := 299622, stop := 299632 ), 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 := 299655, root := ~, star\ t := 299647, stop := 299654 ), 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 := 299699, root := ~, start :\ = 299680, stop := 299698 ), 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 :\ = 299709, root := ~, start := 299699, stop := 299708 ), 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 := 299742, root\ := ~, start := 299734, stop := 299741 ), 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 := 299785, root := \ ~, start := 299770, stop := 299784 ), 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 := 299811, root := ~, start := 299795, stop \ := 299810 ), rec( content := ";\n ", count := [ 7, 2, 6, 5 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], name := "Ite\ m", next := 299826, root := ~, start := 299579, stop := 299825 ), 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 := 299867, root := ~, start := 299848, stop := 29986\ 6 ), 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 :\ = 299887, root := ~, start := 299876, stop := 299886 ), 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 := 299909, root := ~, star\ t := 299901, stop := 299908 ), 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 := 299986, root := ~, start := 299957, stop := 299985 ), 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\ := 300021, root := ~, start := 300002, stop := 300020 ), 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 := 300034, root := ~, start := 300021, stop := \ 300033 ), 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 := 300104, root := \ ~, start := 300069, stop := 300103 ), 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 := 300139, root := ~, start := 30\ 0124, stop := 300138 ), 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 := 300173, root := ~, start := 300157, stop \ := 300172 ), rec( content := ";\n ", count := [ 7, 2, 6, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], name := "Ite\ m", next := 300188, root := ~, start := 299833, stop := 300187 ) ], count := [ 7, 2, 6, 4 ], mathmode := "Text", name := "List", next := 300200, root := ~, start := 299566, stop := 300199 ), 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 := 300232, root := ~, start := 300213, stop := 300231 ), 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 := 300253, root := ~, start := 300234, stop := 300252 ), 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 := 300269, root := ~, start := 300258, stop := 300268 ), 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 := 300347, root := ~, start := 300343, stop := 300346 ), 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 := 300413, root := ~, start := 300360, stop := 300412 ), 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 := 300538, root := ~, start := 300509, stop := 300537 ), 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 := 300638, root := ~, start := 300613, stop := 300637 ), 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 := 300691, root := ~, start := 300677, stop := 300690 ), 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 := 300715, root := ~, start := 300707, stop := 300714 ), 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 := 300822, root := ~, start := 300803, stop := 300821 ), 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 := 300840, root := ~, start := 300826, stop := 300839 ), 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 := 300915, root := ~, start := 300901, stop := 300914 ), 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 := 3010\ 06, root := ~, start := 300992, stop := 301005 ), rec( content := ")", count := [ 7, 2, 6, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "C", next := 301011, root := ~, start := 300971, stop := 301010 ), 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 := 301105, root := ~, start := 301091, stop := 301104 ), 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 := 301169, root := ~, start := 301115, stop := 301168 ), 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 := 301262, root := ~, start := 301179, stop := 301261 ), 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 := 301333, root := ~, start := 301329, stop := 301332 ), 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 := 301356, root := ~, start := 301342, stop := 301355 ), 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 := 301494, root := ~, start := 301480, stop := 301493 ), 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 := 301500, root := ~, start := 301496, stop := 301499 ), 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 := 301523, root := ~, start := 301509, stop := 301522 ), 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 := 301679, root := ~, start := 301665, stop := 301678 ), 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 := 301750, root := ~, start := 301736, stop := 301749 ), 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 := 301876, root := ~, start := 301787, stop := 301875 ), 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 := 301907, root := ~, start := 301903, stop := 301906 ), 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 := 302227, root := ~, start := 302135, stop := 302226 ), 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 := 302270, root := ~, start := 302234, stop := 302269 ), 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 := 302302, root := ~, start := 302272, stop := 302301 ), 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 := 302343, root := ~, start := 302313, stop := 302342 ), 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 := 302890, root := ~, start := 302348, stop := 302889 ), rec( content := "\n ", count := [ 7, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 302907, root := ~, start := 299194, stop := 302906 ) ], count := [ 7, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 302921, root := ~, start := 299002, stop := 302920 ), 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 := 303016, root := ~, start := 302942, stop := 303015 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "NautyCanonicalL\ abelling" ), content := 0, count := [ 7, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 303093, root := ~, start := 303019, stop := 303092 ), 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 := 303160, root := ~, start := 303096, stop := 303159 ), 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 := 303203, root := ~, start := 303192, stop := 303202 ), 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 := 303283, root := ~, start := 303242, stop := 303282 ), 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 := 303350, root := ~, start := 303342, stop := 303349 ), 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 := 303363, root := ~, start := 303355, stop := 303362 ), 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 := 303369, root := ~, start := 303365, stop := 303368 ), 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 := 303417, root := ~, start := 303403, stop := 303416 ), 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 := 303430, root := ~, star\ t := 303422, stop := 303429 ), 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 := 303476, root := ~, start := 303\ 388, stop := 303475 ), 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\ 03520, root := ~, start := 303498, stop := 303519 ), 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 := 303544, root := ~ , star\ t := 303536, stop := 303543 ), 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 := 303557, root := ~, start := 303549, stop := 30355\ 6 ), 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 := 303607, root := ~, start := 303483, stop := 303606 ) ], count := [ 7, 2, 7, 7 ], mathmode := "Text", name := "List", next := 303619, root := ~, start := 303375, stop := 303618 ), 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 := 303653, root := ~, start := 303627, stop := 303652 ), 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 := 303675, root := ~, start := 303667, stop := 303674 ), 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 := 303694, root := ~, start := 303683, stop := 303693 ), 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 := 303729, root := ~, start := 303721, stop := 303728 ), 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 := 303770, root := ~, start := 303739, stop := 303769 ), 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 := 303790, root := ~, start := 303776, stop := 303789 ), 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 := 303852, root := ~, start := 303797, stop := 303851 ), 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 := 303914, root := ~, start := 303910, stop := 303913 ), 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 := 303950, root := ~, start := 303920, stop := 303949 ), 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 := 304014, root := ~, start := 304000, stop := 304013 ), 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 := 304106, root := ~, start := 304076, stop := 304105 ), 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 := 304170, root := ~, start := 304156, stop := 304169 ), 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 := 304369, root := ~, start := 304365, stop := 304368 ), 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 := 304405, root := ~, start := 304375, stop := 304404 ), 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 := 304444, root := ~, start := 304430, stop := 304443 ), 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 := 304505, root := ~, start := 304476, stop := 304504 ), 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 := 304511, root := ~, start := 304507, stop := 304510 ), 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 := 305007, root := ~, start := 304517, stop := 305006 ), rec( content := "\n ", count := [ 7, 2, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "Description", next := 305024, root := ~, start := 303163, stop := 305023 ) ], count := [ 7, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 305038, root := ~, start := 302927, stop := 305037 ), 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 := 305157, root := ~, start := 305059, stop := 305156 ), 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 := 305258, root := ~, start := 305160, stop := 305257 ), 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 := 305294, root := ~, start := 305261, stop := 305293 ), 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 := 305337, root := ~, start := 305326, stop := 305336 ), 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 := 305433, root := ~, start := 305398, stop := 305432 ), 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 := 305509, root := ~, start := 305501, stop := 305508 ), 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 := 305522, root := ~, start := 305514, stop := 305521 ), 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 := 305571, root := ~, start := 305557, stop := 305570 ), 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 := 305584, root := ~, star\ t := 305576, stop := 305583 ), 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 := 305639, root := ~, start := 305\ 542, stop := 305638 ), 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\ 683, root := ~, start := 305661, stop := 305682 ), 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 := 305707, root := ~, star\ t := 305699, stop := 305706 ), 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 := 305720, root := ~, start := 305712, stop := 305719 ), 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 := 305779, root := ~, start := 305646, stop := 305778 ) ], count := [ 7, 2, 8, 6 ], mathmode := "Text", name := "List", next := 305791, root := ~, start := 305529, stop := 305790 ), 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 := 305825, root := ~, start := 305799, stop := 305824 ), 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 := 305856, root := ~, start := 305848, stop := 305855 ), 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 := 305879, root := ~, start := 305868, stop := 305878 ), 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 := 305910, root := ~, start := 305902, stop := 305909 ), 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 := 305957, root := ~, start := 305920, stop := 305956 ), 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 := 305973, root := ~, start := 305959, stop := 305972 ), 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 := 306063, root := ~, start := 305980, stop := 306062 ), 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 := 306134, root := ~, start := 306130, stop := 306133 ), 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 := 306216, root := ~, start := 306202, stop := 306215 ), 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 := 306253, root := ~, start := 306239, stop := 306252 ), 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 := 306268, root := ~, start := 306260, stop := 306267 ), 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 := 306316, root := ~, start := 306302, stop := 306315 ), 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 := 306346, root := ~, start := 306332, stop := 306345 ), 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 := 306396, root := ~, start := 306392, stop := 306395 ), 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 := 306448, root := ~, start := 306440, stop :\ = 306447 ), 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 := 306479, root := ~, start := 306465, stop := 306478 ) , 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 := 306489, root := ~, start := 30647\ 9, stop := 306488 ), 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 := 306530, root := ~, start := 306522, stop \ := 306529 ), 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 := 306565, root := ~, start := 306550, stop := 306564 ), 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\ := 306599, root := ~, start := 306583, stop := 306598 ), rec( content := "; o\ r\n ", count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 15 ], name := "Item", next := 306617, root := ~, start := 306\ 415, stop := 306616 ), 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 := 306730, root := ~, start := 306\ 716, stop := 306729 ), rec( content := ")", count := [ 7, 2, 8, 17 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], mathmod\ e := "Text", name := "C", next := 306735, root := ~, start := 306697, stop := \ 306734 ), 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 := 306769, root := ~, start := 306755, stop := 306768 ) , 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 := 306779, root := ~, start := 30676\ 9, stop := 306778 ), 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 := 306836, root := ~ , star\ t := 306828, stop := 306835 ), rec( content := ".\n ", count := [ 7, 2, 8\ , 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], name := "Ite\ m", next := 306851, root := ~, start := 306624, stop := 306850 ) ], count := [ 7, 2, 8, 14 ], mathmode := "Text", name := "List", next := 306863, root := ~, start := 306402, stop := 306862 ), 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 := 306886, root := ~, start := 306872, stop := 306885 ), 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 := 306905, root := ~, start := 306891, stop := 306904 ), 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 := 306974, root := ~, start := 306944, stop := 306973 ), 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 := 307038, root := ~, start := 307024, stop := 307037 ), 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 := 307141, root := ~, start := 307111, stop := 307140 ), 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 := 307205, root := ~, start := 307191, stop := 307204 ), 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 := 307405, root := ~, start := 307401, stop := 307404 ), 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 := 307441, root := ~, start := 307411, stop := 307440 ), 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 := 307480, root := ~, start := 307466, stop := 307479 ), 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 := 307541, root := ~, start := 307512, stop := 307540 ), 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 := 307588, root := ~, start := 307574, stop := 307587 ), 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 := 307682, root := ~, start := 307668, stop := 307681 ), 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 := 307721, root := ~, start := 307707, stop := 307720 ), 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 := 307758, root := ~, start := 307750, stop := 307757 ), 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 := 307844, root := ~, start := 307786, stop := 307843 ), 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 := 307889, root := ~, start := 307850, stop := 307888 ), 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 := 307989, root := ~, start := 307981, stop := 307988 ), 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 := 308008, root := ~, start := 307994, stop := 308007 ), 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 := 308029, root := ~, start := 308021, stop := 308028 ), 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 := 308072, root := ~, start := 308060, stop := 308071 ), 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 := 308130, root := ~, start := 308122, stop := 308129 ), 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 := 308146, root := ~, start := 308138, stop := 308145 ), 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 := 308240, root := ~, start := 308226, stop := 308239 ), 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 := 308278, root := ~, start := 308264, stop := 308277 ), 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 := 308464, root := ~, start := 308396, stop := 308463 ), 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 := 308506, root := ~, start := 308492, stop := 308505 ), 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 := 308633, root := ~, start := 308592, stop := 308632 ), 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 := 308643, root := ~, start := 308639, stop := 308642 ), 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 := 309354, root := ~, start := 308649, stop := 309353 ), rec( content := "\n ", count := [ 7, 2, 8, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "Description", next := 309371, root := ~, start := 305297, stop := 309370 ) ], count := [ 7, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 309385, root := ~, start := 305044, stop := 309384 ), 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 := 309456, root := ~, start := 309406, stop := 309455 ), rec( attributes := rec( Arg := "digraph", Name := "NautyCanonicalD\ igraph" ), content := 0, count := [ 7, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 309509, root := ~, start := 309459, stop := 309508 ), 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 := 309541, root := ~, start := 309512, stop := 309540 ), 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 := 309606, root := ~, start := 309576, stop := 309605 ), 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 := 309731, root := ~, start := 309666, stop := 309730 ), 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 := 309861, root := ~, start := 309847, stop := 309860 \ ), rec( content := ")", count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ C", next := 309866, root := ~, start := 309822, stop := 309865 ), 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 := 309885, root := \ ~, start := 309871, stop := 309884 ), 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 := 3099\ 39, root := ~, start := 309807, stop := 309938 ), 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 := 30997\ 3, root := ~, start := 309964, stop := 309972 ), 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\ := 310026, root := ~, start := 309994, stop := 310025 ), 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 := 310078, ro\ ot := ~, start := 310064, stop := 310077 ), rec( content := ")", count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "C", next := 310083, root := ~, start \ := 310039, stop := 310082 ), 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 := 310126, root := ~, star\ t := 310117, stop := 310125 ), 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 := 310145, root := ~, start := 310131, stop := 310144 ), 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 := 310187, root := ~, start := 309946, stop := 310186 ) ], count := [ 7, 2, 9, 6 ], mathmode := "Text", name := "List", next := 310199, root := ~, start := 309794, stop := 310198 ), 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 := 310262, root := ~, start := 310232, stop := 310261 ), 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 := 310387, root := ~, start := 310322, stop := 310386 ), 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 := 310397, root := ~, start := 310393, stop := 310396 ), 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 := 310433, root := ~, start := 310419, stop := 310432 ), 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 := 310826, root := ~, start := 310535, stop := 310825 ), rec( content := "\n ", count := [ 7, 2, 9, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Description", next := 310843, root := ~, start := 309544, stop := 310842 ) ], count := [ 7, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 310857, root := ~, start := 309391, stop := 310856 ), 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 := 310919, root := ~, start := 310878, stop := 310918 ), 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 := 310961, root := ~, start := 310922, stop := 310960 ), 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 := 310999, root := ~, start := 310985, stop := 310998 ), 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 := 311145, root := ~, start := 311126, stop := 311144 ), 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 := 311164, root := ~, start := 311150, stop := 311163 ), 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 := 311270, root := ~, start := 311251, stop := 311269 ), 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 := 311330, root := ~, start := 311316, stop := 311329 ), 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 := 311367, root := ~, start := 311353, stop := 311366 ), 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 := 311470, root := ~, start := 311466, stop := 311469 ), 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 := 311565, root := ~, start := 311532, stop := 311564 ), 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 := 311603, root := ~, start := 311567, stop := 311602 ), 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 := 311642, root := ~, start := 311609, stop := 311641 ), 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 := 311793, root := ~, start := 311774, stop := 311792 ), 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 := 313263, root := ~, start := 311836, stop := 313262 ), rec( content := "\n ", count := [ 7, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 313280, root := ~, start := 310964, stop := 313279 ) ], count := [ 7, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 313294, root := ~, start := 310863, stop := 313293 ), 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 := 313357, root := ~, start := 313315, stop := 313356 ), 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 := 313426, root := ~, start := 313360, stop := 313425 ), 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 := 313467, root := ~, start := 313447, stop := 313466 ), 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 := 313538, root := ~, start := 313512, stop := 313537 ), 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 := 313579, root := ~, start := 313565, stop := 313578 ), 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 := 313968, root := ~, start := 313586, stop := 313967 ), rec( content := "\n ", count := [ 7, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Description", next := 313985, root := ~, start := 313429, stop := 313984 ) ], count := [ 7, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 313999, root := ~, start := 313300, stop := 313998 ), 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 := 314065, root := ~, start := 314020, stop := 314064 ), 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 := 314125, root := ~, start := 314068, stop := 314124 ), 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 := 314169, root := ~, start := 314146, stop := 314168 ), 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 := 314261, root := ~, start := 314235, stop := 314260 ), 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 := 314306, root := ~, start := 314292, stop := 314305 ), 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 := 314827, root := ~, start := 314313, stop := 314826 ), rec( content := "\n ", count := [ 7, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 314844, root := ~, start := 314128, stop := 314843 ) ], count := [ 7, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 314858, root := ~, start := 314005, stop := 314857 ), 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 := 314929, root := ~, start := 314879, stop := 314928 ), 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 := 314989, root := ~, start := 314932, stop := 314988 ), 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 := 315038, root := ~, start := 315010, stop := 315037 ), 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 := 315083, root := ~, start := 315061, stop := 315082 ), 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 := 315135, root := ~, start := 315109, stop := 315134 ), 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 := 315180, root := ~, start := 315166, stop := 315179 ), 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 := 315222, root := ~, start := 315212, stop := 315221 ), 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 := 3152\ 86, root := ~, start := 315272, stop := 315285 ), rec( content := ")", count := [ 7, 2, 13, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 315291, root := ~, start := 315251, stop := 315290 ), 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\ := 315343, root := ~, start := 315322, stop := 315342 ) ], count := [ 7, 2, 13, 4 ], name := "Mar\ k", next := 315350, root := ~, start := 315316, stop := 315349 ), 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 := 315393, root := ~, start := 315385, stop \ := 315392 ), 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 := 315467, root := ~, start := 31\ 5453, stop := 315466 ), 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 := 315485, root := ~, start := 31\ 5433, stop := 315484 ), rec( content := ".\n ", count := [ 7, 2, 13, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 5 ], name := "Ite\ m", next := 315500, root := ~, start := 315357, stop := 315499 ), 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\ := 315535, root := ~, start := 315514, stop := 315534 ) ], count := [ 7, 2, 13, 6 ], name := "Mar\ k", next := 315542, root := ~, start := 315508, stop := 315541 ), 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 := 315600, root := ~, star\ t := 315577, stop := 315599 ), 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 := 315698, root := ~, start := 315687, stop \ := 315697 ), 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 := 315750, root := ~, start := 315736, stop := 315749 ), rec\ ( content := ")", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "C", next\ := 315755, root := ~, start := 315720, stop := 315754 ), rec( content := ".\n\ ", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 13, 7 ], name := "Item", next := 315770, root := ~, start := 315\ 549, stop := 315769 ) ], count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "List", next := 315782, root := ~, start := 315303, stop := 315781 ), 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 := 316559, root := ~, start := 315788, stop := 316558 ), rec( content := "\n ", count := [ 7, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Description", next := 316576, root := ~, start := 314992, stop := 316575 ) ], count := [ 7, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 316590, root := ~, start := 314864, stop := 316589 ), 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 := 316660, root := ~, start := 316611, stop := 316659 ), 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 := 316710, root := ~, start := 316663, stop := 316709 ), 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 := 316755, root := ~, start := 316731, stop := 316754 ), 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 := 316801, root := ~, start := 316793, stop := 316800 ), 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 := 316859, root := ~, start := 316833, stop := 316858 ), 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 := 316904, root := ~, start := 316890, stop := 316903 ), 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 := 317280, root := ~, start := 316911, stop := 317279 ), rec( content := "\n ", count := [ 7, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 317297, root := ~, start := 316713, stop := 317296 ) ], count := [ 7, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 317311, root := ~, start := 316596, stop := 317310 ), 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 := 317412, root := ~, start := 317332, stop := 317411 ), 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 := 317435, root := ~, start := 317424, stop := 317434 ), 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 := 317451, root := ~, start := 317439, stop := 317450 ), rec( content := ".", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Returns", next := 317462, root := ~, start := 317415, stop := 317461 ), 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 := 317517, root := ~, start := 317506, stop := 317516 ), 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 := 317585, root := ~, start := 317570, stop := 317584 ), 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 := 317616, root := ~, start := 317601, stop := 317615 ), 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 := 317684, root := ~, start := 317623, stop := 317683 ), 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 := 317746, root := ~, start := 317742, stop := 317745 ), 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 := 318054, root := ~, start := 318024, stop := 318053 ), 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 := 318090, root := ~, start := 318060, stop := 318089 ), 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 := 318692, root := ~, start := 318097, stop := 318691 ), rec( content := "\n ", count := [ 7, 2, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Description", next := 318709, root := ~, start := 317465, stop := 318708 ) ], count := [ 7, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 318723, root := ~, start := 317317, stop := 318722 ), 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 := 318870, root := ~, start := 318744, stop := 318869 ), 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 := 318893, root := ~, start := 318882, stop := 318892 ), 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 := 318909, root := ~, start := 318897, stop := 318908 ), rec( content := ".", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Returns", next := 318920, root := ~, start := 318873, stop := 318919 ), 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 := 319082, root := ~, start := 319067, stop := 319081 ), 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 := 319120, root := ~, start := 319105, stop := 319119 ), 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 := 319135, root := ~, start := 319127, stop := 319134 ), 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 := 319184, root := ~, start := 319169, stop := 319183 ), 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 := 319215, root := ~, start := 319200, stop := 319214 ), 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 := 319312, root := ~, start := 319304, stop :\ = 319311 ), 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 := 319343, root := ~, start := 319329, stop := 319342 ) , 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 := 319353, root := ~, start := 31934\ 3, stop := 319352 ), 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 := 319394, root := ~, start := 319386, stop \ := 319393 ), 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 := 319429, root := ~, start := 319414, stop := 319428 ), 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\ := 319463, root := ~, start := 319447, stop := 319462 ), rec( content := "; o\ r\n ", count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 16, 5 ], name := "Item", next := 319481, root := ~, start := 319\ 279, stop := 319480 ), 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 := 319594, root := ~, start := 319\ 580, stop := 319593 ), rec( content := ")", count := [ 7, 2, 16, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], mathmod\ e := "Text", name := "C", next := 319599, root := ~, start := 319561, stop := \ 319598 ), 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 := 319633, root := ~, start := 319619, stop := 319632 ) , 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 := 319643, root := ~, start := 31963\ 3, stop := 319642 ), 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 := 319700, root := ~ , star\ t := 319692, stop := 319699 ), rec( content := ".\n ", count := [ 7, 2, 1\ 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], name := "Ite\ m", next := 319715, root := ~, start := 319488, stop := 319714 ) ], count := [ 7, 2, 16, 4 ], mathmode := "Text", name := "List", next := 319727, root := ~, start := 319266, stop := 319726 ), 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 := 319751, root := ~, start := 319736, stop := 319750 ), 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 := 319771, root := ~, start := 319756, stop := 319770 ), 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 := 319832, root := ~, start := 319817, stop := 319831 ), 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 := 319852, root := ~, start := 319837, stop := 319851 ), 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 := 319886, root := ~, start := 319871, stop := 319885 ), 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 := 319910, root := ~, start := 319895, stop := 319909 ), 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 := 319965, root := ~, start := 319954, stop := 319964 ), 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 := 320130, root := ~, start := 320047, stop := 320129 ), 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 := 320197, root := ~, start := 320193, stop := 320196 ), 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 := 320505, root := ~, start := 320475, stop := 320504 ), 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 := 320541, root := ~, start := 320511, stop := 320540 ), 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 := 321154, root := ~, start := 320548, stop := 321153 ), rec( content := "\n ", count := [ 7, 2, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 321171, root := ~, start := 318923, stop := 321170 ) ], count := [ 7, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 321185, root := ~, start := 318729, stop := 321184 ), 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 := 321286, root := ~, start := 321206, stop := 321285 ), 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 := 321355, root := ~, start := 321344, stop := 321354 ), rec( content := ".", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Returns", next := 321366, root := ~, start := 321289, stop := 321365 ), 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 := 321461, root := ~, start := 321446, stop := 321460 ), 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 := 321485, root := ~, start := 321470, stop := 321484 ), 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 := 321540, root := ~, start := 321529, stop := 321539 ), 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 := 321550, root := ~, start := 321546, stop := 321549 ), 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 := 321577, root := ~, start := 321559, stop := 321576 ), 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 := 321608, root := ~, start := 321593, stop := 321607 ), 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 := 321641, root := ~, start := 321626, stop := 321640 ), 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 := 321665, root := ~, start := 321657, stop := 321664 ), 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 := 321706, root := ~, start := 321691, stop := 321705 ), 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 := 321741, root := ~, start := 321726, stop := 321740 ), 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 := 321800, root := ~, start := 321792, stop := 321799 ), 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 := 321813, root := ~, start := 321805, stop := 321812 ), 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 := 321832, root := ~, start := 321817, stop := 321831 ), 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 := 321851, root := ~, start := 321838, stop := 321850 ), 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 := 321881, root := ~, start := 321866, stop := 321880 ), 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 := 321924, root := ~, start := 321897, stop := 321923 ), 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 := 321954, root := ~, start := 321939, stop := 321953 ), 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 := 321960, root := ~, start := 321956, stop := 321959 ), 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 := 322094, root := ~, start := 322086, stop := 322093 ), 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 := 322133, root := ~, start := 322118, stop := 322132 ), 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 := 322148, root := ~, start := 322144, stop := 322147 ), 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 := 322183, root := ~, start := 322168, stop := 32218\ 2 ), rec( content := ", p) = digraph2", count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "C", next := 322202, root := ~, start := 322154, stop := 322201 ), 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 := 322511, root := ~, start := 322481, stop := 322510 ), 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 := 322547, root := ~, start := 322517, stop := 322546 ), 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 := 323170, root := ~, start := 322554, stop := 323169 ), rec( content := "\n ", count := [ 7, 2, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 323187, root := ~, start := 321369, stop := 323186 ) ], count := [ 7, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 323201, root := ~, start := 321191, stop := 323200 ), 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 := 323348, root := ~, start := 323222, stop := 323347 ), 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 := 323390, root := ~, start := 323379, stop := 323389 ), rec( content := ".", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Returns", next := 323401, root := ~, start := 323351, stop := 323400 ), 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 := 323574, root := ~, start := 323559, stop := 323573 ), 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 := 323612, root := ~, start := 323597, stop := 323611 ), 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 := 323627, root := ~, start := 323619, stop := 323626 ), 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 := 323676, root := ~, start := 323661, stop := 323675 ), 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 := 323707, root := ~, start := 323692, stop := 323706 ), 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 := 323804, root := ~, start := 323796, stop :\ = 323803 ), 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 := 323835, root := ~, start := 323821, stop := 323834 ) , 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 := 323845, root := ~, start := 32383\ 5, stop := 323844 ), 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 := 323886, root := ~, start := 323878, stop \ := 323885 ), 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 := 323921, root := ~, start := 323906, stop := 323920 ), 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\ := 323955, root := ~, start := 323939, stop := 323954 ), rec( content := "; o\ r\n ", count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 18, 5 ], name := "Item", next := 323973, root := ~, start := 323\ 771, stop := 323972 ), 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 := 324086, root := ~, start := 324\ 072, stop := 324085 ), rec( content := ")", count := [ 7, 2, 18, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], mathmod\ e := "Text", name := "C", next := 324091, root := ~, start := 324053, stop := \ 324090 ), 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 := 324125, root := ~, start := 324111, stop := 324124 ) , 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 := 324135, root := ~, start := 32412\ 5, stop := 324134 ), 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 := 324192, root := ~ , star\ t := 324184, stop := 324191 ), rec( content := ".\n ", count := [ 7, 2, 1\ 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], name := "Ite\ m", next := 324207, root := ~, start := 323980, stop := 324206 ) ], count := [ 7, 2, 18, 4 ], mathmode := "Text", name := "List", next := 324219, root := ~, start := 323758, stop := 324218 ), 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 := 324246, root := ~, start := 324228, stop := 324245 ), 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 := 324423, root := ~, start := 324363, stop := 324422 ), 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 := 324509, root := ~, start := 324501, stop := 324508 ), 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 := 324576, root := ~, start := 324561, stop := 324575 ), 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 := 324612, root := ~, start := 324597, stop := 324611 ), 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 := 324644, root := ~, start := 324629, stop := 324643 ), 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 := 324680, root := ~, start := 324665, stop := 324679 ), 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 := 324699, root := ~, start := 324691, stop := 324698 ), 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 := 324757, root := ~, start := 324742, stop := 324756 ), 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 := 324786, root := ~, start := 324778, stop := 324785 ), 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 := 324802, root := ~, start := 324794, stop := 324801 ), 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 := 324850, root := ~, start := 324835, stop := 324849 ), 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 := 324870, root := ~, start := 324855, stop := 324869 ), 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 := 324953, root := ~, start := 324938, stop := 32495\ 2 ), 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 := 3249\ 80, root := ~, start := 324965, stop := 324979 ), rec( content := "[i ^ p]", count := [ 7, 2, 18, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 324991, root := ~, start := 324935, stop := 324990 ), 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 := 325017, root := ~, start := 325009, stop := 325016 ), 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 := 325036, root := ~, start := 325021, stop := 325035 ), 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 := 325046, root := ~, start := 325042, stop := 325045 ), 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 := 325141, root := ~, start := 325130, stop := 325140 ), 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 := 325147, root := ~, start := 325143, stop := 325146 ), 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 := 325455, root := ~, start := 325425, stop := 325454 ), 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 := 325491, root := ~, start := 325461, stop := 325490 ), 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 := 326474, root := ~, start := 325498, stop := 326473 ), rec( content := "\n ", count := [ 7, 2, 18, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Description", next := 326491, root := ~, start := 323404, stop := 326490 ) ], count := [ 7, 2, 18, 1 ], mathmode := "Text", name := "ManSection", next := 326505, root := ~, start := 323207, stop := 326504 ), 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 := 326582, root := ~, start := 326526, stop := 326581 ), 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 := 326631, root := ~, start := 326585, stop := 326630 ), 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 := 326693, root := ~, start := 326683, stop := 326692 ), 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 := 326718, root := ~, start := 326697, stop := 326717 ), 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 := 326803, root := ~, start := 326777, stop := 326802 ), 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 := 326855, root := ~, start := 326841, stop := 326854 ), 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 := 326861, root := ~, start := 326857, stop := 326860 ), 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 := 326900, root := ~, start := 326889, stop := 326899 ), 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 := 326965, root := ~, start := 326957, stop := 326964 ), 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 := 326990, root := ~, start := 326977, stop := 326989 ), 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 := 327052, root := ~, start := 327038, stop := 327051 ), 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 := 327071, root := ~, start := 327063, stop := 327070 ), 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 := 327093, root := ~, start := 327079, stop := 327092 ), 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 := 327103, root := ~, start := 327099, stop := 327102 ), 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 := 327138, root := ~, start := 327112, stop := 327137 ), 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 := 327183, root := ~, start := 327156, stop := 327182 ), 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 := 328076, root := ~, start := 327206, stop := 328075 ), rec( content := "\n ", count := [ 7, 2, 19, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Description", next := 328093, root := ~, start := 326634, stop := 328092 ) ], count := [ 7, 2, 19, 1 ], mathmode := "Text", name := "ManSection", next := 328107, root := ~, start := 326511, stop := 328106 ), 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 := 328245, root := ~, start := 328128, stop := 328244 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphIsomorphism" ), content := 0, count := [ 7, 2, 20, 4 ], mathmode := "Text", name := "Oper", next := 328313, root := ~, start := 328248, stop := 328312 ), 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 := 328436, root := ~, start := 328316, stop := 328435 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphAutomo\ rphism" ), content := 0, count := [ 7, 2, 20, 8 ], mathmode := "Text", name := "Oper", next := 328497, root := ~, start := 328439, stop := 328496 ), 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 := 328520, root := ~, start := 328509, stop := 328519 ), 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 := 328536, root := ~, start := 328524, stop := 328535 ), rec( content := ".", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Returns", next := 328547, root := ~, start := 328500, stop := 328546 ), 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 := 328595, root := ~, start := 328568, stop := 328594 ), 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 := 328615, root := ~, start := 328604, stop := 328614 ), 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 := 328665, root := ~, start := 328657, stop := 328664 ), 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 := 328711, root := ~, start := 328701, stop := 328710 ), 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 := 328741, root := ~, start := 328731, stop := 328740 ), 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 := 328751, root := ~, start := 328747, stop := 328750 ), 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 := 328785, root := ~, start := 328757, stop := 328784 ), 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 := 328805, root := ~, start := 328794, stop := 328804 ), 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 := 328855, root := ~, start := 328847, stop := 328854 ), 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 := 328904, root := ~, start := 328890, stop := 328903 ), 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 := 328914, root := ~, start := 328910, stop := 328913 ), 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 := 328960, root := ~, start := 328952, stop := 328959 ), 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 := 328985, root := ~, start := 328967, stop := 328984 ), 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 := 329015, root := ~, start := 329005, stop := 329014 ), 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 := 329039, root := ~, start := 329029, stop := 329038 ), 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 := 329103, root := ~, start := 329095, stop := 329102 ), 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 := 329150, root := ~, start := 329140, stop := 329149 ), 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\ := 329181, root := ~, start := 329171, stop := 329180 ), rec( content := ";\n\ ", count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 13 ], name := "Item", next := 329196, root := ~, start := 329\ 080, stop := 329195 ), 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 := 329234, root := ~, s\ tart := 329226, stop := 329233 ), 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 := 329248, root := ~, start := 329240, stop :\ = 329247 ), rec( content := "]", count := [ 7, 2, 20, 15 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Tex\ t", name := "C", next := 329253, root := ~, start := 329218, stop := 329252 ), 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 := 329286, root := ~, start := 329276, stop := 329285 ), 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\ := 329315, root := ~, start := 329302, stop := 329314 ), 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 :\ = 329348, root := ~, start := 329338, stop := 329347 ), rec( content := "; and\ \n ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 15 ], name := "Item", next := 329367, root := ~, start := 329\ 203, stop := 329366 ), 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 := 329397, root :\ = ~, start := 329389, stop := 329396 ), 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 := 329418, root := ~, start := 32941\ 0, stop := 329417 ), 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 := 329454, root := ~, start := 32944\ 4, stop := 329453 ), rec( content := ".\n ", count := [ 7, 2, 20, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17 ], name := "Ite\ m", next := 329469, root := ~, start := 329374, stop := 329468 ) ], count := [ 7, 2, 20, 12 ], mathmode := "Text", name := "List", next := 329481, root := ~, start := 329067, stop := 329480 ), 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 := 329555, root := ~, start := 329495, stop := 329554 ), 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 := 329560, root := ~, start := 329556, stop := 329559 ), 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 := 329584, root := ~, start := 329573, stop := 329583 ), 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 := 329600, root := ~, start := 329589, stop := 329599 ), 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 := 329615, root := ~, start := 329605, stop := 329614 ), 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 := 329761, root := ~, start := 329685, stop := 329760 ), 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 := 329944, root := ~, start := 329916, stop \ := 329943 ), 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 := 329970, root := ~, start := 329962, stop := 329969 ), 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 := 329984, root :\ = ~, start := 329974, stop := 329983 ), 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 := 330025, root :\ = ~, start := 329998, stop := 330024 ), rec( content := ". ", count := [ 7, 2,\ 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ] , name := "Ite\ m", next := 330034, root := ~, start := 329901, stop := 330033 ), 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 := 330082, root := ~, start := 330056, stop \ := 330081 ), 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 := 330108, root := ~, start := 330100, stop := 330107 ), 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 := 330126, root :\ = ~, start := 330112, stop := 330125 ), 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 := 330168, root :\ = ~, start := 330140, stop := 330167 ), rec( content := ". ", count := [ 7, 2,\ 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ] , name := "Ite\ m", next := 330177, root := ~, start := 330041, stop := 330176 ) ], count := [ 7, 2, 20, 21 ], mathmode := "Text", name := "List", next := 330189, root := ~, start := 329888, stop := 330188 ), 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 := 330266, root := ~, start := 330238, stop := 330265 ), 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 := 330338, root := ~, start := 330324, stop := 330337 ), 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 := 331281, root := ~, start := 330345, stop := 331280 ), rec( content := "\n ", count := [ 7, 2, 20, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Description", next := 331298, root := ~, start := 328550, stop := 331297 ) ], count := [ 7, 2, 20, 1 ], mathmode := "Text", name := "ManSection", next := 331312, root := ~, start := 328113, stop := 331311 ), 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 := 331390, root := ~, start := 331337, stop := 331389 ), 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 := 331481, root := ~, start := 331395, stop := 331480 ), 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 := 331507, root := ~, start := 331496, stop := 331506 ), 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 := 331523, root := ~, start := 331511, stop := 331522 ), rec( content := ". ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Returns", next := 331535, root := ~, start := 331486, stop := 331534 ), 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 := 331599, root := ~, start := 331574, stop := 331598 ), 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 := 331651, root := ~, start := 331640, stop := 331650 ), 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 := 331717, root := ~, start := 331703, stop := 331716 ), 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 := 331730, root := ~, start := 331726, stop := 331729 ), 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 := 331756, root := ~, start := 331745, stop := 331755 ), 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 := 331792, root := ~, start := 331769, stop := 331791 ), 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 := 331826, root := ~, start := 331812, stop := 331825 ), 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 := 331841, root := ~, start := 331830, stop := 331840 ), 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 := 331904, root := ~, start := 331893, stop := 331903 ), 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 := 331952, root := ~, start := 331938, stop := 331951 ), 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 := 331992, root := ~, start := 331981, stop := 331991 ), 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 := 332010, root := ~, start := 331996, stop := 332009 ), 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 := 332022, root := ~, start := 332014, stop := 332021 ), 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 := 332035, root := ~, start := 332027, stop := 332034 ), 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 := 332075, root := ~, start := 332064, stop := 33207\ 4 ), 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 := 3320\ 99, root := ~, start := 332088, stop := 332098 ), rec( content := "[v]", count := [ 7, 2, 21, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 332106, root := ~, start := 332061, stop := 332105 ), 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 := 332118, root := ~, start := 332114, stop := 332117 ), 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 := 332151, root := ~, start := 332143, stop := 332150 ), 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 := 332214, root := ~, start := 332200, stop := 332213 ), 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 := 332256, root := ~, start := 332248, stop := 33225\ 5 ), 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 := 3322\ 96, root := ~, start := 332282, stop := 332295 ), rec( content := "))", count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "C", next := 332302, root := ~, start := 332219, stop := 332301 ), 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 := 332349, root := ~, start := 332335, stop := 332348 ), 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 := 332355, root := ~, start := 332351, stop := 332354 ), 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 := 332407, root := ~, start := 332372, stop := 332406 ), 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 := 332843, root := ~, start := 332422, stop := 332842 ), rec( content := "\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Description", next := 332862, root := ~, start := 331540, stop := 332861 ) ], count := [ 7, 2, 21, 1 ], mathmode := "Text", name := "ManSection", next := 332878, root := ~, start := 331320, stop := 332877 ), rec( content := "\n\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 332892, root := ~, start := 288970, stop := 332891 ) , 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 := 332949, root := ~, start := 332905, stop := 332948 ), 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 := 333430, root := ~, start := 333272, stop := 333429 ), 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 := 333472, root := ~, start := 333455, stop := 333471 ), rec( content := ".", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 333483, root := ~, start := 333433, stop := 333482 ), 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 := 333564, root := ~, start := 333555, stop := 333563 ), 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 := 333593, root := ~, start := 333584, stop := 333592 ), 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 := 333678, root := ~, start := 333674, stop := 333677 ), 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 := 333695, root := ~, start := 333687, stop := 333694 ), 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 := 333708, root := ~, start := 333700, stop := 333707 ), 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 := 333773, root := ~, start := 333740, stop := 333772 ), 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 := 333788, root := ~, start := 333780, stop := 333787 ), 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 := 333825, root := ~, start := 333817, stop := 333824 ), 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 := 333878, root := ~, start := 333869, stop := 333877 ), 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 := 333900, root := ~, start := 333886, stop := 333899 ), 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\ 3937, root := ~, start := 333926, stop := 333936 ) ], count := [ 7, 3, 1, 5 ], name := "Mar\ k", next := 333944, root := ~, start := 333920, stop := 333943 ), 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 := 334015, root := \ ~, start := 334004, stop := 334014 ), 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\ := 334020, root := ~, start := 334016, stop := 334019 ), 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 := 334044, root := \ ~, start := 334033, stop := 334043 ), 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 := 334117, root := ~, start := 334100, stop := 334116 ), 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 := 334159, root := ~, start := 334151, stop := 334158 ), 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 := 334196, root := ~, start := 334185,\ stop := 334195 ), 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 := 334214, root := ~, start := 334197, stop := 334213 ), rec( \ content := ", t)", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next\ := 334222, root := ~, start := 334182, stop := 334221 ), 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 := 334279, root := ~, start := 334271, stop := 334278 ), 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 := 334333, root := ~, start := 334300, stop := 334332 ), 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\ := 334370, root := ~, start := 334359, stop := 334369 ), 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\ := 334418, root := ~, start := 334385, stop := 334417 ), 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 := 334685, root := \ ~, start := 334652, stop := 334684 ), 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 := 334699, root := ~, start := 334695, stop := 334698 ), 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\ 34723, root := ~, start := 334712, stop := 334722 ), 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 := 334738, root := ~, start := 334727, stop \ := 334737 ), 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 := 334871, root := ~, start :\ = 334838, stop := 334870 ), 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 := 334892, root := ~, start := 334875, stop := 334891 ), 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 := 334958, root := ~, start := 333951, stop := 334957 ), 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\ := 334989, root := ~, start := 334972, stop := 334988 ) ], count := [ 7, 3, 1, 9 ], name := "Mar\ k", next := 334996, root := ~, start := 334966, stop := 334995 ), 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 := 335032, root := ~, start := 335021, stop := 3\ 35031 ), 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 := 335070, root := ~, start := 335053, stop := 33506\ 9 ), 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\ := 335132, root := ~, start := 335115, stop := 335131 ), 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 := 335198, root := ~, start := 335187, stop := 33\ 5197 ), 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 := 335230, root := ~, start := 335213, stop := 335229 ), 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 :=\ 335294, root := ~, start := 335283, stop := 335293 ), 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 := 335312, root := ~, s\ tart := 335295, stop := 335311 ), rec( content := ", t)", count := [ 7, 3, 1, \ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1,\ 10 ], mathmode := "Text", name := "C", next := 335320, root := ~, start := 33\ 5280, stop := 335319 ), 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 := 335356, root := ~, start := 3353\ 39, stop := 335355 ), 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 := 335395, root := ~, start := 3353\ 78, stop := 335394 ), 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 := 335416, root := ~, start := 335408, stop := 33541\ 5 ), 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 := 335422, root := ~, star\ t := 335418, stop := 335421 ), 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 := 335459, root := ~, start := 3354\ 48, stop := 335458 ), 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 := 335474, root := ~, start := 335463, stop := 335473 ), 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 := 335519, root := ~, start := 335502, stop := 335518 ), 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 := 335557, root := ~, start := 335003, stop := 335556 ), 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\ := 335589, root := ~, start := 335571, stop := 335588 ) ], count := [ 7, 3, 1, 12 ], name := "Mar\ k", next := 335596, root := ~, start := 335565, stop := 335595 ), 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 :\ = 335679, root := ~, start := 335664, stop := 335678 ), 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\ := 335722, root := ~, start := 335689, stop := 335721 ), 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 := 335780, root := ~, start := 335762, stop := 335779 ), 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 := 335868, root := ~, start := 335\ 603, stop := 335867 ), 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 := 335893, root := ~, start := 335882, stop \ := 335892 ) ], count := [ 7, 3, 1, 14 ], name := "Mark", next := 335900, root := ~, start := 335\ 876, stop := 335899 ), 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 := 335979, root := ~, start := 335968, stop \ := 335978 ), 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 := 335985, root :=\ ~, start := 335981, stop := 335984 ), 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 := 336009, root := ~, start := 3359\ 98, stop := 336008 ), 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 \ := 336082, root := ~, start := 336071, stop := 336081 ), 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 := 336097, root := ~, start := 336093, stop \ := 336096 ), 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 := 336121, root := ~, start := 336110, stop := 336120 ), 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 := 336136, root :=\ ~, start := 336125, stop := 336135 ), 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 := 336222, root := ~, start := 335907, stop := 336221 ), 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\ := 336252, root := ~, start := 336236, stop := 336251 ) ], count := [ 7, 3, 1, 18 ], name := "Mar\ k", next := 336259, root := ~, start := 336230, stop := 336258 ), 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 := 336313, root := ~, start := 336305, stop \ := 336312 ), 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\ := 336323, root := ~, start := 336315, stop := 336322 ), 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 := 336336, root := ~, star\ t := 336328, stop := 336335 ), 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 := 336364, root := ~, start := 336356, stop \ := 336363 ), 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 := 336415, root := ~, start := 336407, stop \ := 336414 ), 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\ 491, root := ~, start := 336483, stop := 336490 ), 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 := 336564, root := ~, start := 336560, stop := 336563 ), 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 := 336619, root := ~, start := 336603, stop := 336618\ ), 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 := 336644, root := ~, start := 336632, stop := 336643 ), 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\ 36667, root := ~, start := 336656, stop := 336666 ), 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 := \ 336707, root := ~, start := 336699, stop := 336706 ), 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 := 336720, root := ~, start := 336712, stop \ := 336719 ), 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 := 3367\ 94, root := ~, start := 336266, stop := 336793 ), 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 := 336820, root := ~, start := 336\ 808, stop := 336819 ) ], count := [ 7, 3, 1, 21 ], name := "Mark", next := 3368\ 27, root := ~, start := 336802, stop := 336826 ), 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 := 336920, root := ~, star\ t := 336911, stop := 336919 ), 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 := 336963, root := ~, star\ t := 336930, stop := 336962 ), 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 := 337011, root := \ ~, start := 337002, stop := 337010 ), 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 := 337040, root := ~, start := 337031, stop \ := 337039 ), 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 := 337085, root := ~, start := 337073, stop \ := 337084 ), rec( content := ".\n ", count := [ 7, 3, 1, 22 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ], name := "Ite\ m", next := 337100, root := ~, start := 336834, stop := 337099 ), 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\ := 337132, root := ~, start := 337114, stop := 337131 ) ], count := [ 7, 3, 1, 23 ], name := "Mar\ k", next := 337139, root := ~, start := 337108, stop := 337138 ), 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 := 337213, root\ := ~, start := 337204, stop := 337212 ), 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 := 337226, root := ~, start := 337217, stop \ := 337225 ), 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 := 337320, root := ~, start := 337311, stop := 337319 ), 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 := 337399, root := ~, start := 337390, stop \ := 337398 ), 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 := 337434, root := ~, start := 337401, stop := 3\ 37433 ), 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 := 337496, root := ~, start \ := 337478, stop := 337495 ), rec( content := " (if any).\n ", count := [ \ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], name := "Ite\ m", next := 337520, root := ~, start := 337146, stop := 337519 ), 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\ := 337554, root := ~, start := 337540, stop := 337553 ) ], count := [ 7, 3, 1, 25 ], name := "Mar\ k", next := 337561, root := ~, start := 337534, stop := 337560 ), 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 := 337679, root := ~, start := 337670, stop \ := 337678 ), 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 := \ 337781, root := ~, start := 337694, stop := 337780 ), 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 := 337861, root := ~, start := 337568, stop := 337860 ), 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\ := 337895, root := ~, start := 337881, stop := 337894 ) ], count := [ 7, 3, 1, 27 ], name := "Mar\ k", next := 337902, root := ~, start := 337875, stop := 337901 ), 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 := 338020, root := ~, start := 338011, stop \ := 338019 ), 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 := \ 338122, root := ~, start := 338035, stop := 338121 ), 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 := 338202, root := ~, start := 337909, stop := 338201 ), 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 :\ = 338227, root := ~, start := 338215, stop := 338226 ) ], count := [ 7, 3, 1, 29 ], name := "Mar\ k", next := 338234, root := ~, start := 338209, stop := 338233 ), 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 := 338290, root := ~, start := 338\ 278, stop := 338289 ), 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 := 338344, root := \ ~, start := 338335, stop := 338343 ), 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 := 338591, root := ~, start := 338554, stop := 338590 ), 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 := 338667, root := ~, start := 338241, stop := 338666 ), 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 := 338725, root := ~, start := 3\ 38711, stop := 338724 ), 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 :=\ 338793, root := ~, start := 338784, stop := 338792 ), 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 := 338927, root := ~, start := 338913, stop := 338926 ), 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\ 9026, root := ~, start := 339017, stop := 339025 ), 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 := 3390\ 75, root := ~, start := 338674, stop := 339074 ) ], count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "List", next := 339087, root := ~, start := 333907, stop := 339086 ), 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 := 340642, root := ~, start := 339093, stop := 340641 ), rec( content := "\n ", count := [ 7, 3, 1, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 340659, root := ~, start := 333486, stop := 340658 ) ], count := [ 7, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 340673, root := ~, start := 333257, stop := 340672 ), 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 := 340753, root := ~, start := 340694, stop := 340752 ), 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 := 340798, root := ~, start := 340787, stop := 340797 ), rec( content := ".", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 340809, root := ~, start := 340756, stop := 340808 ), 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 := 340865, root := ~, start := 340850, stop := 340864 ), 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 := 340884, root := ~, start := 340869, stop := 340883 ), 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 := 340940, root := ~, start := 340925, stop := 340939 ), 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 := 340991, root := ~, start := 340976, stop := 340990 ), 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 := 341038, root := ~, start := 341026, stop := 341037 ), 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 := 341072, root := ~, start := 341061, stop := 341071 ), 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 := 341117, root := ~, start := 341105, stop := 341116 ), 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 := 341151, root := ~, start := 341140, stop := 341150 ), 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 := 341236, root := ~, start := 341232, stop := 341235 ), 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 := 341268, root := ~, start := 341242, stop := 341267 ), 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 := 341326, root := ~, start := 341311, stop := 341325 ), 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 := 341346, root := ~, start := 341331, stop := 341345 ), 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 := 341397, root := ~, start := 341386, stop := 341396 ), 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 := 341745, root := ~, start := 341399, stop := 341744 ), rec( content := "\n ", count := [ 7, 3, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 341762, root := ~, start := 340812, stop := 341761 ) ], count := [ 7, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 341776, root := ~, start := 340679, stop := 341775 ), 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 := 341858, root := ~, start := 341797, stop := 341857 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "HomomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 3, 4 ], mathmode := "Text", name := "Oper", next := 341937, root := ~, start := 341861, stop := 341936 ), 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 := 341986, root := ~, start := 341940, stop := 341985 ), 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 := 342050, root := ~, start := 342007, stop := 342049 ), 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 := 342100, root := ~, start := 342067, stop := 342099 ), 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 := 342124, root := ~, start := 342109, stop := 342123 ), 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 := 342148, root := ~, start := 342133, stop := 342147 ), 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 := 342255, root := ~, start := 342202, stop := 342254 ), 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 := 342274, root := ~, start := 342259, stop := 342273 ), 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 := 342323, root := ~, start := 342315, stop := 342322 ), 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 := 342347, root := ~, start := 342332, stop := 342346 ), 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 := 342371, root := ~, start := 342356, stop := 342370 ), 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 := 342422, root := ~, start := 342406, stop := 342421 ), 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 := 342442, root := ~, start := 342434, stop := 342441 ), 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 := 342500, root := ~, start := 342457, stop := 342499 ), 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 := 342517, root := ~, start := 342509, stop := 342516 ), 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 := 342555, root := ~, start := 342540, stop := 342554 ), 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 := 342565, root := ~, start := 342561, stop := 342564 ), 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 := 342599, root := ~, start := 342571, stop := 342598 ), 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 := 342653, root := ~, start := 342638, stop := 342652 ), 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 := 342673, root := ~, start := 342658, stop := 342672 ), 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 := 343279, root := ~, start := 342679, stop := 343278 ), rec( content := "\n ", count := [ 7, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Description", next := 343296, root := ~, start := 341989, stop := 343295 ) ], count := [ 7, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 343310, root := ~, start := 341782, stop := 343309 ), 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 := 343390, root := ~, start := 343331, stop := 343389 ), 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 := 343434, root := ~, start := 343423, stop := 343433 ), rec( content := ".", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 343445, root := ~, start := 343393, stop := 343444 ), 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 := 343492, root := ~, start := 343466, stop := 343491 ), 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 := 343526, root := ~, start := 343510, stop := 343525 ), 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 := 343566, root := ~, start := 343527, stop := 343565 ), 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 := 343590, root := ~, start := 343575, stop := 343589 ), 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 := 343610, root := ~, start := 343595, stop := 343609 ), 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 := 343662, root := ~, start := 343651, stop := 343661 ), 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 := 343959, root := ~, start := 343668, stop := 343958 ), rec( content := "\n ", count := [ 7, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 343976, root := ~, start := 343448, stop := 343975 ) ], count := [ 7, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 343990, root := ~, start := 343316, stop := 343989 ), 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 := 344072, root := ~, start := 344011, stop := 344071 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "MonomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 5, 4 ], mathmode := "Text", name := "Oper", next := 344151, root := ~, start := 344075, stop := 344150 ), 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 := 344199, root := ~, start := 344154, stop := 344198 ), 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 := 344291, root := ~, start := 344256, stop := 344290 ), 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 := 344350, root := ~, start := 344300, stop := 344349 ), 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 := 344396, root := ~, start := 344380, stop := 344395 ), 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 := 344793, root := ~, start := 344416, stop := 344792 ), rec( content := "\n ", count := [ 7, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 344810, root := ~, start := 344202, stop := 344809 ) ], count := [ 7, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 344824, root := ~, start := 343996, stop := 344823 ), 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 := 344903, root := ~, start := 344845, stop := 344902 ), 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 := 344947, root := ~, start := 344936, stop := 344946 ), rec( content := ".", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 344958, root := ~, start := 344906, stop := 344957 ), 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 := 345004, root := ~, start := 344979, stop := 345003 ), 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 := 345039, root := ~, start := 345022, stop := 345038 ), 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 := 345079, root := ~, start := 345040, stop := 345078 ), 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 := 345103, root := ~, start := 345088, stop := 345102 ), 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 := 345123, root := ~, start := 345108, stop := 345122 ), 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 := 345175, root := ~, start := 345164, stop := 345174 ), 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 := 345515, root := ~, start := 345181, stop := 345514 ), rec( content := "\n ", count := [ 7, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 345532, root := ~, start := 344961, stop := 345531 ) ], count := [ 7, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 345546, root := ~, start := 344830, stop := 345545 ), 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 := 345627, root := ~, start := 345567, stop := 345626 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "EpimorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 7, 4 ], mathmode := "Text", name := "Oper", next := 345705, root := ~, start := 345630, stop := 345704 ), 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 := 345753, root := ~, start := 345708, stop := 345752 ), 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 := 345845, root := ~, start := 345810, stop := 345844 ), 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 := 345904, root := ~, start := 345854, stop := 345903 ), 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 := 345951, root := ~, start := 345934, stop := 345950 ), 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 := 347038, root := ~, start := 345971, stop := 347037 ), rec( content := "\n ", count := [ 7, 3, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Description", next := 347055, root := ~, start := 345756, stop := 347054 ) ], count := [ 7, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 347069, root := ~, start := 345552, stop := 347068 ), 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 := 347146, root := ~, start := 347090, stop := 347145 ), 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 := 347191, root := ~, start := 347180, stop := 347190 ), rec( content := ".", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 347202, root := ~, start := 347149, stop := 347201 ), 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 := 347264, root := ~, start := 347249, stop := 347263 ), 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 := 347305, root := ~, start := 347290, stop := 347304 ), 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 := 347344, root := ~, start := 347311, stop := 347343 ), 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 := 347365, root := ~, start := 347350, stop := 347364 ), 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 := 347388, root := ~, start := 347373, stop := 347387 ), 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 := 347464, root := ~, start := 347451, stop := 347463 ), 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 := 347497, root := ~, start := 347484, stop := 347496 ), 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 := 347516, root := ~, start := 347501, stop := 347515 ), 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 := 347568, root := ~, start := 347555, stop := 347567 ), 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 := 347599, root := ~, start := 347588, stop := 347598 ), 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 := 347622, root := ~, start := 347607, stop := 347621 ), 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 := 347627, root := ~, start := 347623, stop := 347626 ), 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 := 347670, root := ~, start := 347662, stop := 347669 ), 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 := 347713, root := ~, start := 347698, stop := 347712 ), 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 := 347752, root := ~, start := 347721, stop := 347751 ), 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 := 347771, root := ~, start := 347756, stop := 347770 ), 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 := 347800, root := ~, start := 347792, stop := 347799 ), 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 := 347806, root := ~, start := 347802, stop := 347805 ), 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 := 347835, root := ~, start := 347812, stop := 347834 ), 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 := 347914, root := ~, start := 347903, stop := 347913 ), 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 := 348168, root := ~, start := 347920, stop := 348167 ), rec( content := "\n ", count := [ 7, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 348185, root := ~, start := 347205, stop := 348184 ) ], count := [ 7, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 348199, root := ~, start := 347075, stop := 348198 ), 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 := 348266, root := ~, start := 348220, stop := 348265 ), rec( attributes := rec( Arg := "D1, D2", Name := "EmbeddingsDigra\ phsRepresentatives" ), content := 0, count := [ 7, 3, 9, 4 ], mathmode := "Text", name := "Oper", next := 348330, root := ~, start := 348269, stop := 348329 ), 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 := 348378, root := ~, start := 348333, stop := 348377 ), 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 := 348470, root := ~, start := 348435, stop := 348469 ), 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 := 348529, root := ~, start := 348479, stop := 348528 ), 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 := 348582, root := ~, start := 348573, stop := 348581 ), 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 := 348597, root := ~, start := 348588, stop := 348596 ), 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 := 348602, root := ~, start := 348598, stop := 348601 ), 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 := 348649, root := ~, start := 348617, stop := 348648 ), 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 := 349363, root := ~, start := 348655, stop := 349362 ), rec( content := "\n ", count := [ 7, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 349380, root := ~, start := 348381, stop := 349379 ) ], count := [ 7, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 349394, root := ~, start := 348205, stop := 349393 ), 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 := 349535, root := ~, start := 349415, stop := 349534 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphHomomorphism" ), content := 0, count := [ 7, 3, 10, 4 ], mathmode := "Text", name := "Oper", next := 349604, root := ~, start := 349538, stop := 349603 ), 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 := 349725, root := ~, start := 349607, stop := 349724 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEpimorphism" ), content := 0, count := [ 7, 3, 10, 8 ], mathmode := "Text", name := "Oper", next := 349793, root := ~, start := 349728, stop := 349792 ), 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 := 349915, root := ~, start := 349796, stop := 349914 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphMonomorphism" ), content := 0, count := [ 7, 3, 10, 12 ], mathmode := "Text", name := "Oper", next := 349984, root := ~, start := 349918, stop := 349983 ), 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 := 350105, root := ~, start := 349987, stop := 350104 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphEndomo\ rphism" ), content := 0, count := [ 7, 3, 10, 16 ], mathmode := "Text", name := "Oper", next := 350166, root := ~, start := 350108, stop := 350165 ), 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 := 350189, root := ~, start := 350178, stop := 350188 ), 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 := 350205, root := ~, start := 350193, stop := 350204 ), rec( content := ".", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Returns", next := 350216, root := ~, start := 350169, stop := 350215 ), 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 := 350265, root := ~, start := 350237, stop := 350264 ), 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 := 350285, root := ~, start := 350274, stop := 350284 ), 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 := 350335, root := ~, start := 350327, stop := 350334 ), 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 := 350385, root := ~, start := 350375, stop := 350384 ), 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 := 350411, root := ~, start := 350401, stop := 350410 ), 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 := 350421, root := ~, start := 350417, stop := 350420 ), 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 := 350454, root := ~, start := 350427, stop := 350453 ), 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 := 350474, root := ~, start := 350463, stop := 350473 ), 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 := 350524, root := ~, start := 350516, stop := 350523 ), 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 := 350585, root := ~, start := 350575, stop := 350584 ), 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 := 350611, root := ~, start := 350601, stop := 350610 ), 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 := 350621, root := ~, start := 350617, stop := 350620 ), 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 := 350655, root := ~, start := 350627, stop := 350654 ), 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 := 350675, root := ~, start := 350664, stop := 350674 ), 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 := 350725, root := ~, start := 350717, stop := 350724 ), 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 := 350786, root := ~, start := 350776, stop := 350785 ), 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 := 350812, root := ~, start := 350802, stop := 350811 ), 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 := 350822, root := ~, start := 350818, stop := 350821 ), 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 := 350856, root := ~, start := 350828, stop := 350855 ), 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 := 350876, root := ~, start := 350865, stop := 350875 ), 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 := 350926, root := ~, start := 350918, stop := 350925 ), 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 := 350979, root := ~, start := 350965, stop := 350978 ), 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 := 350989, root := ~, start := 350985, stop := 350988 ), 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 := 351035, root := ~, start := 351027, stop := 351034 ), 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 := 351060, root := ~, start := 351041, stop := 351059 ), 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 := 351090, root := ~, start := 351080, stop := 351089 ), 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 := 351114, root := ~, start := 351104, stop := 351113 ), 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 := 351186, root := ~, start := 351178, stop := 351185 ), 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 := 351200, root\ := ~, start := 351192, stop := 351199 ), rec( content := "]", count := [ 7, 3\ , 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 3, 10, 23 ], mathmode := "Text", name := "C", next := 351205, root := ~, star\ t := 351170, stop := 351204 ), 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 := 351238, root := ~, start := 351\ 228, stop := 351237 ), 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 := 351261, root := ~, start := 351248, stop := 351260 ), 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 := 351294, root := ~, start := 351284, stop := 351293 ), rec( conten\ t := "; and ", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 3, 10, 23 ], name := "Item", next := 351307, root := ~, start := 351\ 155, stop := 351306 ), 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 := 351337, root :\ = ~, start := 351329, stop := 351336 ), 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 := 351358, root := ~, start := 35135\ 0, stop := 351357 ), 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 := 351394, root := ~, start := 35138\ 4, stop := 351393 ), rec( content := ".\n ", count := [ 7, 3, 10, 25 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25 ], name := "Ite\ m", next := 351409, root := ~, start := 351314, stop := 351408 ) ], count := [ 7, 3, 10, 22 ], mathmode := "Text", name := "List", next := 351421, root := ~, start := 351142, stop := 351420 ), 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 := 351479, root := ~, start := 351435, stop := 351478 ), 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 := 351484, root := ~, start := 351480, stop := 351483 ), 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 := 351505, root := ~, start := 351494, stop := 351504 ), 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 := 351521, root := ~, start := 351510, stop := 351520 ), 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 := 351536, root := ~, start := 351526, stop := 351535 ), 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 := 351682, root := ~, start := 351606, stop := 351681 ), 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 := 351862, root := ~, start := 351836, stop \ := 351861 ), 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 := 351888, root := ~, start := 351880, stop := 351887 ), 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 := 351906, root :\ = ~, start := 351892, stop := 351905 ), 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\ := 351959, root := ~, start := 351931, stop := 351958 ), rec( content := ".", coun\ t := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ] , name := "Ite\ m", next := 351967, root := ~, start := 351821, stop := 351966 ), 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 := 352017, root := ~, start := 351989, stop \ := 352016 ), 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 := 352043, root := ~, start := 352035, stop := 352042 ), 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 := 352057, root :\ = ~, start := 352047, stop := 352056 ), 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 := 3521\ 11, root := ~, start := 351974, stop := 352110 ) ], count := [ 7, 3, 10, 29 ], mathmode := "Text", name := "List", next := 352123, root := ~, start := 351808, stop := 352122 ), 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 := 352177, root := ~, start := 352138, stop := 352176 ), 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 := 353379, root := ~, start := 352188, stop := 353378 ), rec( content := "\n ", count := [ 7, 3, 10, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Description", next := 353396, root := ~, start := 350219, stop := 353395 ) ], count := [ 7, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 353410, root := ~, start := 349400, stop := 353409 ), 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 := 353547, root := ~, start := 353431, stop := 353546 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEmbedding" ), content := 0, count := [ 7, 3, 11, 4 ], mathmode := "Text", name := "Oper", next := 353613, root := ~, start := 353550, stop := 353612 ), 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 := 353636, root := ~, start := 353625, stop := 353635 ), 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 := 353652, root := ~, start := 353640, stop := 353651 ), rec( content := ".", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 353663, root := ~, start := 353616, stop := 353662 ), 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 := 353709, root := ~, start := 353684, stop := 353708 ), 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 := 353729, root := ~, start := 353718, stop := 353728 ), 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 := 353779, root := ~, start := 353771, stop := 353778 ), 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 := 353824, root := ~, start := 353814, stop := 353823 ), 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 := 353852, root := ~, start := 353842, stop := 353851 ), 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 := 353901, root := ~, start := 353890, stop := 353900 ), 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 := 353917, root := ~, start := 353906, stop := 353916 ), 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 := 353936, root := ~, start := 353932, stop := 353935 ), 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 := 353982, root := ~, start := 353974, stop := 353981 ), 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 := 354004, root := ~, start := 353988, stop := 354003 ), 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 := 354032, root := ~, start := 354022, stop := 354031 ), 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 := 354058, root := ~, start := 354048, stop := 354057 ), 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 := 354070, root := ~, start := 354062, stop := 354069 ), 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 := 354108, root := ~, start := 354098, stop := 354107 ), 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 := 354122, root := ~, start := 354112, stop := 354121 ), 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 := 354150, root := ~, start := 354142, stop := 354149 ), 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 := 354206, root := ~, start := 354196, stop := 354205 ), 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 := 354239, root := ~, start := 354231, stop := 354238 ), 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 := 354257, root := ~, start := 354247, stop := 354256 ), 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 := 354308, root := ~, start := 354273, stop := 354307 ), 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 := 354313, root := ~, start := 354309, stop := 354312 ), 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 := 354962, root := ~, start := 354319, stop := 354961 ), rec( content := "\n ", count := [ 7, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 354979, root := ~, start := 353666, stop := 354978 ) ], count := [ 7, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 354993, root := ~, start := 353416, stop := 354992 ), 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 := 355088, root := ~, start := 355018, stop := 355087 ), 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 := 355114, root := ~, start := 355103, stop := 355113 ), 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 := 355130, root := ~, start := 355118, stop := 355129 ), rec( content := ". ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 355142, root := ~, start := 355093, stop := 355141 ), 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 := 355213, root := ~, start := 355181, stop := 355212 ), 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 := 355287, root := ~, start := 355279, stop := 355286 ), 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 := 355336, root := ~, start := 355325, stop := 355335 ), 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 := 355352, root := ~, start := 355341, stop := 355351 ), 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 := 355379, root := ~, start := 355369, stop := 355378 ), 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 := 355396, root := ~, start := 355384, stop := 355395 ), 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 := 355445, root := ~, start := 355413, stop := 355444 ), 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 := 355465, root := ~, start := 355454, stop := 355464 ), 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 := 355512, root := ~, start := 355504, stop := 355511 ), 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 := 355526, root := ~, start := 355516, stop := 355525 ), 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 := 355556, root := ~, start := 355528, stop := 355555 ), 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 := 355568, root := ~, start := 355564, stop := 355567 ), 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 := 355927, root := ~, start := 355576, stop := 355926 ), rec( content := "\n ", count := [ 7, 3, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 355946, root := ~, start := 355147, stop := 355945 ) ], count := [ 7, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 355962, root := ~, start := 355001, stop := 355961 ), 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 := 356061, root := ~, start := 355983, stop := 356060 ), rec( attributes := rec( Arg := "digraph", Name := "GeneratorsOfEnd\ omorphismMonoidAttr" ), content := 0, count := [ 7, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 356127, root := ~, start := 356064, stop := 356126 ), 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 := 356176, root := ~, start := 356130, stop := 356175 ), 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 := 356230, root := ~, start := 356216, stop := 356229 ), 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 := 356286, root := ~, start := 356253, stop := 356285 ), 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 := 356306, root := ~, start := 356292, stop := 356305 ), 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 := 356365, root := ~, start := 356328, stop := 356364 ), 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 := 356484, root := ~, start := 356470, stop := 356483 ), 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 := 356503, root := ~, start := 356489, stop := 356502 ), 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 := 356554, root := ~, start := 356516, stop := 356553 ), 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 := 356615, root := ~, start := 356578, stop := 356614 ), 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 := 356663, root := ~, start := 356659, stop := 356662 ), 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 := 356689, root := ~, start := 356676, stop := 356688 ), 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 := 356759, root := ~, start := 356722, stop := 356758 ), 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 := 356891, root := ~, start := 356878, stop := 356890 ), 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 := 356924, root := ~, start := 356920, stop := 356923 ), 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 := 357042, root := ~, start := 357028, stop := 357041 ), 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 := 357\ 063, root := ~, start := 357050, stop := 357062 ), 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 := 357073, root := ~, start := 357063, stop := 357072 ), 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 := 357114, root := ~, start := 357106, stop := 357113 ), rec( cont\ ent := ".\n ", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], name := "Item", next := 357129, root := ~, start := 356\ 943, stop := 357128 ), 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 := 357191, root := ~, start := 357178, stop := 357190 ), 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 := 357201, root := ~, start := 357\ 191, stop := 357200 ), 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 := 357256, root :=\ ~, start := 357248, stop := 357255 ), rec( content := ".\n ", count := [ \ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 11 ] , name := "Ite\ m", next := 357271, root := ~, start := 357136, stop := 357270 ) ], count := [ 7, 3, 13, 8 ], mathmode := "Text", name := "List", next := 357283, root := ~, start := 356930, stop := 357282 ), 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 := 357308, root := ~, start := 357296, stop := 357307 ), 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 := 357383, root := ~, start := 357371, stop := 357382 ), 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 := 357417, root := ~, start := 357405, stop := 357416 ), 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 := 357467, root := ~, start := 357452, stop := 357466 ), 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 := 357473, root := ~, start := 357469, stop := 357472 ), 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 := 358342, root := ~, start := 357479, stop := 358341 ), rec( content := "\n ", count := [ 7, 3, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Description", next := 358359, root := ~, start := 356179, stop := 358358 ) ], count := [ 7, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 358373, root := ~, start := 355968, stop := 358372 ), 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 := 358492, root := ~, start := 358394, stop := 358491 ), 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 := 358537, root := ~, start := 358526, stop := 358536 ), rec( content := ".", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 358548, root := ~, start := 358495, stop := 358547 ), 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 := 358594, root := ~, start := 358571, stop := 358593 ), 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 := 358729, root := ~, start := 358721, stop := 35872\ 8 ), rec( content := "-colouring", count := [ 7, 3, 14, 3 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "E", next := 358743, root := ~, start := 358705, stop := 358742 ), 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 := 358792, root := ~, start := 358784, stop := 358791 ), 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 := 358839, root := ~, start := 358831, stop := 358838 ), 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 := 358923, root := ~, start := 358891, stop := 358922 ), 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 := 358983, root := ~, start := 358975, stop := 358982 ), 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 := 359034, root := ~, start := 358999, stop := 359033 ), 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 := 359104, root := ~, start := 359069, stop := 359103 ), 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 := 359137, root := ~, start := 359129, stop := 359136 ), 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 := 359174, root := ~, start := 359166, stop := 359173 ), 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 := 359184, root := ~, start := 359180, stop := 359183 ), 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 := 359207, root := ~, start := 359193, stop := 359206 ), 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 := 359233, root := ~, start := 359225, stop := 359232 ), 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 := 3593\ 04, root := ~, start := 359290, stop := 359303 ), 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 := 3593\ 14, root := ~, start := 359306, stop := 359313 ), rec( content := ")", count := [ 7, 3, 14, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "C", next := 359319, root := ~, start := 359270, stop := 359318 ), 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 := 359366, root := ~, start := 359352, stop := 359365 ), 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 := 359406, root := ~, start := 359398, stop := 359405 ), 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 := 359462, root := ~, start := 359451, stop := 359461 ), 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 := 359468, root := ~, start := 359464, stop := 359467 ), 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 := 359541, root := ~, start := 359483, stop := 359540 ), 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 := 359555, root := ~, start := 359551, stop := 359554 ), 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 := 359698, root := ~, start := 359666, stop := 359697 ), 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 := 360071, root := ~, start := 359705, stop := 360070 ), rec( content := "\n ", count := [ 7, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 360088, root := ~, start := 358551, stop := 360087 ) ], count := [ 7, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 360102, root := ~, start := 358379, stop := 360101 ), 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 := 360238, root := ~, start := 360123, stop := 360237 ), 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 := 360364, root := ~, start := 360241, stop := 360363 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "DigraphGreedyCo\ louring" ), content := 0, count := [ 7, 3, 15, 6 ], mathmode := "Text", name := "Attr", next := 360458, root := ~, start := 360367, stop := 360457 ), 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 := 360503, root := ~, start := 360492, stop := 360502 ), rec( content := ".", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Returns", next := 360514, root := ~, start := 360461, stop := 360513 ), 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 := 360560, root := ~, start := 360537, stop := 360559 ), 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 := 360735, root := ~, start := 360706, stop := 360734 ), 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 := 360786, root := ~, start := 360782, stop := 360785 ), 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 := 360813, root := ~, start := 360799, stop := 360812 ), 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 := 360843, root := ~, start := 360831, stop := 360842 ), 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 := 360916, root := ~, start := 360902, stop := 360915 ), 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 := 360973, root := ~, start := 360944, stop := 360972 ), 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 := 361080, root := ~, start := 361066, stop := 361079 ), 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 := 361139, root := ~, start := 361135, stop := 361138 ), 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 := 361166, root := ~, start := 361152, stop := 361165 ), 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 := 361195, root := ~, start := 361184, stop := 361194 ), 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 := 361286, root := ~, start := 361274, stop := 361285 ), 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 := 3613\ 38, root := ~, start := 361324, stop := 361337 ), 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\ 51, root := ~, start := 361340, stop := 361350 ), rec( content := ")", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 361356, root := ~, start := 361298, stop := 361355 ), 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 := 3614\ 09, root := ~, start := 361395, stop := 361408 ), 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 := 3614\ 22, root := ~, start := 361411, stop := 361421 ), 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 := 3614\ 37, root := ~, start := 361423, stop := 361436 ), rec( content := "))", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 361443, root := ~, start := 361369, stop := 361442 ), 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 := 361453, root := ~, start := 361449, stop := 361452 ), 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 := 361587, root := ~, start := 361550, stop := 361586 ), 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 := 361617, root := ~, start := 361613, stop := 361616 ), 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 := 361726, root := ~, start := 361640, stop := 361725 ), 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 := 361736, root := ~, start := 361732, stop := 361735 ), 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 := 361981, root := ~, start := 361742, stop := 361980 ), rec( content := "\n ", count := [ 7, 3, 15, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Description", next := 361998, root := ~, start := 360517, stop := 361997 ) ], count := [ 7, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 362012, root := ~, start := 360108, stop := 362011 ), 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 := 362085, root := ~, start := 362033, stop := 362084 ), 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 := 362131, root := ~, start := 362088, stop := 362130 ), 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 := 362182, root := ~, start := 362152, stop := 362181 ), 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 := 362255, root := ~, start := 362241, stop := 362254 ), 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 := 362364, root := ~, start := 362360, stop := 362363 ), 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 := 362614, root := ~, start := 362370, stop := 362613 ), rec( content := "\n ", count := [ 7, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "Description", next := 362631, root := ~, start := 362134, stop := 362630 ) ], count := [ 7, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 362645, root := ~, start := 362018, stop := 362644 ), 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 := 362710, root := ~, start := 362666, stop := 362709 ), 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 := 362756, root := ~, start := 362713, stop := 362755 ), 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 := 362802, root := ~, start := 362779, stop := 362801 ), 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 := 363017, root := ~, start := 362979, stop := 363016 ), 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 := 363062, root := ~, start := 363058, stop := 363061 ), 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 := 363085, root := ~, start := 363071, stop := 363084 ), 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 := 363153, root := ~, start := 363118, stop := 363152 ), 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 := 363182, root := ~, start := 363160, stop := 363181 ), 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 := 363234, root := ~, start := 363226, stop := 363233 ), 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 := 363294, root := ~, start := 363280, stop := 363293 ), 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 := 363308, root := ~, start := 363300, stop := 363307 ), 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 := 363401, root := ~, start := 363379, stop := 363400 ), 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 := 363435, root := ~, start := 363427, stop := 363434 ), 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 := 3634\ 84, root := ~, start := 363470, stop := 363483 ), rec( content := ", n)", count := [ 7, 3, 17, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 363492, root := ~, start := 363450, stop := 363491 ), 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 := 363524, root := ~, start := 363513, stop := 363523 ), 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 := 363612, root := ~, start := 363530, stop := 363611 ), 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 := 364011, root := ~, start := 363619, stop := 364010 ), rec( content := "\n ", count := [ 7, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 364028, root := ~, start := 362759, stop := 364027 ) ], count := [ 7, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 364042, root := ~, start := 362651, stop := 364041 ), 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 := 364105, root := ~, start := 364067, stop := 364104 ), 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 := 364157, root := ~, start := 364110, stop := 364156 ), 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 := 364193, root := ~, start := 364185, stop := 364192 ), 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 := 364231, root := ~, start := 364213, stop := 364230 ), 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 := 364297, root := ~, start := 364286, stop := 364296 ), 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 := 364309, root := ~, start := 364301, stop := 364308 ), 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 := 364358, root := ~, start := 364350, stop := 364357 ), 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 := 364417, root := ~, start := 364409, stop := 364416 ), 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 := 364422, root := ~, start := 364418, stop := 364421 ), 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 := 364443, root := ~, start := 364432, stop := 364442 ), 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 := 364465, root := ~, start := 364457, stop := 364464 ), 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 := 364500, root := ~, start := 364492, stop := 364499 ), 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 := 364516, root := ~, start := 364508, stop := 364515 ), 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 := 364557, root := ~, start := 364549, stop := 364556 ), 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 := 365041, root := ~, start := 364621, stop := 365040 ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 365060, root := ~, start := 364162, stop := 365059 ) ], count := [ 7, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 365076, root := ~, start := 364050, stop := 365075 ), rec( content := "\n\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 365090, root := ~, start := 332896, stop := 365089 ) , rec( content := "\n\n", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 365102, root := ~, start := 284068, stop := 365101 ), 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 := 365241, root := ~, start := 365186, stop := 365240 ) , 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 := 365277, root := ~, start := 365264, stop := 365276 ), 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 := 365379, root := ~, start := 365357, stop := 365378 ), 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 := 365466, root := ~, start := 365445, stop := 365465 ), 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 := 365576, root := ~, start := 365540, stop := 365575 ), 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 := 365884, root := ~, start := 365854, stop := 365883 ), 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 := 365894, root := ~, start := 365890, stop := 365893 ), 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 := 366532, root := ~, start := 366528, stop := 366531 ), 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 := 366788, root := ~, start := 366753, stop := 366787 ), 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 := 366821, root := ~, start := 366793, stop := 366820 ), 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 := 366916, root := ~, start := 366902, stop := 366915 ), 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 := 366968, root := ~, start := 366954, stop := 366967 ), 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 := 367124, root := ~, start := 367068, stop := 367123 ), 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 := 367138, root := ~, start := 367124, stop := 367137 ), 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 := 367147, root := ~, start := 367138, stop := 367146 ), rec( content := ".\n ", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], name := "Item", next := 367162, root := ~, start := 367053, stop := 367161 ) ], count := [ 8, 0, 0, 6 ], mathmode := "Text", name := "List", next := 367174, root := ~, start := 367040, stop := 367173 ), 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 := 367216, root := ~, start := 367184, stop := 367215 ), 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 := 367270, root := ~, start := 367221, stop := 367269 ), 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 := 367446, root := ~, start := 367434, stop := 367445 ), 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 := 367504, root := ~, start := 367492, stop := 367503 ), 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 := 367575, root := ~, start := 367561, stop := 367574 ), 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 := 367710, root := ~, start := 367664, stop := 367709 ), 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 := 367724, root := ~, start := 367710, stop := 367723 ), 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 := 367732, root := ~, start := 367724, stop := 367731 ), rec( content := ".\n ", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], name := "Item", next := 367747, root := ~, start := 367649, stop := 367746 ) ], count := [ 8, 0, 0, 10 ], mathmode := "Text", name := "List", next := 367759, root := ~, start := 367636, stop := 367758 ), 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 := 367821, root := ~, start := 367769, stop := 367820 ), 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 := 367923, root := ~, start := 367863, stop := 367922 ), 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 := 368003, root := ~, start := 367989, stop := 368002 ), 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 := 368281, root := ~, start := 368267, stop := 368280 ), 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 := 368436, root := ~, start := 368381, stop := 368435 ), 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 := 368461, root := ~, start := 368457, stop := 368460 ), 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 := 368583, root := ~, start := 368546, stop := 368582 ), 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 := 368648, root := ~, start := 368634, stop := 368647 ), 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 := 368835, root := ~, start := 368746, stop := 368834 ), 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 := 368849, root := ~, start := 368835, stop := 368848 ), 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 := 368859, root := ~, start := 368849, stop := 368858 ), rec( content := ".\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], name := "Item", next := 368874, root := ~, start := 368731, stop := 368873 ) ], count := [ 8, 0, 0, 15 ], mathmode := "Text", name := "List", next := 368886, root := ~, start := 368718, stop := 368885 ), 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 := 369001, root := ~, start := 368987, stop := 369000 ), 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 := 369108, root := ~, start := 369083, stop := 369107 ), 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 := 369348, root := ~, start := 369334, stop := 369347 ), 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 := 369481, root := ~, start := 369457, stop := 369480 ), 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 := 369549, root := ~, start := 369545, stop := 369548 ), 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 := 369711, root := ~, start := 369697, stop := 369710 ), 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 := 369794, root := ~, start := 369780, stop := 369793 ), 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 := 369886, root := ~, start := 369882, stop := 369885 ), 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 := 369933, root := ~, start := 369899, stop := 369932 ), 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 := 369991, root := ~, start := 369951, stop := 369990 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalClique\ " ), content := 0, count := [ 8, 1, 1, 4 ], mathmode := "Text", name := "Oper", next := 370039, root := ~, start := 369992, stop := 370038 ), 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 := 370060, root := ~, start := 370049, stop := 370059 ), 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 := 370076, root := ~, start := 370064, stop := 370075 ), rec( content := ".", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Returns", next := 370087, root := ~, start := 370040, stop := 370086 ), 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 := 370121, root := ~, start := 370107, stop := 370120 ), 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 := 370147, root := ~, start := 370139, stop := 370146 ), 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 := 370204, root := ~, start := 370190, stop := 370203 ), 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 := 370229, root := ~, start := 370213, stop := 370228 ), 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 := 370243, root := ~, start := 370229, stop := 370242 ), 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 := 370251, root := ~, start := 370243, stop := 370250 ), 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 := 370259, root := ~, start := 370251, stop := 370258 ), 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 := 370267, root := ~, start := 370259, stop := 370266 ), 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 := 370287, root := ~, start := 370276, stop := 370286 ), 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 := 370303, root := ~, start := 370295, stop := 370302 ), 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 := 370322, root := ~, start := 370309, stop := 370321 ), 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 := 370340, root := ~, start := 370326, stop := 370339 ), 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 := 370357, root := ~, start := 370345, stop := 370356 ), 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 := 370415, root := ~, start := 370392, stop := 370414 ), 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 := 370429, root := ~, start := 370415, stop := 370428 ), 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 := 370437, root := ~, start := 370429, stop := 370436 ), 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 := 370445, root := ~, start := 370437, stop := 370444 ), 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 := 370453, root := ~, start := 370445, stop := 370452 ), 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 := 370477, root := ~, start := 370466, stop := 370476 ), 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 := 370489, root := ~, start := 370481, stop := 370488 ), 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 := 370516, root := ~, start := 370495, stop := 370515 ), 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 := 370534, root := ~, start := 370520, stop := 370533 ), 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 := 370555, root := ~, start := 370543, stop := 370554 ), 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 := 370575, root := ~, start := 370571, stop := 370574 ), 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 := 370596, root := ~, start := 370583, stop := 370595 ), 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 := 370692, root := ~, start := 370671, stop := 370691 ), 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 := 371314, root := ~, start := 370854, stop := 371313 ), rec( content := "\n ", count := [ 8, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 371331, root := ~, start := 370088, stop := 371330 ) ], count := [ 8, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 371345, root := ~, start := 369938, stop := 371344 ), 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 := 371472, root := ~, start := 371366, stop := 371471 ), 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 := 371514, root := ~, start := 371497, stop := 371513 ), rec( content := ".", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Returns", next := 371525, root := ~, start := 371475, stop := 371524 ), 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 := 371603, root := ~, start := 371589, stop := 371602 ), 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 := 371791, root := ~, start := 371787, stop := 371790 ), 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 := 371809, root := ~, start := 371801, stop := 371808 ), 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 := 371897, root := ~, start := 371883, stop := 371896 ), 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 := 371976, root := ~, start := 371917, stop := 371975 ), 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 := 372039, root := ~, start := 371981, stop := 372038 ), 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\ 2077, root := ~, start := 372066, stop := 372076 ) ], count := [ 8, 1, 2, 5 ], name := "Mar\ k", next := 372084, root := ~, start := 372060, stop := 372083 ), 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 := 372155, root := \ ~, start := 372144, stop := 372154 ), 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\ := 372160, root := ~, start := 372156, stop := 372159 ), 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 := 372184, root := \ ~, start := 372173, stop := 372183 ), 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 := 372259, root := ~, start := 372242, stop := 372258 ), 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\ := 372293, root := ~, start := 372285, stop := 372292 ), 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 := 372330, root := ~, start := 372319, stop\ := 372329 ), 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 := 372348, root := ~, start := 372331, stop := 372347 ), 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 :\ = 372356, root := ~, start := 372316, stop := 372355 ), 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 := 372407, root := ~, start := 372399, stop := 372406 ), 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 \ := 372440, root := ~, start := 372420, stop := 372439 ), 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 := 372445, root := ~, start := 372441, stop := 372444 ), 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\ 72469, root := ~, start := 372458, stop := 372468 ), 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 := 372484, root := ~, start := 372473, stop \ := 372483 ), 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 := 372589, root := ~, start := 372569, stop := 372588 ), 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 \ := 372618, root := ~, start := 372601, stop := 372617 ), 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 := 372668, root := ~, start := 372091, stop := 372667 ), 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\ := 372699, root := ~, start := 372682, stop := 372698 ) ], count := [ 8, 1, 2, 9 ], name := "Mar\ k", next := 372706, root := ~, start := 372676, stop := 372705 ), 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 := 372742, root := ~, start := 372731, stop := 3\ 72741 ), 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 := 372780, root := ~, start := 372763, stop := 37277\ 9 ), 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\ := 372842, root := ~, start := 372825, stop := 372841 ), 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 := 372909, root := ~, start := 372898, stop := 3\ 72908 ), 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 := 372941, root := ~, start := 372924, stop := 372940 ) , 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 := 372985, root := ~, \ start := 372974, stop := 372984 ), 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 := 373003, root := ~, start := 372986, stop\ := 373002 ), rec( content := ", c)", count := [ 8, 1, 2, 10 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "C", next := 373011, root := ~, start := 372971, stop := 373010\ ), 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 := 373061, root := ~, start := 373053, stop \ := 373060 ), 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 := 373091, root := ~, start := 373074, stop := 373090\ ), 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 := 373097, root := ~, star\ t := 373093, stop := 373096 ), 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 := 373134, root := ~, start := 3731\ 23, stop := 373133 ), 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 := 373149, root := ~, start := 373138, stop := 373148 ), 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 := 373194, root := ~, start := 373177, stop := 373193 ), 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 := 3732\ 24, root := ~, start := 372713, stop := 373223 ), 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 := 373250, root := ~, start := 373\ 238, stop := 373249 ) ], count := [ 8, 1, 2, 12 ], name := "Mark", next := 3732\ 57, root := ~, start := 373232, stop := 373256 ), 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 := 373340, root := ~, start := 373\ 325, stop := 373339 ), 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 := 373370, root := ~, start := 373350, stop := 37\ 3369 ), 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 := 373422, root := ~, start := 373410, stop \ := 373421 ), 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 := 373471, root := ~, start := 373264, stop := 373470 ), 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\ := 373499, root := ~, start := 373485, stop := 373498 ), 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 := 373518, root := \ ~, start := 373504, stop := 373517 ) ], count := [ 8, 1, 2, 14 ], name := "Mar\ k", next := 373525, root := ~, start := 373479, stop := 373524 ), 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 := 373654, root := ~, start := 3\ 73640, stop := 373653 ), 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 := 373737, root := ~, start := 373723, stop := 373736 ), 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 := 373744, root := ~, start := 373740, stop \ := 373743 ), 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 := 373774, root := ~, start := 373754, stop := 3737\ 73 ), 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\ 3858, root := ~, start := 373844, stop := 373857 ), 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 := 373920, root := ~, start := 373906, stop := 373919 ), 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 := 373926, root := ~, start := 373922, stop \ := 373925 ), 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 := 374056, root := ~, start := 374048, stop \ := 374055 ), 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 := 374091, root := ~, start := 373532, stop := 374090 ), 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 := \ 374115, root := ~, start := 374105, stop := 374114 ) ], count := [ 8, 1, 2, 18 ], name := "Mar\ k", next := 374122, root := ~, start := 374099, stop := 374121 ), 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 := 374179, root := ~, start := 3741\ 68, stop := 374178 ), 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 := 374195, root := ~, start := 374183, stop := 374194 ), 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 := 374211, root :=\ ~, start := 374201, stop := 374210 ), 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 := 374253, root := ~, star\ t := 374233, stop := 374252 ), 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 := 374289, root := ~, start := 3\ 74275, stop := 374288 ), 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 := 374320, root := ~, start := 374310, stop \ := 374319 ), 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 := 374336, root := ~, start := 374324, stop := 374335 ), 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 := 374397, root := ~, start := 374129, stop := 374396 ), 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 :=\ 374422, root := ~, start := 374411, stop := 374421 ) ], count := [ 8, 1, 2, 20 ], name := "Mar\ k", next := 374429, root := ~, start := 374405, stop := 374428 ), 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 := 374486, root := ~, start := 3744\ 75, stop := 374485 ), 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 := 374532, root := ~, start := 3745\ 21, stop := 374531 ), 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 := 374580, root := ~, star\ t := 374560, stop := 374579 ), 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 := 374652, root := ~, start := 374641, stop := 374651 ), 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\ := 374685, root := ~, start := 374674, stop := 374684 ), 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 := 374700, root := ~, start := 3746\ 89, stop := 374699 ), 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 := 374753, root := ~, start := 374\ 436, stop := 374752 ), 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 := 374778, root := ~, start := 374767, stop \ := 374777 ) ], count := [ 8, 1, 2, 22 ], name := "Mark", next := 374785, root := ~, start := 374\ 761, stop := 374784 ), 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 := 374842, root := ~, start := 374831, stop := 374841 ), 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 := 374858, root :=\ ~, start := 374846, stop := 374857 ), 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\ := 374863, root := ~, start := 374859, stop := 374862 ), 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 := 374887, root :=\ ~, start := 374876, stop := 374886 ), 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 := 374902, root := ~, start := 374891, stop \ := 374901 ), 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 := 374936, root := ~, start := 374922, stop := 374935 ) , 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\ := 374963, root := ~, start := 374949, stop := 374962 ), 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 := 375034, root := ~, start := 375026, stop \ := 375033 ), 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 := 375091, root := ~, star\ t := 375071, stop := 375090 ), 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 := 375199, root := ~, start := 375191, stop \ := 375198 ), 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\ := 375285, root := ~, start := 375201, stop := 375284 ), 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 := 375309, root :=\ ~, start := 375298, stop := 375308 ), 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 := 375344, root := ~, star\ t := 375324, stop := 375343 ), 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 := 375395, root := ~, start :=\ 375391, stop := 375394 ), 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 := 375437, root := ~, star\ t := 375429, stop := 375436 ), 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 := 3755\ 81, root := ~, start := 374792, stop := 375580 ) ], count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "List", next := 375593, root := ~, start := 372047, stop := 375592 ), 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 := 377318, root := ~, start := 375664, stop := 377317 ), rec( content := "\n ", count := [ 8, 1, 2, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 377335, root := ~, start := 371528, stop := 377334 ) ], count := [ 8, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 377349, root := ~, start := 371351, stop := 377348 ), 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 := 377442, root := ~, start := 377370, stop := 377441 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalClique" ), content := 0, count := [ 8, 1, 3, 4 ], mathmode := "Text", name := "Func", next := 377528, root := ~, start := 377445, stop := 377527 ), 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 := 377594, root := ~, start := 377583, stop := 377593 ), rec( content := ".", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Returns", next := 377605, root := ~, start := 377531, stop := 377604 ), 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 := 377643, root := ~, start := 377629, stop := 377642 ), 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 := 377717, root := ~, start := 377703, stop := 377716 ), 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 := 377793, root := ~, start := 377782, stop := 377792 ), 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 := 377887, root := ~, start := 377865, stop := 377886 ), 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 := 377921, root := ~, start := 377892, stop := 377920 ), 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 := 377926, root := ~, start := 377922, stop := 377925 ), 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 := 377969, root := ~, start := 377955, stop := 377968 ), 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 := 377988, root := ~, start := 377974, stop := 377987 ), 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 := 378074, root := ~, start := 378060, stop := 378073 ), 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 := 378117, root := ~, start := 378106, stop := 378116 ), 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 := 378177, root := ~, start := 378163, stop := 378176 ), 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 := 378196, root := ~, start := 378182, stop := 378195 ), 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 := 378270, root := ~, start := 378256, stop := 378269 ), 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 := 378315, root := ~, start := 378301, stop := 378314 ), 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 := 378371, root := ~, start := 378357, stop := 378370 ), 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 := 378404, root := ~, start := 378393, stop := 378403 ), 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 := 378500, root := ~, start := 378489, stop := 378499 ), 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 := 378514, root := ~, start := 378510, stop := 378513 ), 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 := 378537, root := ~, start := 378523, stop := 378536 ), 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 := 378594, root := ~, start := 378583, stop := 378593 ), 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 := 378747, root := ~, start := 378714, stop := 378746 ), 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 :=\ 378832, root := ~, start := 378812, stop := 378831 ), 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 := \ 378872, root := ~, start := 378845, stop := 378871 ), 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 :=\ 378915, root := ~, start := 378901, stop := 378914 ), 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 := 379113, root :\ = ~, start := 379078, stop := 379112 ), 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 := 379148, root := ~, start := 378\ 754, stop := 379147 ), 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 := 379183 , root := ~, start := 379\ 155, stop := 379182 ), 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 := 379263, root := ~, start := 379243, stop := 37\ 9262 ), 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 := 379315, root := ~, start := 3793\ 01, stop := 379314 ), 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 := 379395, root\ := ~, start := 379391, stop := 379394 ), 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 := 379455, root := ~, start := 379428, stop := 379454 ), 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 := 379532, root := ~, start := \ 379503, stop := 379531 ), 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\ := 379546, root := ~, start := 379532, stop := 379545 ), 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 := 379563, root := ~, start := 379546, stop \ := 379562 ), 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 := 379577, root\ := ~, start := 379563, stop := 379576 ), 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 := 379586, root := ~, start := 379577, stop := 379585 ), 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 := 379600, root := ~, start := 3\ 79586, stop := 379599 ), 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 :\ = 379611, root := ~, start := 379600, stop := 379610 ), 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\ := 379651, root := ~, start := 379640, stop := 379650 ), rec( content := ".\n\ ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 8, 1, 3, 11 ], name := "Item", next := 379666, root := ~, start := 379\ 190, stop := 379665 ), 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 := 379700, root := ~, start := 379\ 673, stop := 379699 ), 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 := 379779, root := ~, start := 379759, stop := 37\ 9778 ), 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 := 379841, root := ~, start\ := 379819, stop := 379840 ), 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\ := 379855, root := ~, start := 379841, stop := 379854 ), 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 := 379874, root := ~, start := 379855, stop \ := 379873 ), 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 := 379888, root\ := ~, start := 379874, stop := 379887 ), 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 := 379897, root := ~, start := 379888, stop := 379896 ), 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 := 379911, root := ~, start := 3\ 79897, stop := 379910 ), 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 := 379931, root := ~, start := 379911, stop := 379930 ), 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 := 379942, root := ~, start := 379931, stop \ := 379941 ), 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 := 379950, root :=\ ~, start := 379942, stop := 379949 ), 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 := 379982, root := ~, start := 3799\ 71, stop := 379981 ), 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 := 380027, root := ~, start := 380023, stop :\ = 380026 ), 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 := 380077, root := ~, start := 380050, stop \ := 380076 ), 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\ := 380162, root := ~, start := 380133, stop := 380161 ), 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 := 380176, root := ~, start := 380162, stop \ := 380175 ), 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 := 380185, root :=\ ~, start := 380176, stop := 380184 ), 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 := 380199, root := ~, start := 380185, stop := 380198 ), 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 := 380216, root := ~, start \ := 380199, stop := 380215 ), 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\ := 380230, root := ~, start := 380216, stop := 380229 ), 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 := 380242, root := ~, start := 380230, stop :=\ 380241 ), 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 := 380253, root :=\ ~, start := 380242, stop := 380252 ), 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 := 380261, root := ~, start := 380253, stop := 380260 ), 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 := 380301, root := ~, start := 380290, stop := 380300 ), rec( conten\ t := ".\n ", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 3, 14 ], name := "Item", next := 380316, root := ~, start := 379\ 707, stop := 380315 ) ], count := [ 8, 1, 3, 8 ], mathmode := "Text", name := "List", next := 380328, root := ~, start := 378701, stop := 380327 ), 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 := 381033, root := ~, start := 380334, stop := 381032 ), rec( content := "\n ", count := [ 8, 1, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Description", next := 381050, root := ~, start := 377608, stop := 381049 ) ], count := [ 8, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 381064, root := ~, start := 377355, stop := 381063 ), 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 := 381174, root := ~, start := 381085, stop := 381173 ), 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 := 381274, root := ~, start := 381177, stop := 381273 ), 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 := 381363, root := ~, start := 381277, stop := 381362 ), 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 := 381456, root := ~, start := 381366, stop := 381455 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesAttr" ), content := 0, count := [ 8, 1, 4, 10 ], mathmode := "Text", name := "Attr", next := 381513, root := ~, start := 381459, stop := 381512 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesRepsAttr" ), content := 0, count := [ 8, 1, 4, 12 ], mathmode := "Text", name := "Attr", next := 381574, root := ~, start := 381516, stop := 381573 ), 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 := 381644, root := ~, start := 381577, stop := 381643 ), 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 := 381682, root := ~, start := 381668, stop := 381681 ), 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 := 381768, root := ~, start := 381735, stop := 381767 ), 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 := 381804, root := ~, start := 381790, stop := 381803 ), 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 := 381903, root := ~, start := 381875, stop := 381902 ), 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 := 381937, root := ~, start := 381908, stop := 381936 ), 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 := 381942, root := ~, start := 381938, stop := 381941 ), 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 := 381985, root := ~, start := 381971, stop := 381984 ), 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 := 382004, root := ~, start := 381990, stop := 382003 ), 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 := 382075, root := ~, start := 382061, stop := 382074 ), 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 := 382115, root := ~, start := 382103, stop := 382114 ), 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 := 382168, root := ~, start := 382153, stop := 382167 ), 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 := 382211, root := ~, start := 382200, stop := 382210 ), 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 := 382282, root := ~, start := 382268, stop := 382281 ), 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 := 382301, root := ~, start := 382287, stop := 382300 ), 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 := 382352, root := ~, start := 382340, stop := 382351 ), 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 := 382378, root := ~, start := 382363, stop := 382377 ), 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 := 382384, root := ~, start := 382380, stop := 382383 ), 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 := 382487, root := ~, start := 382475, stop := 382486 ), 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 := 382578, root := ~, start := 382564, stop := 382577 ), 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 := 382639, root := ~, start := 382625, stop := 382638 ), 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 := 382666, root := ~, start := 382655, stop := 382665 ), 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 := 382743, root := ~, start := 382732, stop := 382742 ), 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 := 382805, root := ~, start := 382791, stop := 382804 ), 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 := 382922, root := ~, start := 382918, stop := 382921 ), 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 := 382940, root := ~, start := 382932, stop := 382939 ), 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 := 383024, root := ~, start := 383010, stop := 383023 ), 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 := 383103, root := ~, start := 383044, stop := 383102 ), 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 := 383166, root := ~, start := 383108, stop := 383165 ), 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 := 383216, root := ~, start := 383187, stop := 383215 ), 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 := 383266, root := ~, start := 383238, stop \ := 383265 ), 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 := 383292, root := ~, start := 383271, stop := 383291 ), 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 := 383358, root := ~, start := 383346, stop \ := 383357 ), 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 := 383384, root := ~, start := 383370, stop := 383383 ), 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\ 83424, root := ~, start := 383420, stop := 383423 ), 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 := 383495, root := ~, start := 383481, stop := 383494 ) , 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\ := 383522, root := ~, start := 383508, stop := 383521 ), 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\ := 383565, root := ~, start := 383557, stop := 383564 ), 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 := 383608, root := ~, start := 383223, stop := 383607 ), 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 := 383661, root := ~, start := 383616, stop := 383660 ), 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 := 383722, root := ~, star\ t := 383690, stop := 383721 ), 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 := 383751, root := ~, start := 383726, stop \ := 383750 ), 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 := 383789, root := ~, start := 383775, stop \ := 383788 ), 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 := 383808, root := ~, start := 383794, stop := 383807 ), 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 := 383868, root := ~, start := 383860, stop \ := 383867 ), 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\ 83893, root := ~, start := 383889, stop := 383892 ), 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 := 383961, root := ~, start := 383929, stop := 383960 ), 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 := 383999, root := ~, start := 383974, stop := 383998 ), 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 := 384065, root := ~, start := 384053, stop \ := 384064 ), 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 := 384118, root := ~, start := 384110, stop \ := 384117 ), 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 := 384173, root := ~, start := 3\ 84159, stop := 384172 ), 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 := 384213, root := ~, start := 384209, stop := 384212 ), 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 := 384329, root := ~, start := 384319, stop := 384328 ), 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 := 384456, root := ~, start := 383668, stop := 384455 ) ], count := [ 8, 1, 4, 17 ], mathmode := "Text", name := "List", next := 384468, root := ~, start := 383174, stop := 384467 ), 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 := 385483, root := ~, start := 384474, stop := 385482 ), rec( content := "\n ", count := [ 8, 1, 4, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Description", next := 385500, root := ~, start := 381647, stop := 385499 ) ], count := [ 8, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 385514, root := ~, start := 381070, stop := 385513 ), 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 := 385576, root := ~, start := 385535, stop := 385575 ), 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 := 385621, root := ~, start := 385579, stop := 385620 ), 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 := 385659, root := ~, start := 385645, stop := 385658 ), 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 := 385709, root := ~, start := 385695, stop := 38570\ 8 ), rec( content := ")", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 385714, root := ~, start := 385679, stop := 385713 ), 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 := 385755, root := ~, start := 385747, stop := 385754 ), 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 := 385780, root := ~, start := 385766, stop := 385779 ), 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 := 385816, root := ~, start := 385808, stop := 385815 ), 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 := 385860, root := ~, start := 385856, stop := 385859 ), 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 := 385881, root := ~, start := 385868, stop := 385880 ), 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 := 386359, root := ~, start := 386063, stop := 386358 ), rec( content := "\n", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 386374, root := ~, start := 385624, stop := 386373 ) ], count := [ 8, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 386388, root := ~, start := 385520, stop := 386387 ), rec( content := "\n\n ", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 386402, root := ~, start := 369890, stop := 386401 ) , 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 := 386458, root := ~, start := 386415, stop := 386457 ), 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 := 386526, root := ~, start := 386478, stop := 386525 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalIndepe\ ndentSet" ), content := 0, count := [ 8, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 386584, root := ~, start := 386529, stop := 386583 ), 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 := 386607, root := ~, start := 386596, stop := 386606 ), 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 := 386623, root := ~, start := 386611, stop := 386622 ), rec( content := ".", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 386634, root := ~, start := 386587, stop := 386633 ), 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 := 386672, root := ~, start := 386658, stop := 386671 ), 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 := 386698, root := ~, start := 386690, stop := 386697 ), 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 := 386757, root := ~, start := 386743, stop := 386756 ), 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 := 386792, root := ~, start := 386768, stop := 386791 ), 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 := 386806, root := ~, start := 386792, stop := 386805 ), 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 := 386814, root := ~, start := 386806, stop := 386813 ), 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 := 386822, root := ~, start := 386814, stop := 386821 ), 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 := 386830, root := ~, start := 386822, stop := 386829 ), 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 := 386854, root := ~, start := 386843, stop := 386853 ), 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 := 386866, root := ~, start := 386858, stop := 386865 ), 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 := 386895, root := ~, start := 386873, stop := 386894 ), 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 := 386913, root := ~, start := 386899, stop := 386912 ), 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 := 386934, root := ~, start := 386922, stop := 386933 ), 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 := 386996, root := ~, start := 386965, stop := 386995 ), 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 := 387010, root := ~, start := 386996, stop := 387009 ), 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 := 387018, root := ~, start := 387010, stop := 387017 ), 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 := 387026, root := ~, start := 387018, stop := 387025 ), 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 := 387034, root := ~, start := 387026, stop := 387033 ), 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 := 387058, root := ~, start := 387047, stop := 387057 ), 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 := 387070, root := ~, start := 387062, stop := 387069 ), 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 := 387106, root := ~, start := 387076, stop := 387105 ), 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 := 387128, root := ~, start := 387114, stop := 387127 ), 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 := 387145, root := ~, start := 387133, stop := 387144 ), 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 := 387165, root := ~, start := 387161, stop := 387164 ), 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 := 387196, root := ~, start := 387174, stop := 387195 ), 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 := 387305, root := ~, start := 387275, stop := 387304 ), 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 := 387849, root := ~, start := 387500, stop := 387848 ), rec( content := "\n ", count := [ 8, 2, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 387866, root := ~, start := 386637, stop := 387865 ) ], count := [ 8, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 387880, root := ~, start := 386463, stop := 387879 ), 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 := 387985, root := ~, start := 387901, stop := 387984 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalIndependentSet" ), content := 0, count := [ 8, 2, 2, 4 ], mathmode := "Text", name := "Func", next := 388083, root := ~, start := 387988, stop := 388082 ), 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 := 388149, root := ~, start := 388138, stop := 388148 ), rec( content := ".", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Returns", next := 388160, root := ~, start := 388086, stop := 388159 ), 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 := 388198, root := ~, start := 388184, stop := 388197 ), 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 := 388282, root := ~, start := 388268, stop := 388281 ), 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 := 388358, root := ~, start := 388347, stop := 388357 ), 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 := 388470, root := ~, start := 388440, stop := 388469 ), 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 := 388518, root := ~, start := 388475, stop := 388517 ), 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 := 388523, root := ~, start := 388519, stop := 388522 ), 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 := 388566, root := ~, start := 388552, stop := 388565 ), 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 := 388585, root := ~, start := 388571, stop := 388584 ), 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 := 388671, root := ~, start := 388657, stop := 388670 ), 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 := 388714, root := ~, start := 388703, stop := 388713 ), 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 := 388774, root := ~, start := 388760, stop := 388773 ), 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 := 388793, root := ~, start := 388779, stop := 388792 ), 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 := 388877, root := ~, start := 388863, stop := 388876 ), 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 := 388926, root := ~, start := 388912, stop := 388925 ), 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 := 388978, root := ~, start := 388964, stop := 388977 ), 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 := 389011, root := ~, start := 389000, stop := 389010 ), 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 := 389116, root := ~, start := 389105, stop := 389115 ), 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 := 389130, root := ~, start := 389126, stop := 389129 ), 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 := 389153, root := ~, start := 389139, stop := 389152 ), 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 := 389224, root := ~, start := 389213, stop := 389223 ), 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 := 389374, root := ~, start := 389341, stop := 389373 ), 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\ := 389467, root := ~, start := 389439, stop := 389466 ), 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\ := 389515, root := ~, start := 389480, stop := 389514 ), 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 := 389575, root := ~, start := 389561, stop := 389574 ), 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 := 389792, root := ~, start := 389739, stop := 389791 ), 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 := 3898\ 27, root := ~, start := 389381, stop := 389826 ), 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 := 3898\ 62, root := ~, start := 389834, stop := 389861 ), 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 := 389950, root := ~, start :=\ 389922, stop := 389949 ), 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 := 3900\ 11, root := ~, start := 389997, stop := 390010 ), 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 := 390100, root := ~, start := 390096, stop := 390099 ), 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 := 390168, root := ~, start := 3901\ 33, stop := 390167 ), 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 := 390270, root := ~, start := 390233, stop := 3902\ 69 ), 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 := 390284, root :=\ ~, start := 390270, stop := 390283 ), 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 := 390301, root := ~, start := 390284, stop := 390300 ), 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 := 390315, root := ~, start := 3\ 90301, stop := 390314 ), 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 := \ 390324, root := ~, start := 390315, stop := 390323 ), 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 := 390338, root := ~, start := 390324, stop := \ 390337 ), 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 := 390349, root :=\ ~, start := 390338, stop := 390348 ), 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 := 390389, root :=\ ~, start := 390378, stop := 390388 ), rec( content := ".\n ", count := [ \ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 11 ], name := "Ite\ m", next := 390404, root := ~, start := 389869, stop := 390403 ), 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 := 390438, root := ~, start := 390411, stop := 390437 ), 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 \ := 390525, root := ~, start := 390497, stop := 390524 ), 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 := 390612, root := ~, start \ := 390582, stop := 390611 ), 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\ := 390626, root := ~, start := 390612, stop := 390625 ), 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 := 390635, root := ~, start := 390626, stop := 39\ 0634 ), 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 := 390649, root :=\ ~, start := 390635, stop := 390648 ), 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 := 390666, root := ~, start := 390649, stop := 390665 ), 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 := 390680, root := ~, start := 3\ 90666, stop := 390679 ), 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 \ := 390692, root := ~, start := 390680, stop := 390691 ), 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 := 390703, root := ~, start := 390692, stop := \ 390702 ), 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 := 390711, root := \ ~, start := 390703, stop := 390710 ), 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 := 390751, root :=\ ~, start := 390740, stop := 390750 ), 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 := 390797, root := ~, start \ := 390793, stop := 390796 ), 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 := 390863, root :=\ ~, start := 390828, stop := 390862 ), 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 := 390965, root := ~, start := 3\ 90928, stop := 390964 ), 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\ := 390979, root := ~, start := 390965, stop := 390978 ), 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 := 390998, root := ~, start := 390979, stop \ := 390997 ), 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 := 391012, root\ := ~, start := 390998, stop := 391011 ), 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 := 391021, root := ~, start := 391012, stop := 391020 ), 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 := 391035, root := ~, start := 3\ 91021, stop := 391034 ), 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 := 391055, root := ~, start := 391035, stop := 391054 ), 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 := 391066, root := ~, start := 391055, stop \ := 391065 ), 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 := 391074, root :=\ ~, start := 391066, stop := 391073 ), 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 := 391106, root := ~, start := 3910\ 95, stop := 391105 ), rec( content := ".\n ", count := [ 8, 2, 2, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], name := "Ite\ m", next := 391121, root := ~, start := 390445, stop := 391120 ) ], count := [ 8, 2, 2, 8 ], mathmode := "Text", name := "List", next := 391133, root := ~, start := 389328, stop := 391132 ), 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 := 392286, root := ~, start := 391138, stop := 392285 ), rec( content := "\n ", count := [ 8, 2, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Description", next := 392303, root := ~, start := 388163, stop := 392302 ) ], count := [ 8, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 392317, root := ~, start := 387886, stop := 392316 ), 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 := 392439, root := ~, start := 392338, stop := 392438 ), 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 := 392547, root := ~, start := 392442, stop := 392546 ), 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 := 392644, root := ~, start := 392550, stop := 392643 ), 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 := 392745, root := ~, start := 392647, stop := 392744 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsAttr" ), content := 0, count := [ 8, 2, 3, 10 ], mathmode := "Text", name := "Attr", next := 392810, root := ~, start := 392748, stop := 392809 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsRepsAttr" ), content := 0, count := [ 8, 2, 3, 12 ], mathmode := "Text", name := "Attr", next := 392879, root := ~, start := 392813, stop := 392878 ), 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 := 392949, root := ~, start := 392882, stop := 392948 ), 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 := 392987, root := ~, start := 392973, stop := 392986 ), 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 := 393073, root := ~, start := 393040, stop := 393072 ), 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 := 393118, root := ~, start := 393104, stop := 393117 ), 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 := 393240, root := ~, start := 393203, stop := 393239 ), 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 := 393281, root := ~, start := 393245, stop := 393280 ), 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 := 393286, root := ~, start := 393282, stop := 393285 ), 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 := 393329, root := ~, start := 393315, stop := 393328 ), 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 := 393348, root := ~, start := 393334, stop := 393347 ), 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 := 393419, root := ~, start := 393405, stop := 393418 ), 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 := 393459, root := ~, start := 393447, stop := 393458 ), 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 := 393512, root := ~, start := 393497, stop := 393511 ), 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 := 393555, root := ~, start := 393544, stop := 393554 ), 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 := 393626, root := ~, start := 393612, stop := 393625 ), 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 := 393645, root := ~, start := 393631, stop := 393644 ), 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 := 393696, root := ~, start := 393684, stop := 393695 ), 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 := 393722, root := ~, start := 393707, stop := 393721 ), 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 := 393728, root := ~, start := 393724, stop := 393727 ), 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 := 393840, root := ~, start := 393828, stop := 393839 ), 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 := 393940, root := ~, start := 393926, stop := 393939 ), 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 := 394001, root := ~, start := 393987, stop := 394000 ), 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 := 394026, root := ~, start := 394015, stop := 394025 ), 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 := 394103, root := ~, start := 394092, stop := 394102 ), 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 := 394166, root := ~, start := 394152, stop := 394165 ), 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 := 394305, root := ~, start := 394301, stop := 394304 ), 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 := 394323, root := ~, start := 394315, stop := 394322 ), 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 := 394388, root := ~, start := 394335, stop := 394387 ), 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 := 394437, root := ~, start := 394400, stop := 394436 ), 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 := 394483, root := ~, start := 394469, stop := 394482 ), 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 := 394598, root := ~, start := 394560, stop := 394597 ), 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 := 394656, root := ~, start := 39\ 4620, stop := 394655 ), 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 := 394690, root := ~, start := 394661, stop \ := 394689 ), 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 := 394756, root :=\ ~, start := 394744, stop := 394755 ), 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 := 394799, root :=\ ~, start := 394785, stop := 394798 ), 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 := 394831, root := ~, start := 394827, stop \ := 394830 ), 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 := 394902, root :=\ ~, start := 394888, stop := 394901 ), 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 := 394929, root := ~, start := 3\ 94915, stop := 394928 ), 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 := 394972, root := ~, start := 394964, stop \ := 394971 ), 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 := 395015, root := ~, start := 394605, stop := 395014 ), 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 := 395090, root := ~, start := 395\ 023, stop := 395089 ), 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 := 395159, root := ~, start := 395119, stop := 395158\ ), 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 := 395204, root := ~, start := 395171, stop :=\ 395203 ), 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 := 395234, root := ~, start := 395220, stop := 395233 ) , 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\ := 395261, root := ~, start := 395247, stop := 395260 ), 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 := 395313, root := ~, start := 395305, stop := 395312 ), 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 := 395346, root\ := ~, start := 395342, stop := 395345 ), 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 := 395422, root := ~, start := 395382, stop := 395421 ), 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 := 395468, root := ~, start := 395435, stop := 395467 ), 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 := 395519, root := ~, start := 395507, stop := 395518 ), 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 := 395580, root := ~, start := 395572, stop \ := 395579 ), 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 := 395639, root := ~, start := 3\ 95625, stop := 395638 ), 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 := 395680, root := ~, start := 395676, stop :=\ 395679 ), 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 := 395785, root := ~, start := 395775, stop := 395784 ), 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 := 395828, root := ~, start := 395818, stop := 395827 ), 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 := 395972, root := ~, start := 395097, stop := 395971 ) ], count := [ 8, 2, 3, 17 ], mathmode := "Text", name := "List", next := 395984, root := ~, start := 394547, stop := 395983 ), 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 := 397012, root := ~, start := 395990, stop := 397011 ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Description", next := 397029, root := ~, start := 392952, stop := 397028 ) ], count := [ 8, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 397043, root := ~, start := 392323, stop := 397042 ), rec( content := "\n\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 397057, root := ~, start := 386406, stop := 397056 ) , rec( content := "\n", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 397068, root := ~, start := 365129, stop := 397067 ), 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 := 397162, root := ~, start := 397125, stop := 397161 ) , 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 := 397215, root := ~, start := 397175, stop := 397214 ), 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 := 397278, root := ~, start := 397239, stop := 397277 ), 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 := 397310, root := ~, start := 397283, stop := 397309 ), 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 := 397390, root := ~, start := 397380, stop := 397389 ), 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 := 397485, root := ~, start := 397481, stop := 397484 ), 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 := 397514, root := ~, start := 397504, stop := 397513 ), 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 := 397552, root := ~, start := 397542, stop := 397551 ), 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 := 397574, root := ~, start := 397562, stop := 397573 ), 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 := 397623, root := ~, start := 397608, stop := 397622 ), 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 := 397649, root := ~, start := 397634, stop := 397648 ), 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 := 397766, root := ~, start := 397727, stop := 397765 ), 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 := 397811, root := ~, start := 397777, stop := 397810 ), 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 := 397816, root := ~, start := 397812, stop := 397815 ), 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 := 397960, root := ~, start := 397936, stop := 397959 ), 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 := 397965, root := ~, start := 397961, stop := 397964 ), 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 := 398013, root := ~, start := 398002, stop := 398012 ), 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 := 398141, root := ~, start := 398124, stop := 398140 ), 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 := 398268, root := ~, start := 398255, stop \ := 398267 ), 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 := 398342 , root\ := ~, start := 398331, stop := 398341 ), rec( content := ".\n ", count\ := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 8 ], name := "Ite\ m", next := 398359, root := ~, start := 398150, stop := 398358 ), 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 := 398391, root := ~, start := 398369, stop := 398390 ), 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 := 398504, root := ~, start := 398493, stop := 398503 ), 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\ := 398533, root := ~, start := 398520, stop := 398532 ), 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 := 398632, root := ~, start := 39\ 8628, stop := 398631 ), 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 := 398696, root\ := ~, start := 398680, stop := 398695 ), 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 := 398732, root :=\ ~, start := 398721, stop := 398731 ), 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 := 398820, root := ~, start := 398773, stop := 398819 ), 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 := 398855, root := ~, start := 398400, stop := 398854 ), 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 := 398886, root := ~, start := 398865, stop := 398885 ), 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 := 398996, root := ~, start := 398986, stop := 398995 ), 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 := 399080, root := ~, star\ t := 399061, stop := 399079 ), rec( content := ".\n ", count := [ 9, 1,\ 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 13 ], name := "Ite\ m", next := 399097, root := ~, start := 398895, stop := 399096 ), 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 := 399126, root := ~, start := 399107, stop := 399125 ), 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 := 399278, root :=\ ~, start := 399263, stop := 399277 ), 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 := 399297, root := ~, start := 399282, stop \ := 399296 ), rec( content := ".\n ", count := [ 9, 1, 1, 15 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 15 ], name := "Ite\ m", next := 399314, root := ~, start := 399135, stop := 399313 ), 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 := 399341, root := ~, start := 399324, stop := 399340 ), 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 \ := 399428, root := ~, start := 399418, stop := 399427 ), 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 := 399471, root := ~, start := 39946\ 1, stop := 399470 ), 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 := 399533, root :=\ ~, start := 399523, stop := 399532 ), 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 := 399583, root :=\ ~, start := 399568, stop := 399582 ), 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 := 399611, root := ~, start :=\ 399597, stop := 399610 ), 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 := 399710, root := ~, start := 399700, stop := 3\ 99709 ), 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 := 399732, root := ~, start := 399717, stop := 399731 ), rec( cont\ ent := ".\n ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], name := "Item", next := 39974\ 9, root := ~, start := 399350, stop := 399748 ), 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 := 399778, root := ~, start := 399\ 759, stop := 399777 ), 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 := \ 399898, root := ~, start := 399888, stop := 399897 ), 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\ := 399943, root := ~, start := 399931, stop := 399942 ), 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 := 399969, root :=\ ~, start := 399955, stop := 399968 ), 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 := 399985, root := ~, start := 399971, stop \ := 399984 ), 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 := 400001, root := ~, start := 399987, stop := 400000 ), 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 := 400015, root :=\ ~, start := 400003, stop := 400014 ), 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 := 400040, root := ~, start := \ 400027, stop := 400039 ), 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 := 400064, root := ~, start := 400046, stop := 40\ 0063 ), 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 := 400208, root :=\ ~, start := 400196, stop := 400207 ), 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\ := 400265, root := ~, start := 400248, stop := 400264 ), rec( content := ".\n\ ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 19 ], name := "Item", next := 400282, root := ~, start := 399\ 787, stop := 400281 ), 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 := 400313, root := ~, start := 400\ 292, stop := 400312 ), 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 := 400422, root :=\ ~, start := 400409, stop := 400421 ), 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\ := 400525, root := ~, start := 400513, stop := 400524 ), rec( content := " is\ used.\n ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 1, 21 ], name := "Item", next := 400550, root := ~, start := 400\ 322, stop := 400549 ) ], count := [ 9, 1, 1, 7 ], mathmode := "Text", name := "List", next := 400564, root := ~, start := 398109, stop := 400563 ), 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 := 400689, root := ~, start := 400685, stop := 400688 ), 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 := 400764, root := ~, start := 400697, stop := 400763 ), rec( content := "\n ", count := [ 9, 1, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 400783, root := ~, start := 397315, stop := 400782 ) ], count := [ 9, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 400799, root := ~, start := 397222, stop := 400798 ), 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 := 400863, root := ~, start := 400824, stop := 400862 ), rec( attributes := rec( Arg := "digraph", Name := "DotVertexLabell\ edDigraph" ), content := 0, count := [ 9, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 400921, root := ~, start := 400868, stop := 400920 ), 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 := 400954, root := ~, start := 400926, stop := 400953 ), 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 := 400996, root := ~, start := 400979, stop := 400995 ), 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 := 401068, root := ~, start := 401054, stop := 401067 ), 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 := 401152, root := ~, start := 401138, stop := 401151 ), 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 := 401296, root := ~, start := 401292, stop := 401295 ), 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 := 401335, root := ~, start := 401304, stop := 401334 ), 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 := 401366, root := ~, start := 401349, stop := 401365 ), 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 := 401433, root := ~, start := 401400, stop := 401432 ), 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 := 401527, root := ~, start := 401519, stop := 401526 ), 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 := 401580, root := ~, start := 401576, stop := 401579 ), 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 := 401615, root := ~, start := 401605, stop := 401614 ), 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 := 401653, root := ~, start := 401638, stop := 401652 ), 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 := 401806, root := ~, start := 401772, stop := 401805 ), 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 := 401812, root := ~, start := 401808, stop := 401811 ), 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 := 401860, root := ~, start := 401843, stop := 401859 ), 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 := 401901, root := ~, start := 401870, stop := 401900 ), 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 := 401993, root := ~, start := 401951, stop := 401992 ), 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 := 401998, root := ~, start := 401994, stop := 401997 ), 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 := 402305, root := ~, start := 402006, stop := 402304 ), rec( content := "\n ", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 402324, root := ~, start := 400959, stop := 402323 ) ], count := [ 9, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 402340, root := ~, start := 400807, stop := 402339 ), 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 := 402413, root := ~, start := 402365, stop := 402412 ), 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 := 402446, root := ~, start := 402418, stop := 402445 ), 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 := 402566, root := ~, start := 402552, stop := 402565 ), 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 := 402594, root := ~, start := 402568, stop := 402593 ), 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 := 402639, root := ~, start := 402625, stop := 402638 ), 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 := 402710, root := ~, start := 402678, stop := 402709 ), 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 := 402715, root := ~, start := 402711, stop := 402714 ), 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 := 402805, root := ~, start := 402791, stop := 402804 ), 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 := 402827, root := ~, start := 402813, stop := 402826 ), 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 := 402966, root := ~, start := 402940, stop := 402965 ), 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 := 403085, root := ~, start := 403081, stop := 403084 ), 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 := 403120, root := ~, start := 403110, stop := 403119 ), 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 := 403158, root := ~, start := 403143, stop := 403157 ), 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 := 403317, root := ~, start := 403283, stop := 403316 ), 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 := 403323, root := ~, start := 403319, stop := 403322 ), 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 := 403380, root := ~, start := 403354, stop := 403379 ), 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 := 403472, root := ~, start := 403430, stop := 403471 ), 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 := 403477, root := ~, start := 403473, stop := 403476 ), 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 := 403714, root := ~, start := 403485, stop := 403713 ), rec( content := "\n ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 403733, root := ~, start := 402451, stop := 403732 ) ], count := [ 9, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 403749, root := ~, start := 402348, stop := 403748 ), 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 := 403825, root := ~, start := 403774, stop := 403824 ), 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 := 403858, root := ~, start := 403830, stop := 403857 ), 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 := 403980, root := ~, start := 403966, stop := 403979 ), 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 := 404011, root := ~, start := 403982, stop := 404010 ), 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 := 404056, root := ~, start := 404042, stop := 404055 ), 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 := 404136, root := ~, start := 404093, stop := 404135 ), 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 := 404141, root := ~, start := 404137, stop := 404140 ), 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 := 404169, root := ~, start := 404155, stop := 404168 ), 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 := 404276, root := ~, start := 404247, stop := 404275 ), 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 := 404314, root := ~, start := 404290, stop := 404313 ), 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 := 404377, root := ~, start := 404328, stop := 404376 ), 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 := 404395, root := ~, start := 404381, stop := 404394 ), 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 := 404400, root := ~, start := 404396, stop := 404399 ), 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 := 404435, root := ~, start := 404425, stop := 404434 ), 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 := 404473, root := ~, start := 404458, stop := 404472 ), 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 := 404632, root := ~, start := 404598, stop := 404631 ), 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 := 404638, root := ~, start := 404634, stop := 404637 ), 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 := 404698, root := ~, start := 404669, stop := 404697 ), 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 := 404790, root := ~, start := 404748, stop := 404789 ), 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 := 404795, root := ~, start := 404791, stop := 404794 ), 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 := 404981, root := ~, start := 404803, stop := 404980 ), rec( content := "\n ", count := [ 9, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 405000, root := ~, start := 403863, stop := 404999 ) ], count := [ 9, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 405016, root := ~, start := 403757, stop := 405015 ), 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 := 405088, root := ~, start := 405041, stop := 405087 ), rec( attributes := rec( Arg := "digraph", Name := "DotQuasiorderDi\ graph" ), content := 0, count := [ 9, 1, 5, 4 ], mathmode := "Text", name := "Attr", next := 405142, root := ~, start := 405093, stop := 405141 ), 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 := 405175, root := ~, start := 405147, stop := 405174 ), 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 := 405292, root := ~, start := 405278, stop := 405291 ), 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 := 405319, root := ~, start := 405294, stop := 405318 ), 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 := 405364, root := ~, start := 405350, stop := 405363 ), 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 := 405435, root := ~, start := 405396, stop := 405434 ), 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 := 405440, root := ~, start := 405436, stop := 405439 ), 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 := 405656, root := ~, start := 405627, stop := 405655 ), 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 := 405671, root := ~, start := 405663, stop := 405670 ), 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 := 405722, root := ~, start := 405708, stop := 405721 ), 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 := 405852, root := ~, start := 405817, stop := 405851 ), 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 := 405889, root := ~, start := 405881, stop := 405888 ), 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 := 405953, root := ~, start := 405939, stop := 405952 ), 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 := 406000, root := ~, start := 405975, stop := 405999 ), 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 := 406070, root := ~, start := 406021, stop := 406069 ), 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 := 406082, root := ~, start := 406074, stop := 406081 ), 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 := 406190, root := ~, start := 406176, stop := 406189 ), 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 := 406246, root := ~, start := 406242, stop := 406245 ), 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 := 406281, root := ~, start := 406271, stop := 406280 ), 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 := 406319, root := ~, start := 406304, stop := 406318 ), 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 := 406478, root := ~, start := 406444, stop := 406477 ), 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 := 406484, root := ~, start := 406480, stop := 406483 ), 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 := 406540, root := ~, start := 406515, stop := 406539 ), 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 := 406632, root := ~, start := 406590, stop := 406631 ), 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 := 406637, root := ~, start := 406633, stop := 406636 ), 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 := 406842, root := ~, start := 406645, stop := 406841 ), rec( content := "\n ", count := [ 9, 1, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Description", next := 406861, root := ~, start := 405180, stop := 406860 ) ], count := [ 9, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 406877, root := ~, start := 405024, stop := 406876 ), 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 := 406980, root := ~, start := 406902, stop := 406979 ), 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 := 407013, root := ~, start := 406985, stop := 407012 ), 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 := 407066, root := ~, start := 407038, stop := 407065 ), 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 := 407137, root := ~, start := 407123, stop := 407136 ), 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 := 407182, root := ~, start := 407170, stop := 407181 ), 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 := 407250, root := ~, start := 407236, stop := 407249 ), 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 := 407307, root := ~, start := 407293, stop := 407306 ), 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 := 407344, root := ~, start := 407330, stop := 407343 ), 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 := 407369, root := ~, start := 407355, stop := 407368 ), 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 := 407388, root := ~, start := 407374, stop := 407387 ), 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 := 407442, root := ~, start := 407414, stop := 407441 ), 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 := 407481, root := ~, start := 407477, stop := 407480 ), 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 := 407527, root := ~, start := 407499, stop := 407526 ), 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 := 407578, root := ~, start := 407564, stop := 407577 ), 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 := 407597, root := ~, start := 407583, stop := 407596 ), 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 := 407702, root := ~, start := 407678, stop := 407701 ), 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 := 407738, root := ~, start := 407734, stop := 407737 ), 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 := 407953, root := ~, start := 407746, stop := 407952 ), rec( content := "\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 407972, root := ~, start := 407018, stop := 407971 ) ], count := [ 9, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 407988, root := ~, start := 406885, stop := 407987 ), rec( content := "\n\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 408002, root := ~, start := 397166, stop := 408001 ) , 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 := 408072, root := ~, start := 408015, stop := 408071 ), 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 := 408221, root := ~, start := 408200, stop := 408220 ), 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 := 408251, root := ~, start := 408236, stop := 408250 ), 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 := 408372, root := ~, start := 408353, stop := 408371 ), 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 := 408705, root := ~, start := 408636, stop := 408704 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], name := "Item", next := 408720, root := ~, start := 408228, stop := 408719 ), 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 := 408750, root := ~, start := 408728, stop := 408749 ), 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 := 408781, root := ~, start := 408765, stop := 408780 ), 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 := 408937, root := ~, start := 408918, stop := 408936 ), 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 := 409140, root := ~, start := 409124, stop := 409139 ), 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 := 409200, root := ~, start := 409185, stop := 409199 ), 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 := 409422, root := ~, start := 409345, stop := 409421 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], name := "Item", next := 409437, root := ~, start := 408757, stop := 409436 ), 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 := 409468, root := ~, start := 409445, stop := 409467 ), 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 := 409498, root := ~, start := 409483, stop := 409497 ), 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 := 409539, root := ~, start := 409524, stop := 409538 ), 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 := 409741, root := ~, start := 409475, stop := 409740 ), 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 := 409773, root := ~, start := 409749, stop := 409772 ), 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 := 409804, root := ~, start := 409788, stop := 409803 ), 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 := 409846, root := ~, start := 409830, stop := 409845 ), 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 := 410094, root := ~, start := 410078, stop := 410093 ), 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 := 410153, root := ~, start := 409780, stop := 410152 ) ], count := [ 9, 2, 0, 4 ], mathmode := "Text", name := "List", next := 410165, root := ~, start := 408187, stop := 410164 ), 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 := 410220, root := ~, start := 410185, stop := 410219 ), rec( attributes := rec( Arg := "digraph", Name := "PrintString" ), content := 0, count := [ 9, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 410263, root := ~, start := 410223, stop := 410262 ), 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 := 410294, root := ~, start := 410266, stop := 410293 ), 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 := 410345, root := ~, start := 410332, stop := 410344 ), 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 := 410381, root := ~, start := 410356, stop := 410380 ), 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 := 410412, root := ~, start := 410398, stop := 410411 ), 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 := 410491, root := ~, start := 410451, stop := 410490 ), 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 := 410501, root := ~, start := 410497, stop := 410500 ), 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 := 410546, root := ~, start := 410533, stop := 410545 ), 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 := 410598, root := ~, start := 410585, stop := 410597 ), 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 := 410710, root := ~, start := 410696, stop := 410709 ), 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 := 410720, root := ~, start := 410716, stop := 410719 ), 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 := 410759, root := ~, start := 410746, stop := 410758 ), 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 := 410848, root := ~, start := 410834, stop := 410847 ), 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 := 410937, root := ~, start := 410902, stop := 410936 ), 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 := 410965, root := ~, start := 410953, stop := 410964 ), 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 := 411016, root := ~, start := 410998, stop := 411015 ), 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 := 411050, root := ~, start := 411037, stop := 411049 ), 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 := 411525, root := ~, start := 411056, stop := 411524 ), rec( content := "\n ", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 411542, root := ~, start := 410297, stop := 411541 ) ], count := [ 9, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 411556, root := ~, start := 410170, stop := 411555 ), 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 := 411636, root := ~, start := 411577, stop := 411635 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDigr\ aph6String" ), content := 0, count := [ 9, 2, 2, 4 ], mathmode := "Text", name := "Oper", next := 411698, root := ~, start := 411639, stop := 411697 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromSpar\ se6String" ), content := 0, count := [ 9, 2, 2, 6 ], mathmode := "Text", name := "Oper", next := 411760, root := ~, start := 411701, stop := 411759 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDiSp\ arse6String" ), content := 0, count := [ 9, 2, 2, 8 ], mathmode := "Text", name := "Oper", next := 411822, root := ~, start := 411763, stop := 411821 ), 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 := 411854, root := ~, start := 411825, stop := 411853 ), 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 := 411888, root := ~, start := 411878, stop := 411887 ), 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 := 412231, root := ~, start := 412227, stop := 412230 ), 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 := 412305, root := ~, start := 412294, stop := 412304 ), 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 := 412364, root := ~, start := 412334, stop := 412363 ), 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 := 412404, root := ~, start := 412372, stop := 412403 ), 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 := 413333, root := ~, start := 412588, stop := 413332 ), rec( content := "\n ", count := [ 9, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "Description", next := 413352, root := ~, start := 411857, stop := 413351 ) ], count := [ 9, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 413366, root := ~, start := 411562, stop := 413365 ), 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 := 413428, root := ~, start := 413387, stop := 413427 ), rec( attributes := rec( Arg := "digraph", Name := "Digraph6String"\ ), content := 0, count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 413474, root := ~, start := 413431, stop := 413473 ), rec( attributes := rec( Arg := "digraph", Name := "Sparse6String" ), content := 0, count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "Oper", next := 413519, root := ~, start := 413477, stop := 413518 ), rec( attributes := rec( Arg := "digraph", Name := "DiSparse6String\ " ), content := 0, count := [ 9, 2, 3, 8 ], mathmode := "Text", name := "Oper", next := 413566, root := ~, start := 413522, stop := 413565 ), 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 := 413597, root := ~, start := 413569, stop := 413596 ), 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 := 413720, root := ~, start := 413706, stop := 413719 ), 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 := 413726, root := ~, start := 413722, stop := 413725 ), 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 := 413861, root := ~, start := 413847, stop := 413860 ), 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 := 413886, root := ~, start := 413870, stop := 413885 ), 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 := 413936, root := ~, start := 413932, stop := 413935 ), 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 := 413973, root := ~, start := 413946, stop := 413972 ), 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 := 414160, root := ~, start := 413979, stop := 414159 ), rec( content := "\n ", count := [ 9, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "Description", next := 414177, root := ~, start := 413600, stop := 414176 ) ], count := [ 9, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 414191, root := ~, start := 413372, stop := 414190 ), 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 := 414271, root := ~, start := 414212, stop := 414270 ), 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 := 414311, root := ~, start := 414274, stop := 414310 ), 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 := 414350, root := ~, start := 414335, stop := 414349 ), 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 := 414423, root := ~, start := 414405, stop := 414422 ), 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 := 414483, root := ~, start := 414479, stop := 414482 ), 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 := 414526, root := ~, start := 414514, stop := 414525 ), 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 := 414772, root := ~, start := 414760, stop := 414771 ), 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 := 414955, root := ~, start := 414945, stop := 414954 ), 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 := 414967, root := ~, start := 414957, stop := 414966 ), 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 := 414979, root := ~, start := 414969, stop := 414978 ), 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 := 414992, root := ~, start := 414981, stop := 414991 ), 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 := 415005, root := ~, start := 414994, stop := 415004 ), 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 := 415020, root := ~, start := 415011, stop := 415019 ), 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 := 415039, root := ~, start := 415025, stop := 415038 ), 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 := 415100, root := ~, start := 415096, stop := 415099 ), 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 := 415142, root := ~, start := 415131, stop := 415141 ), 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 := 415195, root := ~, start := 415185, stop := 415194 ), 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 := 415219, root := ~, start := 415209, stop := 415218 ), 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 := 415247, root := ~, start := 415237, stop := 415246 ), 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 := 415278, root := ~, start := 415267, stop := 415277 ), 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 := 415312, root := ~, start := 415302, stop := 415311 ), 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 := 415337, root := ~, start := 415333, stop := 415336 ), 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 := 415361, root := ~, start := 415346, stop := 415360 ), 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 := 415388, root := ~, start := 415378, stop := 415387 ), 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 := 415401, root := ~, start := 415390, stop := 415400 ), 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 := 415420, root := ~, start := 415410, stop := 415419 ), 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 := 415561, root := ~, start := 415557, stop := 415560 ), 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 := 415613, root := ~, start := 415595, stop := 415612 ), 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 := 415707, root := ~, start := 415681, stop := 415706 ), 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 := 415744, root := ~, start := 415711, stop := 415743 ), 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 := 415873, root := ~, start := 415847, stop := 415872 ), 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 := 415908, root := ~, start := 415881, stop := 415907 ), 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 := 415984, root := ~, start := 415980, stop := 415983 ), 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 := 416115, root := ~, start := 416090, stop := 416114 ), 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 := 416451, root := ~, start := 416122, stop := 416450 ), 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 := 416484, root := ~, start := 416458, stop := 416483 ), 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 := 4167\ 19, root := ~, start := 416491, stop := 416718 ), 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 := 4167\ 53, root := ~, start := 416726, stop := 416752 ), 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 := 416983, root := ~, start := 416760, stop := 416982 ), 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 := 417019, root := ~, start := 416990, stop := 417018 ), 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 := 417231, root := ~, start := 417\ 026, stop := 417230 ), 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 := 4172\ 68, root := ~, start := 417238, stop := 417267 ), 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 := 417400, root := ~, start := 417355, stop := 417399 ), 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 := 417624, root := ~, start := 417620, stop := 417623 ), 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 := 417672, root := ~, start := 417638, stop := 417671 ), 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 := 417749, root := ~, start := 417745, stop := 417748 ), rec( cont\ ent := "\n ", count := [ 9, 2, 4, 20 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 4, 18 ], name := "Item", next := 417763, root := ~, start := 417\ 275, stop := 417762 ), 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 := 417795, root := \ ~, start := 417786, stop := 417794 ), 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 := 417813, root := ~, start := 417799, stop \ := 417812 ), rec( content := ")", count := [ 9, 2, 4, 21 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 21 ], name := "Mark", next := 4178\ 21, root := ~, start := 417771, stop := 417820 ), 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 := 4179\ 57, root := ~, start := 417931, stop := 417956 ), 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 := 417987, root := ~, start := 417\ 828, stop := 417986 ) ], count := [ 9, 2, 4, 9 ], mathmode := "Text", name := "List", next := 417999, root := ~, start := 416077, stop := 417998 ), 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 := 418374, root := ~, start := 418005, stop := 418373 ), rec( content := "\n ", count := [ 9, 2, 4, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 418393, root := ~, start := 414314, stop := 418392 ) ], count := [ 9, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 418407, root := ~, start := 414197, stop := 418406 ), 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 := 418486, root := ~, start := 418428, stop := 418485 ), 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 := 418541, root := ~, start := 418489, stop := 418540 ), 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 := 418580, root := ~, start := 418565, stop := 418579 ), 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 := 418723, root := ~, start := 418692, stop := 418722 ), 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 := 418749, root := ~, start := 418730, stop := 418748 ), 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 := 418835, root := ~, start := 418820, stop := 418834 ), 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 := 418948, root := ~, start := 418938, stop := 418947 ), 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 := 418961, root := ~, start := 418950, stop := 418960 ), 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 := 418976, root := ~, start := 418966, stop := 418975 ), 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 := 419006, root := ~, start := 418987, stop := 419005 ), 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 := 419031, root := ~, start := 419016, stop := 419030 ), 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 := 419089, root := ~, start := 419085, stop := 419088 ), 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 := 419134, root := ~, start := 419120, stop := 419133 ), 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 := 419237, root := ~, start := 419218, stop := 419236 ), 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 := 419261, root := ~, start := 419247, stop := 419260 ), 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 := 419317, root := ~, start := 419302, stop := 419316 ), 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 := 419322, root := ~, start := 419318, stop := 419321 ), 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 := 419361, root := ~, start := 419353, stop := 419360 ), 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 := 419400, root := ~, start := 419381, stop := 419399 ), 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 := 419425, root := ~, start := 419417, stop := 419424 ), 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 := 419471, root := ~, start := 419456, stop := 419470 ), 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 := 419481, root := ~, start := 419477, stop := 419480 ), 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 := 419526, root := ~, start := 419512, stop := 419525 ), 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 := 419573, root := ~, start := 419554, stop := 419572 ), 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 := 419661, root := ~, start := 419646, stop := 419660 ), 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 := 419736, root := ~, start := 419726, stop := 419735 ), 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 := 419749, root := ~, start := 419738, stop := 419748 ), 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 := 419765, root := ~, start := 419755, stop := 419764 ), 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 := 419825, root := ~, start := 419815, stop := 419824 ), 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 := 419851, root := ~, start := 419832, stop := 419850 ), 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 := 419921, root := ~, start := 419884, stop := 419920 ), 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 := 419931, root := ~, start := 419927, stop := 419930 ), 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 := 420062, root := ~, start := 420037, stop := 420061 ), 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 := 420398, root := ~, start := 420069, stop := 420397 ), 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 := 420431, root := ~, start := 420405, stop := 420430 ), 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 := 4206\ 66, root := ~, start := 420438, stop := 420665 ), 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 := 4207\ 00, root := ~, start := 420673, stop := 420699 ), 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 := 420930, root := ~, start := 420707, stop := 420929 ), 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 := 420966, root := ~, start := 420937, stop := 420965 ), 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 := 421178, root := ~, start := 420\ 973, stop := 421177 ), 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 := 4212\ 15, root := ~, start := 421185, stop := 421214 ), 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 := 421347, root := ~, start := 421302, stop := 421346 ), 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 := 421571, root := ~, start := 421567, stop := 421570 ), 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 := 421619, root := ~, start := 421585, stop := 421618 ), 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 := 421696, root := ~, start := 421692, stop := 421695 ), 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 := 4218\ 41, root := ~, start := 421222, stop := 421840 ), 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 := 421873 , root\ := ~, start := 421864, stop := 421872 ), 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 := 421891, root := ~, start := 42187\ 7, stop := 421890 ), rec( content := ")", count := [ 9, 2, 5, 20 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 20 ], name := "Mar\ k", next := 421899, root := ~, start := 421849, stop := 421898 ), 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 := 422035, root := ~, start := 422009, stop := 42\ 2034 ), 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 := 422065, root := ~, start := 421906, stop := 422064 ) ], count := [ 9, 2, 5, 8 ], mathmode := "Text", name := "List", next := 422077, root := ~, start := 420024, stop := 422076 ), 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 := 423123, root := ~, start := 422083, stop := 423122 ), rec( content := "\n ", count := [ 9, 2, 5, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 423142, root := ~, start := 418544, stop := 423141 ) ], count := [ 9, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 423156, root := ~, start := 418413, stop := 423155 ), 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 := 423250, root := ~, start := 423177, stop := 423249 ), 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 := 423289, root := ~, start := 423274, stop := 423288 ), 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 := 423344, root := ~, start := 423329, stop := 423343 ), 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 := 423423, root := ~, start := 423398, stop := 423422 ), 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 := 423454, root := ~, start := 423434, stop := 423453 ), 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 := 423521, root := ~, start := 423506, stop := 423520 ), 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 := 423593, root := ~, start := 423583, stop := 423592 ), 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 := 423606, root := ~, start := 423595, stop := 423605 ), 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 := 423621, root := ~, start := 423611, stop := 423620 ), 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 := 423981, root := ~, start := 423977, stop := 423980 ), 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 := 424131, root := ~, start := 424112, stop := 424130 ), 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 := 424173, root := ~, start := 424154, stop := 424172 ), 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 := 424178, root := ~, start := 424174, stop := 424177 ), 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 := 424268, root := ~, start := 424243, stop := 424267 ), 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 := 424604, root := ~, start := 424275, stop := 424603 ), 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 := 424637, root := ~, start := 424611, stop := 424636 ), 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 := 4248\ 72, root := ~, start := 424644, stop := 424871 ), 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 := 4249\ 06, root := ~, start := 424879, stop := 424905 ), 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 := 425136, root := ~, start := 424913, stop := 425135 ), 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 := 425172, root := ~, start := 425143, stop := 425171 ), 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 := 425384, root := ~, start := 425\ 179, stop := 425383 ), 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 := 4254\ 22, root := ~, start := 425392, stop := 425421 ), 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 := 425554, root := ~, start := 425509, stop := 425553 ), 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 := 425778, root := ~, start := 425774, stop := 425777 ), 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 := 425826, root := ~, start := 425792, stop := 425825 ), 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 := 425903, root := ~, start := 425899, stop := 425902 ), 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 := 426040, root := ~, start := 425\ 429, stop := 426039 ), 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 := 426071, root := \ ~, start := 426062, stop := 426070 ), 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 := 426089, root := ~, start := 426075, stop \ := 426088 ), rec( content := ")", count := [ 9, 2, 6, 18 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 18 ], name := "Mark", next := 4260\ 97, root := ~, start := 426047, stop := 426096 ), 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 := 4262\ 33, root := ~, start := 426207, stop := 426232 ), 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 := 426263, root := ~, start := 426\ 104, stop := 426262 ) ], count := [ 9, 2, 6, 6 ], mathmode := "Text", name := "List", next := 426275, root := ~, start := 424230, stop := 426274 ), 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 := 427042, root := ~, start := 426281, stop := 427041 ), rec( content := "\n ", count := [ 9, 2, 6, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 427061, root := ~, start := 423253, stop := 427060 ) ], count := [ 9, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 427075, root := ~, start := 423162, stop := 427074 ), 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 := 427161, root := ~, start := 427096, stop := 427160 ), 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 := 427195, root := ~, start := 427164, stop := 427194 ), 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 := 427234, root := ~, start := 427219, stop := 427233 ), 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 := 427347, root := ~, start := 427317, stop := 427346 ), 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 := 427436, root := ~, start := 427395, stop := 427435 ), 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 := 427490, root := ~, start := 427486, stop := 427489 ), 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 := 427535, root := ~, start := 427521, stop := 427534 ), 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 := 427645, root := ~, start := 427615, stop := 427644 ), 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 := 427673, root := ~, start := 427659, stop := 427672 ), 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 := 427725, root := ~, start := 427710, stop := 427724 ), 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 := 427735, root := ~, start := 427731, stop := 427734 ), 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 := 427913, root := ~, start := 427909, stop := 427912 ), 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 := 427988, root := ~, start := 427961, stop := 427987 ), 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 := 428337, root := ~, start := 427995, stop := 428336 ), rec( content := "\n ", count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Description", next := 428354, root := ~, start := 427198, stop := 428353 ) ], count := [ 9, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 428368, root := ~, start := 427081, stop := 428367 ), 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 := 428470, root := ~, start := 428389, stop := 428469 ), rec( attributes := rec( Arg := "delimiter1, [del\ imiter2,] offset", Name := "DigraphPlainTextLineDecoder" ), content := 0, count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "Oper", next := 428554, root := ~, start := 428473, stop := 428553 ), 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 := 428585, root := ~, start := 428557, stop := 428584 ), 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 := 428710, root := ~, start := 428706, stop := 428709 ), 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 := 428750, root := ~, start := 428716, stop := 428749 ), 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 := 428980, root := ~, start := 428963, stop := 428979 ), 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 := 429056, root := ~, start := 429039, stop := 429055 ), 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 := 429093, root := ~, start := 429059, stop := 429092 ), 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 := 429217, root := ~, start := 429213, stop := 429216 ), 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 := 429311, root := ~, start := 429277, stop := 429310 ), 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 := 429421, root := ~, start := 429417, stop := 429420 ), 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 := 429453, root := ~, start := 429440, stop := 429452 ), 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 := 429671, root := ~, start := 429667, stop := 429670 ), 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 := 430138, root := ~, start := 429812, stop := 430137 ), rec( content := "\n ", count := [ 9, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "Description", next := 430155, root := ~, start := 428588, stop := 430154 ) ], count := [ 9, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 430169, root := ~, start := 428374, stop := 430168 ), 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 := 430236, root := ~, start := 430190, stop := 430235 ), 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 := 430268, root := ~, start := 430239, stop := 430267 ), 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 := 430328, root := ~, start := 430318, stop := 430327 ), 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 := 430409, root := ~, start := 430383, stop := 430408 ), 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 := 430471, root := ~, start := 430467, stop := 430470 ), 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 := 430516, root := ~, start := 430506, stop := 430515 ), 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 := 430636, root := ~, start := 430628, stop := 430635 ), 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 := 430727, root := ~, start := 430723, stop := 430726 ), 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 := 430795, root := ~, start := 430782, stop := 430794 ), 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 := 430836, root := ~, start := 430823, stop := 430835 ), 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 := 430866, root := ~, start := 430853, stop := 430865 ), 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 := 430925, root := ~, start := 430912, stop := 430924 ), 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 := 431002, root := ~, start := 430987, stop := 431001 ), 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 := 431008, root := ~, start := 431004, stop := 431007 ), 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 := 431080, root := ~, start := 431067, stop := 431079 ), 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 := 431105, root := ~, start := 431097, stop := 431104 ), 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 := 431134, root := ~, start := 431121, stop := 431133 ), 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 := 431202, root := ~, start := 431189, stop := 431201 ), 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 := 431477, root := ~, start := 431293, stop := 431476 ), rec( content := "\n ", count := [ 9, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Description", next := 431494, root := ~, start := 430271, stop := 431493 ) ], count := [ 9, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 431508, root := ~, start := 430175, stop := 431507 ), 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 := 431593, root := ~, start := 431529, stop := 431592 ), 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 := 431625, root := ~, start := 431596, stop := 431624 ), 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 := 431685, root := ~, start := 431675, stop := 431684 ), 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 := 431794, root := ~, start := 431758, stop := 431793 ), 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 := 431856, root := ~, start := 431852, stop := 431855 ), 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 := 431901, root := ~, start := 431891, stop := 431900 ), 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 := 432014, root := ~, start := 432006, stop := 432013 ), 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 := 432109, root := ~, start := 432105, stop := 432108 ), 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 := 432177, root := ~, start := 432164, stop := 432176 ), 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 := 432218, root := ~, start := 432205, stop := 432217 ), 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 := 432248, root := ~, start := 432235, stop := 432247 ), 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 := 432307, root := ~, start := 432294, stop := 432306 ), 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 := 432384, root := ~, start := 432369, stop := 432383 ), 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 := 432457, root := ~, start := 432444, stop := 432456 ), 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 := 432478, root := ~, start := 432470, stop := 432477 ), 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 := 432625, root := ~, start := 432621, stop := 432624 ), 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 := 432690, root := ~, start := 432677, stop := 432689 ), 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 := 432715, root := ~, start := 432694, stop := 432714 ), 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 := 433160, root := ~, start := 432781, stop := 433159 ), rec( content := "\n ", count := [ 9, 2, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 433177, root := ~, start := 431628, stop := 433176 ) ], count := [ 9, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 433191, root := ~, start := 431514, stop := 433190 ), 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 := 433249, root := ~, start := 433212, stop := 433248 ), 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 := 433281, root := ~, start := 433252, stop := 433280 ), 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 := 433315, root := ~, start := 433305, stop := 433314 ), 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 := 433500, root := ~, start := 433496, stop := 433499 ), 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 := 433577, root := ~, start := 433569, stop := 433576 ), 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 := 433649, root := ~, start := 433641, stop := 433648 ), 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 := 433679, root := ~, start := 433670, stop := 433678 ), 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 := 433750, root := ~, start := 433727, stop := 433749 ), 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 := 433870, root := ~, start := 433860, stop := 433869 ), 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 := 433901, root := ~, start := 433890, stop := 433900 ), 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 := 433915, root := ~, start := 433911, stop := 433914 ), 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 := 434041, root := ~, start := 434031, stop := 434040 ), 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 := 434081, root := ~, start := 434069, stop := 434080 ), 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 := 434449, root := ~, start := 434087, stop := 434448 ), rec( content := "\n ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "Description", next := 434466, root := ~, start := 433284, stop := 434465 ) ], count := [ 9, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 434480, root := ~, start := 433197, stop := 434479 ), 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 := 434545, root := ~, start := 434501, stop := 434544 ), rec( attributes := rec( Arg := "s", Name := "DigraphFromPlai\ nTextString" ), content := 0, count := [ 9, 2, 12, 4 ], mathmode := "Text", name := "Oper", next := 434597, root := ~, start := 434548, stop := 434596 ), 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 := 434628, root := ~, start := 434600, stop := 434627 ), 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 := 434671, root := ~, start := 434649, stop := 434670 ), 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 := 434793, root := ~, start := 434760, stop := 434792 ), 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 := 435124, root := ~, start := 435120, stop := 435123 ), 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 := 435263, root := ~, start := 435259, stop := 435262 ), 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 := 435321, root := ~, start := 435288, stop := 435320 ), 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 := 435389, root := ~, start := 435359, stop := 435388 ), 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 := 435429, root := ~, start := 435397, stop := 435428 ), 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 := 435850, root := ~, start := 435602, stop := 435849 ), rec( content := "\n ", count := [ 9, 2, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "Description", next := 435867, root := ~, start := 434631, stop := 435866 ) ], count := [ 9, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 435881, root := ~, start := 434486, stop := 435880 ), 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 := 435981, root := ~, start := 435902, stop := 435980 ), rec( attributes := rec( Arg := "filename, delimi\ ter, offset, ignore", Name := "ReadPlainTextDigraph" ), content := 0, count := [ 9, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 436061, root := ~, start := 435984, stop := 436060 ), 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 := 436309, root := ~, start := 436293, stop := 436308 ), 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 := 436314, root := ~, start := 436310, stop := 436313 ), 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 := 436335, root := ~, start := 436320, stop := 436334 ), 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 := 436425, root := ~, start := 436412, stop := 436424 ), 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 := 436563, root := ~, start := 436535, stop := 436562 ), 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 := 436592, root := ~, start := 436579, stop := 436591 ), 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 := 436602, root := ~, start := 436593, stop := 436601 ), 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 := 436717, root := ~, start := 436690, stop := 436716 ), 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 := 436765, root := ~, start := 436752, stop := 436764 ), 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 := 436778, root := ~, start := 436770, stop := 436777 ), 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 := 436821, root := ~, start := 436817, stop := 436820 ), 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 := 436840, root := ~, start := 436827, stop := 436839 ), 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 := 437283, root := ~, start := 436926, stop := 437282 ), rec( content := "\n ", count := [ 9, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 437300, root := ~, start := 436064, stop := 437299 ) ], count := [ 9, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 437314, root := ~, start := 435887, stop := 437313 ), 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 := 437392, root := ~, start := 437335, stop := 437391 ), rec( attributes := rec( Arg := "filename", Name := "ReadDIMACSDigra\ ph" ), content := 0, count := [ 9, 2, 14, 4 ], mathmode := "Text", name := "Oper", next := 437442, root := ~, start := 437395, stop := 437441 ), 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 := 437537, root := ~, start := 437523, stop := 437536 ), 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 := 437637, root := ~, start := 437612, stop := 437636 ), 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 := 437686, root := ~, start := 437672, stop := 437685 ), 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 := 437727, root := ~, start := 437713, stop := 437726 ), 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 := 437812, root := ~, start := 437780, stop := 437811 ), 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 := 437856, root := ~, start := 437817, stop := 437855 ), 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 := 437861, root := ~, start := 437857, stop := 437860 ), 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 := 437901, root := ~, start := 437886, stop := 437900 ), 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 := 438058, root := ~, start := 438043, stop := 438057 ), 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 := 438084, root := ~, start := 438074, stop := 438083 ), 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 := 438097, root := ~, start := 438086, stop := 438096 ), 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 := 438116, root := ~, start := 438106, stop := 438115 ), 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 := 438187, root := ~, start := 438183, stop := 438186 ), 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 := 438355, root := ~, start := 438347, stop \ := 438354 ), 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 := 438453, root := ~, start := 438310, stop := 438452 ), 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 := 438505, root := ~, start := 43849\ 7, stop := 438504 ), 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 := 438655, root :=\ ~, start := 438595, stop := 438654 ), 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 := 438689, root :\ = ~, start := 438663, stop := 438688 ), 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 := 438725, root :\ = ~, start := 438702, stop := 438724 ), 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 := 438894, root := ~, start := 438890, stop := 438893 ), 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 :\ = 438987, root := ~, start := 438964, stop := 438986 ), 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 := 439116, root := ~, start := 438460, stop := 439115 ), 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 := 439185, root := ~, start \ := 439157, stop := 439184 ), 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 := 439209, root := ~, start := 4391\ 93, stop := 439208 ), 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 := 439238, root := ~, start := 43922\ 2, stop := 439237 ), 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 := 43930\ 8, root := ~, start := 439265, stop := 439307 ), 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 := 439413, root := ~, star\ t := 439397, stop := 439412 ), 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 := 439442, root := ~, start\ := 439426, stop := 439441 ), 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 := 439547, root := ~, start := 439519, stop := 4395\ 46 ), 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 := 439590, root := ~, start := 439552, stop := 439589 ), 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 := 4396\ 88, root := ~, start := 439123, stop := 439687 ), 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 := 439761, root := ~, start := 439729, stop := 439760\ ), 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 := 439793, root := ~, start := 439777, stop := 4397\ 92 ), 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 := 439864, root := ~, start :\ = 439821, stop := 439863 ), 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 := 439889, root := ~, start := 439869, stop \ := 439888 ), 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 := 43995\ 5, root := ~, start := 439939, stop := 439954 ), 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 :\ = 439998, root := ~, start := 439978, stop := 439997 ), 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 := 440094, root := ~ , star\ t := 440070, stop := 440093 ), 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 := 440133, root := ~, star\ t := 440125, stop := 440132 ), 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 := 4401\ 87, root := ~, start := 439695, stop := 440186 ) ], count := [ 9, 2, 14, 8 ], mathmode := "Text", name := "List", next := 440199, root := ~, start := 438297, stop := 440198 ), 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 := 440328, root := ~, start := 440268, stop := 440327 ), 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 := 440445, root := ~, start := 440441, stop := 440444 ), 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 := 440807, root := ~, start := 440452, stop := 440806 ), rec( content := "\n ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "Description", next := 440824, root := ~, start := 437445, stop := 440823 ) ], count := [ 9, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 440838, root := ~, start := 437320, stop := 440837 ), rec( content := "\n\n ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 440852, root := ~, start := 408006, stop := 440851 ) , rec( content := "\n", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 440863, root := ~, start := 397089, stop := 440862 ), 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 := 440897, root := ~, start := 3969, stop := 440896 ), 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 := 440968, root := ~, start := 440912, stop := 440967 ), 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 := 441213, root := ~, start := 441142, stop := 441212 ), 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 := 441351, root := ~, start := 441241, stop := 441350 ), rec( content := "\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "Alt", next := 441362, root := ~, start := 441218, stop := 441361 ), 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 := 441408, root := ~, start := 441398, stop := 441407 ), 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 := 441477, root := ~, start := 441429, stop := 441476 ), 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 := 441537, root := ~, start := 441486, stop := 441536 ), 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 := 441618, root := ~, start := 441563, stop := 441617 ), rec( content := "\n \ ", count := [ "A", 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 9 ], name := "Item", next := 441634, root := ~, start := 441546, stop := 441633 ) ], count := [ "A", 1, 0, 4 ], name := "Row", next := 441647, root := ~, start := 441415, stop := 441646 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 10 ], name := "HorLine", next := 441664, root := ~, start := 441654, stop := 441663 ), 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 := 441715, root := ~, start := 441703, stop := 441714 ), rec( content := "\n \ ", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 11 ], name := "Item", next := 441731, root := ~, start := 441686, stop := 441730 ), 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 := 441778, root := ~, start := 441757, stop := 441777 ), rec( content := "\n \ ", count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 13 ], name := "Item", next := 441794, root := ~, start := 441740, stop := 441793 ), 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 := 4418\ 87, root := ~, start := 441875, stop := 441886 ), 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 := 441933, root := ~, start := 441900, stop := 4419\ 32 ), 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 := 441973, root := ~, start := 441969, stop := 44197\ 2 ), rec( HTML := true, content := "\n ", count := [ "A", 1, 0, 16 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], name := "Alt", next := 441990, root := ~, start := 441820, stop := 441989 ), rec( content := "\n \ ", count := [ "A", 1, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], name := "Item", next := 442006, root := ~, start := 441803, stop := 442005 ) ], count := [ "A", 1, 0, 10 ], name := "Row", next := 442019, root := ~, start := 441672, stop := 442018 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 17 ], name := "HorLine", next := 442036, root := ~, start := 442026, stop := 442035 ), 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 := 442097, root := ~, start := 442075, stop := 442096 ), rec( content := "\n \ ", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 18 ], name := "Item", next := 442113, root := ~, start := 442058, stop := 442112 ), 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 := 442170, root := ~, start := 442139, stop := 442169 ), rec( content := "\n \ ", count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 20 ], name := "Item", next := 442186, root := ~, start := 442122, stop := 442185 ), 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 := 442285, root := ~, start := 442254, stop := 4422\ 84 ), 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 := 442340, root := ~, start := 442316, stop := 44233\ 9 ), 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 := 442369, root := ~, start := 442212, stop := 442368 ), rec( content := "\n \ ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], name := "Item", next := 442385, root := ~, start := 442195, stop := 442384 ) ], count := [ "A", 1, 0, 17 ], name := "Row", next := 442398, root := ~, start := 442044, stop := 442397 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 23 ], name := "HorLine", next := 442415, root := ~, start := 442405, stop := 442414 ), 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 := 442470, root := ~, start := 442454, stop := 442469 ), rec( content := "\n \ ", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 24 ], name := "Item", next := 442486, root := ~, start := 442437, stop := 442485 ), 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 := 442537, root := ~, start := 442512, stop := 442536 ), 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 := 442619, root := ~, start := 442593, stop := 442618 ), rec( HTML := true , content := "\ \n ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 26 ], name := "Alt", next := 442636, root := ~, start := 442548, stop := 442635 ), rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 26 ], name := "Item", next := 442652, root := ~, start := 442495, stop := 442651 ), 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 := 442795, root := ~, start := 442756, stop := 4427\ 94 ), 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 := 442826, root := ~, start := 442800, stop := 442825 ), 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 := 443087, root := ~, start := 442678, stop := 443086 ), rec( content := "\n \ ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], name := "Item", next := 443103, root := ~, start := 442661, stop := 443102 ) ], count := [ "A", 1, 0, 23 ], name := "Row", next := 443116, root := ~, start := 442423, stop := 443115 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 29 ], name := "HorLine", next := 443133, root := ~, start := 443123, stop := 443132 ), 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 := 443192, root := ~, start := 443172, stop := 443191 ), rec( content := "\n \ ", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 30 ], name := "Item", next := 443208, root := ~, start := 443155, stop := 443207 ), 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 := 443263, root := ~, start := 443234, stop := 443262 ), rec( content := "\n \ ", count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 32 ], name := "Item", next := 443279, root := ~, start := 443217, stop := 443278 ), 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 := 443424, root := ~, start := 443381, stop := 4434\ 23 ), 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 := 443712, root := ~, start := 443666, stop := 4437\ 11 ), rec( HTML := true, content := ".\n ", count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], name := "Alt", next := 443730, root := ~, start := 443305, stop := 443729 ), rec( content := "\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], name := "Item", next := 443746, root := ~, start := 443288, stop := 443745 ) ], count := [ "A", 1, 0, 29 ], name := "Row", next := 443759, root := ~, start := 443141, stop := 443758 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 35 ], name := "HorLine", next := 443776, root := ~, start := 443766, stop := 443775 ), 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 := 443834, root := ~, start := 443815, stop := 443833 ), rec( content := "\n \ ", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 36 ], name := "Item", next := 443850, root := ~, start := 443798, stop := 443849 ), 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 := 443904, root := ~, start := 443876, stop := 443903 ), rec( content := "\n \ ", count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 38 ], name := "Item", next := 443920, root := ~, start := 443859, stop := 443919 ), 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 := 444016, root := ~, start := 443997, stop := 44401\ 5 ), 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 := 444074, root := ~, start := 444046, stop := 444073 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 40 ], name := "Alt", next := 444092, root := ~, start := 443946, stop := 444091 ), rec( content := "\n \ ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], name := "Item", next := 444108, root := ~, start := 443929, stop := 444107 ) ], count := [ "A", 1, 0, 35 ], name := "Row", next := 444121, root := ~, start := 443784, stop := 444120 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 41 ], name := "HorLine", next := 444138, root := ~, start := 444128, stop := 444137 ), 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 := 444195, root := ~, start := 444177, stop := 444194 ), rec( content := "\n \ ", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 42 ], name := "Item", next := 444211, root := ~, start := 444160, stop := 444210 ), 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 := 444264, root := ~, start := 444237, stop := 444263 ), rec( content := "\n \ ", count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 44 ], name := "Item", next := 444280, root := ~, start := 444220, stop := 444279 ), 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 := 444377, root := ~, start := 444350, stop := 4443\ 76 ), 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 := 444428, root := ~, start := 444410, stop := 44442\ 7 ), 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 := 444457, root := ~, start := 444306, stop := 444456 ), rec( content := "\n \ ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], name := "Item", next := 444473, root := ~, start := 444289, stop := 444472 ) ], count := [ "A", 1, 0, 41 ], name := "Row", next := 444486, root := ~, start := 444146, stop := 444485 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 47 ], name := "HorLine", next := 444503, root := ~, start := 444493, stop := 444502 ), 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 := 444561, root := ~, start := 444542, stop := 444560 ), rec( content := "\n \ ", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 48 ], name := "Item", next := 444577, root := ~, start := 444525, stop := 444576 ), 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 := 444636, root := ~, start := 444603, stop := 444635 ), rec( content := "\n \ ", count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 50 ], name := "Item", next := 444652, root := ~, start := 444586, stop := 444651 ), 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 := 444916, root := ~, start := 444678, stop := 444915 ), rec( content := "\n \ ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], name := "Item", next := 444932, root := ~, start := 444661, stop := 444931 ) ], count := [ "A", 1, 0, 47 ], name := "Row", next := 444945, root := ~, start := 444511, stop := 444944 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 53 ], name := "HorLine", next := 444962, root := ~, start := 444952, stop := 444961 ), 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 := 445023, root := ~, start := 445001, stop := 445022 ), rec( content := "\n \ ", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 54 ], name := "Item", next := 445039, root := ~, start := 444984, stop := 445038 ), 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 := 445101, root := ~, start := 445065, stop := 445100 ), rec( content := "\n \ ", count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 56 ], name := "Item", next := 445117, root := ~, start := 445048, stop := 445116 ), 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 := 445381, root := ~, start := 445143, stop := 445380 ), rec( content := "\n \ ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], name := "Item", next := 445397, root := ~, start := 445126, stop := 445396 ) ], count := [ "A", 1, 0, 53 ], name := "Row", next := 445410, root := ~, start := 444970, stop := 445409 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 59 ], name := "HorLine", next := 445427, root := ~, start := 445417, stop := 445426 ), 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 := 445490, root := ~, start := 445466, stop := 445489 ), rec( content := "\n \ ", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 60 ], name := "Item", next := 445506, root := ~, start := 445449, stop := 445505 ), 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 := 445568, root := ~, start := 445532, stop := 445567 ), 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 := 445660, root := ~, start := 445625, stop := 445659 ), rec( HTML := true , content := "\ \n ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 62 ], name := "Alt", next := 445677, root := ~, start := 445579, stop := 445676 ), rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 62 ], name := "Item", next := 445693, root := ~, start := 445515, stop := 445692 ), 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 := 445789, root := ~, start := 445754, stop := 4457\ 88 ), 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 := 445876, root := ~, start := 445840, stop := 4458\ 75 ), 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 := 445938, root := ~, start := 445719, stop := 445937 ), rec( content := "\n \ ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 64 ], name := "Item", next := 445954, root := ~, start := 445702, stop := 445953 ) ], count := [ "A", 1, 0, 59 ], name := "Row", next := 445967, root := ~, start := 445435, stop := 445966 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 65 ], name := "HorLine", next := 445984, root := ~, start := 445974, stop := 445983 ) ], count := [ "A", 1, 0, 4 ], mathmode := "Text", name := "Table", next := 445997, root := ~, start := 441368, stop := 445996 ), rec( content := "\n ", count := [ "A", 1, 0, 66 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 1 ], mathmode := "Text", name := "Section", next := 446010, root := ~, start := 441128, stop := 446009 ), 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 := 446106, root := ~, start := 446028, stop := 446105 ), 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 := 446244, root := ~, start := 446134, stop := 446243 ), rec( content := "\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 446255, root := ~, start := 446111, stop := 446254 ), 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 := 446301, root := ~, start := 446291, stop := 446300 ), 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 := 446371, root := ~, start := 446323, stop := 446370 ), 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 := 446431, root := ~, start := 446380, stop := 446430 ), 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 := 446512, root := ~, start := 446457, stop := 446511 ), rec( content := "\n \ ", count := [ "A", 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 9 ], name := "Item", next := 446528, root := ~, start := 446440, stop := 446527 ) ], count := [ "A", 2, 0, 4 ], name := "Row", next := 446541, root := ~, start := 446309, stop := 446540 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 10 ], name := "HorLine", next := 446558, root := ~, start := 446548, stop := 446557 ), 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 := 446611, root := ~, start := 446597, stop := 446610 ), rec( content := "\n \ ", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 11 ], name := "Item", next := 446627, root := ~, start := 446580, stop := 446626 ), 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 := 446676, root := ~, start := 446653, stop := 446675 ), rec( content := "\n \ ", count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 13 ], name := "Item", next := 446692, root := ~, start := 446636, stop := 446691 ), 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 := 446771, root := ~, start := 446748, stop := 4467\ 70 ), 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 := 446836, root := ~, start := 446822, stop := 44683\ 5 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 15 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], name := "Alt", next := 446854, root := ~, start := 446718, stop := 446853 ), rec( content := "\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], name := "Item", next := 446870, root := ~, start := 446701, stop := 446869 ) ], count := [ "A", 2, 0, 10 ], name := "Row", next := 446883, root := ~, start := 446566, stop := 446882 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 16 ], name := "HorLine", next := 446900, root := ~, start := 446890, stop := 446899 ), 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 := 446956, root := ~, start := 446939, stop := 446955 ), rec( content := "\n \ ", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 17 ], name := "Item", next := 446972, root := ~, start := 446922, stop := 446971 ), 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 := 447029, root := ~, start := 446998, stop := 447028 ), rec( content := "\n \ ", count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 19 ], name := "Item", next := 447045, root := ~, start := 446981, stop := 447044 ), 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 := 447146, root := ~, start := 447115, stop := 4471\ 45 ), 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 := 447196, root := ~, start := 447179, stop := 44719\ 5 ), 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 := 447225, root := ~, start := 447071, stop := 447224 ), rec( content := "\n \ ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], name := "Item", next := 447241, root := ~, start := 447054, stop := 447240 ) ], count := [ "A", 2, 0, 16 ], name := "Row", next := 447254, root := ~, start := 446908, stop := 447253 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 22 ], name := "HorLine", next := 447271, root := ~, start := 447261, stop := 447270 ), 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 := 447334, root := ~, start := 447322, stop := 44733\ 3 ), 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 := 4473\ 44, root := ~, start := 447336, stop := 447343 ), rec( content := ")", count := [ "A", 2, 0, 23 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "C", next := 447349, root := ~, start := 447310, stop := 447348 ), rec( content := "\n \ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], name := "Item", next := 447365, root := ~, start := 447293, stop := 447364 ), 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 := 447402, root := ~, start := 447394, stop := 44740\ 1 ), 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 := 4474\ 36, root := ~, start := 447422, stop := 447435 ), rec( content := ")", count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "C", next := 447441, root := ~, start := 447391, stop := 447440 ), rec( content := "\n \ ", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], name := "Item", next := 447457, root := ~, start := 447374, stop := 447456 ), 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 := 447556, root := ~, start := 447527, stop := 4475\ 55 ), 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 := 4476\ 40, root := ~, start := 447632, stop := 447639 ), 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 := 447734, root := ~, start := 447726, stop := 447733 ), 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 := 447773, root := ~, start := 447759, stop \ := 447772 ), 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 := 447778, root := ~, start := 447723, stop := 44777\ 7 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], name := "Alt", next := 447795, root := ~, start := 447483, stop := 447794 ), rec( content := "\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], name := "Item", next := 447811, root := ~, start := 447466, stop := 447810 ) ], count := [ "A", 2, 0, 22 ], name := "Row", next := 447824, root := ~, start := 447279, stop := 447823 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 28 ], name := "HorLine", next := 447841, root := ~, start := 447831, stop := 447840 ), 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 := 447897, root := ~, start := 447880, stop := 447896 ), rec( content := "\n \ ", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 29 ], name := "Item", next := 447913, root := ~, start := 447863, stop := 447912 ), 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 := 447971, root := ~, start := 447939, stop := 447970 ), rec( content := "\n \ ", count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 31 ], name := "Item", next := 447987, root := ~, start := 447922, stop := 447986 ), 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 := 4480\ 67, root := ~, start := 448050, stop := 448066 ), 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 := 448144, root := ~, start := 448112, stop := 448143 ), 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 := 448173, root := ~, start := 448013, stop := 448172 ), rec( content := "\n \ ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], name := "Item", next := 448189, root := ~, start := 447996, stop := 448188 ) ], count := [ "A", 2, 0, 28 ], name := "Row", next := 448202, root := ~, start := 447849, stop := 448201 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 34 ], name := "HorLine", next := 448219, root := ~, start := 448209, stop := 448218 ), 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 := 448276, root := ~, start := 448258, stop := 448275 ), rec( content := "\n \ ", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 35 ], name := "Item", next := 448292, root := ~, start := 448241, stop := 448291 ), 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 := 448351, root := ~, start := 448318, stop := 448350 ), rec( content := "\n \ ", count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 37 ], name := "Item", next := 448367, root := ~, start := 448301, stop := 448366 ), 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 := 4484\ 48, root := ~, start := 448430, stop := 448447 ), 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 := 448526, root := ~, start := 448493, stop := 448525 ), 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 := 448555, root := ~, start := 448393, stop := 448554 ), rec( content := "\n \ ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], name := "Item", next := 448571, root := ~, start := 448376, stop := 448570 ) ], count := [ "A", 2, 0, 34 ], name := "Row", next := 448584, root := ~, start := 448227, stop := 448583 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 40 ], name := "HorLine", next := 448601, root := ~, start := 448591, stop := 448600 ), 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 := 448655, root := ~, start := 448640, stop := 448654 ), rec( content := "\n \ ", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 41 ], name := "Item", next := 448671, root := ~, start := 448623, stop := 448670 ), 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 := 448726, root := ~, start := 448697, stop := 448725 ), rec( content := "\n \ ", count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 43 ], name := "Item", next := 448742, root := ~, start := 448680, stop := 448741 ), 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 := 448887, root := ~, start := 448858, stop := 448886\ ), 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 := 4489\ 02, root := ~, start := 448894, stop := 448901 ), 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 := 449001, root := ~, start := 448972, stop := 449000 ), 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 := 449049, root := ~, start := 449034, stop := 449048 ), 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 := 449078, root := ~, start := 448768, stop := 449077 ), rec( content := "\n \ ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], name := "Item", next := 449094, root := ~, start := 448751, stop := 449093 ) ], count := [ "A", 2, 0, 40 ], name := "Row", next := 449107, root := ~, start := 448609, stop := 449106 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 46 ], name := "HorLine", next := 449124, root := ~, start := 449114, stop := 449123 ), 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 := 449182, root := ~, start := 449163, stop := 449181 ), rec( content := "\n \ ", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 47 ], name := "Item", next := 449198, root := ~, start := 449146, stop := 449197 ), 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 := 449255, root := ~, start := 449224, stop := 449254 ), 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 := 449337, root := ~, start := 449307, stop := 4493\ 36 ), rec( HTML := true, content := ")\n ", count := [ "A", 2, 0, 49 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 49 ], name := "Alt", next := 449355, root := ~, start := 449266, stop := 449354 ), rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 49 ], name := "Item", next := 449371, root := ~, start := 449207, stop := 449370 ), 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 := 449472, root := ~, start := 449441, stop := 4494\ 71 ), 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 := 449524, root := ~, start := 449505, stop := 44952\ 3 ), 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 := 449553, root := ~, start := 449397, stop := 449552 ), rec( content := "\n \ ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], name := "Item", next := 449569, root := ~, start := 449380, stop := 449568 ) ], count := [ "A", 2, 0, 46 ], name := "Row", next := 449582, root := ~, start := 449132, stop := 449581 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 52 ], name := "HorLine", next := 449599, root := ~, start := 449589, stop := 449598 ), 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 := 449658, root := ~, start := 449638, stop := 449657 ), rec( content := "\n \ ", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 53 ], name := "Item", next := 449674, root := ~, start := 449621, stop := 449673 ), 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 := 449726, root := ~, start := 449700, stop := 449725 ), 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 := 449821, root := ~, start := 449790, stop := 449820 ), 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 := 449859, root := ~, start := 449835, stop := 4498\ 58 ), 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 := 449896, root := ~, start := 449873, stop := 449895 ), 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 := 449940, root := ~, start := 449910, stop := 4499\ 39 ), 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 := 449979, root := ~, start := 449954, stop := 449978 ), rec( HTML := true , content := "\ )\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 55 ], name := "Alt", next := 449997, root := ~, start := 449737, stop := 449996 ), rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 55 ], name := "Item", next := 450013, root := ~, start := 449683, stop := 450012 ), 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 := 450109, root := ~, start := 450083, stop := 4501\ 08 ), 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 := 450162, root := ~, start := 450142, stop := 45016\ 1 ), 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 := 450191, root := ~, start := 450039, stop := 450190 ), rec( content := "\n \ ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], name := "Item", next := 450207, root := ~, start := 450022, stop := 450206 ) ], count := [ "A", 2, 0, 52 ], name := "Row", next := 450220, root := ~, start := 449607, stop := 450219 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 58 ], name := "HorLine", next := 450237, root := ~, start := 450227, stop := 450236 ), 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 := 450290, root := ~, start := 450276, stop := 450289 ), rec( content := "\n \ ", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 59 ], name := "Item", next := 450306, root := ~, start := 450259, stop := 450305 ), 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 := 450361, root := ~, start := 450332, stop := 450360 ), rec( content := "\n \ ", count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 61 ], name := "Item", next := 450377, root := ~, start := 450315, stop := 450376 ), 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 := 450476, root := ~, start := 450447, stop := 4504\ 75 ), 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 := 4505\ 23, root := ~, start := 450509, stop := 450522 ), 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 := 450552, root := ~, start := 450403, stop := 450551 ), rec( content := "\n \ ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 63 ], name := "Item", next := 450568, root := ~, start := 450386, stop := 450567 ) ], count := [ "A", 2, 0, 58 ], name := "Row", next := 450581, root := ~, start := 450245, stop := 450580 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 64 ], name := "HorLine", next := 450598, root := ~, start := 450588, stop := 450597 ), 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 := 450657, root := ~, start := 450637, stop := 450656 ), rec( content := "\n \ ", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 65 ], name := "Item", next := 450673, root := ~, start := 450620, stop := 450672 ), 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 := 450761, root := ~, start := 450729, stop := 4507\ 60 ), 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 := 450812, root := ~, start := 450783, stop := 450811 ), rec( HTML := true , content := "\ \n ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 67 ], name := "Alt", next := 450829, root := ~, start := 450699, stop := 450828 ), 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 := 450901, root := ~, start := 450869, stop := 450900 ), rec( content := "\ \n ", count := [ "A", 2, 0, 67 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "Alt", next := 450918, root := ~, start := 450840, stop := 450917 ), rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 67 ], name := "Item", next := 450934, root := ~, start := 450682, stop := 450933 ), 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 := 451031, root := ~, start := 451011, stop := 45103\ 0 ), 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 := 451093, root := ~, start := 451061, stop := 451092 ), 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 := 451144, root := ~, start := 451115, stop := 4511\ 43 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], name := "Alt", next := 451162, root := ~, start := 450960, stop := 451161 ), rec( content := "\n \ ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], name := "Item", next := 451178, root := ~, start := 450943, stop := 451177 ) ], count := [ "A", 2, 0, 64 ], name := "Row", next := 451191, root := ~, start := 450606, stop := 451190 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 70 ], name := "HorLine", next := 451208, root := ~, start := 451198, stop := 451207 ), 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 := 451263, root := ~, start := 451247, stop := 451262 ), rec( content := "\n \ ", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 71 ], name := "Item", next := 451279, root := ~, start := 451230, stop := 451278 ), 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 := 451340, root := ~, start := 451305, stop := 451339 ), 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 := 451425, root := ~, start := 451391, stop := 4514\ 24 ), rec( HTML := true, content := ")\n ", count := [ "A", 2, 0, 73 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 73 ], name := "Alt", next := 451443, root := ~, start := 451351, stop := 451442 ), rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 73 ], name := "Item", next := 451459, root := ~, start := 451288, stop := 451458 ), 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 := 451564, root := ~, start := 451529, stop := 4515\ 63 ), 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 := 451613, root := ~, start := 451597, stop := 451612\ ), 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 := 451642, root := ~, start := 451485, stop := 451641 ), rec( content := "\n \ ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], name := "Item", next := 451658, root := ~, start := 451468, stop := 451657 ) ], count := [ "A", 2, 0, 70 ], name := "Row", next := 451671, root := ~, start := 451216, stop := 451670 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 76 ], name := "HorLine", next := 451688, root := ~, start := 451678, stop := 451687 ), 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 := 451740, root := ~, start := 451727, stop := 451739 ), rec( content := "\n \ ", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 77 ], name := "Item", next := 451756, root := ~, start := 451710, stop := 451755 ), 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 := 451838, root := ~, start := 451782, stop := 451837 ), rec( content := "\n \ ", count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 79 ], name := "Item", next := 451854, root := ~, start := 451765, stop := 451853 ), 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 := 451995, root := ~, start := 451939, stop := 4519\ 94 ), 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 := 452181, root := ~, start := 452125, stop := 452180 ) , 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 := 4522\ 27, root := ~, start := 452214, stop := 452226 ), 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 := 452256, root := ~, start := 451880, stop := 452255 ), rec( content := "\n \ ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 81 ], name := "Item", next := 452272, root := ~, start := 451863, stop := 452271 ) ], count := [ "A", 2, 0, 76 ], name := "Row", next := 452285, root := ~, start := 451696, stop := 452284 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 82 ], name := "HorLine", next := 452302, root := ~, start := 452292, stop := 452301 ), 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 := 452361, root := ~, start := 452341, stop := 452360 ), rec( content := "\n \ ", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 83 ], name := "Item", next := 452377, root := ~, start := 452324, stop := 452376 ), 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 := 452429, root := ~, start := 452403, stop := 452428 ), rec( content := "\n \ ", count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 85 ], name := "Item", next := 452445, root := ~, start := 452386, stop := 452444 ), 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 := 452541, root := ~, start := 452515, stop := 4525\ 40 ), 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 := 452594, root := ~, start := 452574, stop := 45259\ 3 ), 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 := 452623, root := ~, start := 452471, stop := 452622 ), rec( content := "\n \ ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], name := "Item", next := 452639, root := ~, start := 452454, stop := 452638 ) ], count := [ "A", 2, 0, 82 ], name := "Row", next := 452652, root := ~, start := 452310, stop := 452651 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 88 ], name := "HorLine", next := 452669, root := ~, start := 452659, stop := 452668 ), 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 := 452730, root := ~, start := 452708, stop := 452729 ), rec( content := "\n \ ", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 89 ], name := "Item", next := 452746, root := ~, start := 452691, stop := 452745 ), 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 := 452792, root := ~, start := 452755, stop := 452791 ), 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 := 452951, root := ~, start := 452818, stop := 452950 ), rec( content := "\n \ ", count := [ "A", 2, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], name := "Item", next := 452967, root := ~, start := 452801, stop := 452966 ) ], count := [ "A", 2, 0, 88 ], name := "Row", next := 452980, root := ~, start := 452677, stop := 452979 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 94 ], name := "HorLine", next := 452997, root := ~, start := 452987, stop := 452996 ), 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 := 453051, root := ~, start := 453036, stop := 453050 ), rec( content := "\n \ ", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 95 ], name := "Item", next := 453067, root := ~, start := 453019, stop := 453066 ), 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 := 453184, root := ~, start := 453093, stop := 453183 ), rec( content := "\n \ ", count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 97 ], name := "Item", next := 453200, root := ~, start := 453076, stop := 453199 ), 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 := 453358, root := ~, start := 453270, stop := 4533\ 57 ), 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 := 453406, root := ~, start := 453391, stop := 453405 ), 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 := 453435, root := ~, start := 453226, stop := 453434 ), rec( content := "\n \ ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 99 ], name := "Item", next := 453451, root := ~, start := 453209, stop := 453450 ) ], count := [ "A", 2, 0, 94 ], name := "Row", next := 453464, root := ~, start := 453005, stop := 453463 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 100 ], name := "HorLine", next := 453481, root := ~, start := 453471, stop := 453480 ), 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 := 453535, root := ~, start := 453520, stop := 453534 ), rec( content := "\n \ ", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 101 ], name := "Item", next := 453551, root := ~, start := 453503, stop := 453550 ), 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 := 453606, root := ~, start := 453577, stop := 453605 ), rec( content := "\n \ ", count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 103 ], name := "Item", next := 453622, root := ~, start := 453560, stop := 453621 ), 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 := 453721, root := ~, start := 453692, stop := 4537\ 20 ), 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 := 453769, root := ~, start := 453754, stop := 45376\ 8 ), 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 := 453798, root := ~, start := 453648, stop := 453797 ), rec( content := "\n \ ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], name := "Item", next := 453814, root := ~, start := 453631, stop := 453813 ) ], count := [ "A", 2, 0, 100 ], name := "Row", next := 453827, root := ~, start := 453489, stop := 453826 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 106 ], name := "HorLine", next := 453844, root := ~, start := 453834, stop := 453843 ), 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 := 453895, root := ~, start := 453883, stop := 453894 ), rec( content := "\n \ ", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 107 ], name := "Item", next := 453911, root := ~, start := 453866, stop := 453910 ), 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 := 453973, root := ~, start := 453937, stop := 453972 ), rec( content := "\n \ ", count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 109 ], name := "Item", next := 453989, root := ~, start := 453920, stop := 453988 ), 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 := 454095, root := ~, start := 454059, stop := 4540\ 94 ), 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 := 4541\ 40, root := ~, start := 454128, stop := 454139 ), 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 := 454234, root := ~, start := 454219, stop := 45423\ 3 ), 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 := 4542\ 90, root := ~, start := 454281, stop := 454289 ), 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 := 454331, root := ~, start := 454015, stop := 454330 ), rec( content := "\n \ ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], name := "Item", next := 454347, root := ~, start := 453998, stop := 454346 ) ], count := [ "A", 2, 0, 106 ], name := "Row", next := 454360, root := ~, start := 453852, stop := 454359 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 112 ], name := "HorLine", next := 454377, root := ~, start := 454367, stop := 454376 ), 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 := 454439, root := ~, start := 454416, stop := 454438 ), rec( content := "\n \ ", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 113 ], name := "Item", next := 454455, root := ~, start := 454399, stop := 454454 ), 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 := 454521, root := ~, start := 454481, stop := 454520 ), rec( content := "\n \ ", count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 115 ], name := "Item", next := 454537, root := ~, start := 454464, stop := 454536 ), 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 := 454647, root := ~, start := 454607, stop := 454646 ), 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 := 454703, root := ~, start := 454680, stop := 45470\ 2 ), 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 := 454732, root := ~, start := 454563, stop := 454731 ), rec( content := "\n \ ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], name := "Item", next := 454748, root := ~, start := 454546, stop := 454747 ) ], count := [ "A", 2, 0, 112 ], name := "Row", next := 454761, root := ~, start := 454385, stop := 454760 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 118 ], name := "HorLine", next := 454778, root := ~, start := 454768, stop := 454777 ), 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 := 454835, root := ~, start := 454817, stop := 454834 ), rec( content := "\n \ ", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 119 ], name := "Item", next := 454851, root := ~, start := 454800, stop := 454850 ), 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 := 454909, root := ~, start := 454877, stop := 454908 ), 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 := 454986, root := ~, start := 454954, stop := 4549\ 85 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 121 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 121 ], name := "Alt", next := 455003, root := ~, start := 454920, stop := 455002 ), rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 121 ], name := "Item", next := 455019, root := ~, start := 454860, stop := 455018 ), 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 := 455114, root := ~, start := 455096, stop := 45511\ 3 ), 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 := 455176, root := ~, start := 455144, stop := 455175 ), 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 := 455227, root := ~, start := 455181, stop := 4552\ 26 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 123 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], name := "Alt", next := 455245, root := ~, start := 455045, stop := 455244 ), rec( content := "\n \ ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], name := "Item", next := 455261, root := ~, start := 455028, stop := 455260 ) ], count := [ "A", 2, 0, 118 ], name := "Row", next := 455274, root := ~, start := 454786, stop := 455273 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 124 ], name := "HorLine", next := 455291, root := ~, start := 455281, stop := 455290 ), 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 := 455348, root := ~, start := 455330, stop := 455347 ), rec( content := "\n \ ", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 125 ], name := "Item", next := 455364, root := ~, start := 455313, stop := 455363 ), 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 := 455417, root := ~, start := 455390, stop := 455416 ), 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 := 455489, root := ~, start := 455461, stop := 4554\ 88 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 127 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 127 ], name := "Alt", next := 455506, root := ~, start := 455428, stop := 455505 ), rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 127 ], name := "Item", next := 455522, root := ~, start := 455373, stop := 455521 ), 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 := 455617, root := ~, start := 455599, stop := 45561\ 6 ), 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 := 455675, root := ~, start := 455647, stop := 455674 ), 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 := 455733, root := ~, start := 455692, stop := 4557\ 32 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 129 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], name := "Alt", next := 455751, root := ~, start := 455548, stop := 455750 ), rec( content := "\n \ ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], name := "Item", next := 455767, root := ~, start := 455531, stop := 455766 ) ], count := [ "A", 2, 0, 124 ], name := "Row", next := 455780, root := ~, start := 455299, stop := 455779 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 130 ], name := "HorLine", next := 455797, root := ~, start := 455787, stop := 455796 ), 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 := 455858, root := ~, start := 455836, stop := 455857 ), rec( content := "\n \ ", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 131 ], name := "Item", next := 455874, root := ~, start := 455819, stop := 455873 ), 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 := 455931, root := ~, start := 455900, stop := 455930 ), rec( content := "\n \ ", count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 133 ], name := "Item", next := 455947, root := ~, start := 455883, stop := 455946 ), 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 := 456046, root := ~, start := 456024, stop := 45604\ 5 ), 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 := 456127, root := ~, start := 456096, stop := 4561\ 26 ), 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 := 456204, root := ~, start := 456189, stop := 45620\ 3 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], name := "Alt", next := 456222, root := ~, start := 455973, stop := 456221 ), rec( content := "\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], name := "Item", next := 456238, root := ~, start := 455956, stop := 456237 ) ], count := [ "A", 2, 0, 130 ], name := "Row", next := 456251, root := ~, start := 455805, stop := 456250 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 136 ], name := "HorLine", next := 456268, root := ~, start := 456258, stop := 456267 ) ], count := [ "A", 2, 0, 4 ], mathmode := "Text", name := "Table", next := 456281, root := ~, start := 446261, stop := 456280 ), rec( content := "\n ", count := [ "A", 2, 0, 137 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 1 ], mathmode := "Text", name := "Section", next := 456294, root := ~, start := 446014, stop := 456293 ), 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 := 456397, root := ~, start := 456312, stop := 456396 ), 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 := 456535, root := ~, start := 456425, stop := 456534 ), rec( content := "\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "Alt", next := 456546, root := ~, start := 456402, stop := 456545 ), 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 := 456592, root := ~, start := 456582, stop := 456591 ), 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 := 456662, root := ~, start := 456614, stop := 456661 ), 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 := 456722, root := ~, start := 456671, stop := 456721 ), 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 := 456803, root := ~, start := 456748, stop := 456802 ), rec( content := "\n \ ", count := [ "A", 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 9 ], name := "Item", next := 456819, root := ~, start := 456731, stop := 456818 ) ], count := [ "A", 3, 0, 4 ], name := "Row", next := 456832, root := ~, start := 456600, stop := 456831 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 10 ], name := "HorLine", next := 456849, root := ~, start := 456839, stop := 456848 ), 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 := 456909, root := ~, start := 456888, stop := 456908 ), rec( content := "\n \ ", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 11 ], name := "Item", next := 456925, root := ~, start := 456871, stop := 456924 ), 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 := 456984, root := ~, start := 456951, stop := 456983 ), 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 := 457079, root := ~, start := 457047, stop := 457078 ), 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 := 457138, root := ~, start := 457108, stop := 457137 ), rec( HTML := true , content := "\ )\n ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 13 ], name := "Alt", next := 457156, root := ~, start := 456995, stop := 457155 ), rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 13 ], name := "Item", next := 457172, root := ~, start := 456934, stop := 457171 ), 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 := 457270, root := ~, start := 457249, stop := 45726\ 9 ), 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 := 457333, root := ~, start := 457300, stop := 457332 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 15 ], name := "Alt", next := 457351, root := ~, start := 457198, stop := 457350 ), rec( content := "\n \ ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], name := "Item", next := 457367, root := ~, start := 457181, stop := 457366 ) ], count := [ "A", 3, 0, 10 ], name := "Row", next := 457380, root := ~, start := 456857, stop := 457379 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 16 ], name := "HorLine", next := 457397, root := ~, start := 457387, stop := 457396 ), 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 := 457458, root := ~, start := 457436, stop := 457457 ), rec( content := "\n \ ", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 17 ], name := "Item", next := 457474, root := ~, start := 457419, stop := 457473 ), 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 := 457520, root := ~, start := 457483, stop := 457519 ), 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 := 457618, root := ~, start := 457596, stop := 45761\ 7 ), 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 := 457731, root := ~, start := 457545, stop := 457730 ), rec( content := "\n \ ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], name := "Item", next := 457747, root := ~, start := 457529, stop := 457746 ) ], count := [ "A", 3, 0, 16 ], name := "Row", next := 457760, root := ~, start := 457405, stop := 457759 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 22 ], name := "HorLine", next := 457777, root := ~, start := 457767, stop := 457776 ), 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 := 457839, root := ~, start := 457816, stop := 457838 ), rec( content := "\n \ ", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 23 ], name := "Item", next := 457855, root := ~, start := 457799, stop := 457854 ), 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 := 457901, root := ~, start := 457864, stop := 457900 ), 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 := 458000, root := ~, start := 457977, stop := 45799\ 9 ), 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 := 458113, root := ~, start := 457926, stop := 458112 ), rec( content := "\n \ ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], name := "Item", next := 458129, root := ~, start := 457910, stop := 458128 ) ], count := [ "A", 3, 0, 22 ], name := "Row", next := 458142, root := ~, start := 457785, stop := 458141 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 28 ], name := "HorLine", next := 458159, root := ~, start := 458149, stop := 458158 ), 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 := 458222, root := ~, start := 458198, stop := 458221 ), rec( content := "\n \ ", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 29 ], name := "Item", next := 458238, root := ~, start := 458181, stop := 458237 ), 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 := 458302, root := ~, start := 458264, stop := 458301 ), rec( content := "\n \ ", count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 31 ], name := "Item", next := 458318, root := ~, start := 458247, stop := 458317 ), 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 := 458419, root := ~, start := 458395, stop := 45841\ 8 ), 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 := 458487, root := ~, start := 458449, stop := 4584\ 86 ), rec( HTML := true, content := ".\n ", count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], name := "Alt", next := 458505, root := ~, start := 458344, stop := 458504 ), rec( content := "\n \ ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], name := "Item", next := 458521, root := ~, start := 458327, stop := 458520 ) ], count := [ "A", 3, 0, 28 ], name := "Row", next := 458534, root := ~, start := 458167, stop := 458533 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 34 ], name := "HorLine", next := 458551, root := ~, start := 458541, stop := 458550 ), 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 := 458618, root := ~, start := 458590, stop := 458617 ), rec( content := "\n \ ", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 35 ], name := "Item", next := 458634, root := ~, start := 458573, stop := 458633 ), 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 := 458680, root := ~, start := 458643, stop := 458679 ), 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 := 458784, root := ~, start := 458756, stop := 45878\ 3 ), 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 := 458897, root := ~, start := 458705, stop := 458896 ), rec( content := "\n \ ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], name := "Item", next := 458913, root := ~, start := 458689, stop := 458912 ) ], count := [ "A", 3, 0, 34 ], name := "Row", next := 458926, root := ~, start := 458559, stop := 458925 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 40 ], name := "HorLine", next := 458943, root := ~, start := 458933, stop := 458942 ), 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 := 459011, root := ~, start := 458982, stop := 459010 ), rec( content := "\n \ ", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 41 ], name := "Item", next := 459027, root := ~, start := 458965, stop := 459026 ), 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 := 459073, root := ~, start := 459036, stop := 459072 ), 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 := 459178, root := ~, start := 459149, stop := 45917\ 7 ), 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 := 459291, root := ~, start := 459098, stop := 459290 ), rec( content := "\n \ ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], name := "Item", next := 459307, root := ~, start := 459082, stop := 459306 ) ], count := [ "A", 3, 0, 40 ], name := "Row", next := 459320, root := ~, start := 458951, stop := 459319 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 46 ], name := "HorLine", next := 459337, root := ~, start := 459327, stop := 459336 ), 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 := 459403, root := ~, start := 459376, stop := 459402 ), rec( content := "\n \ ", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 47 ], name := "Item", next := 459419, root := ~, start := 459359, stop := 459418 ), 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 := 459465, root := ~, start := 459428, stop := 459464 ), 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 := 459568, root := ~, start := 459541, stop := 45956\ 7 ), 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 := 459681, root := ~, start := 459490, stop := 459680 ), rec( content := "\n \ ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], name := "Item", next := 459697, root := ~, start := 459474, stop := 459696 ) ], count := [ "A", 3, 0, 46 ], name := "Row", next := 459710, root := ~, start := 459345, stop := 459709 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 52 ], name := "HorLine", next := 459727, root := ~, start := 459717, stop := 459726 ) ], count := [ "A", 3, 0, 4 ], mathmode := "Text", name := "Table", next := 459741, root := ~, start := 456552, stop := 459740 ), rec( content := "\n\n ", count := [ "A", 3, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 1 ], mathmode := "Text", name := "Section", next := 459755, root := ~, start := 456298, stop := 459754 ), 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 := 459842, root := ~, start := 459773, stop := 459841 ), 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 := 459980, root := ~, start := 459870, stop := 459979 ), rec( content := "\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "Alt", next := 459991, root := ~, start := 459847, stop := 459990 ), 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 := 460037, root := ~, start := 460027, stop := 460036 ), 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 := 460107, root := ~, start := 460059, stop := 460106 ), 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 := 460167, root := ~, start := 460116, stop := 460166 ), 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 := 460248, root := ~, start := 460193, stop := 460247 ), rec( content := "\n \ ", count := [ "A", 4, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 9 ], name := "Item", next := 460264, root := ~, start := 460176, stop := 460263 ) ], count := [ "A", 4, 0, 4 ], name := "Row", next := 460277, root := ~, start := 460045, stop := 460276 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 10 ], name := "HorLine", next := 460294, root := ~, start := 460284, stop := 460293 ), 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 := 460358, root := ~, start := 460333, stop := 460357 ), rec( content := "\n \ ", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 11 ], name := "Item", next := 460374, root := ~, start := 460316, stop := 460373 ), 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 := 460439, root := ~, start := 460400, stop := 460438 ), rec( content := "\n \ ", count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 13 ], name := "Item", next := 460455, root := ~, start := 460383, stop := 460454 ), 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 := 460596, root := ~, start := 460571, stop := 46059\ 5 ), 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 := 460676, root := ~, start := 460637, stop := 460675 ), 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 := 460729, root := ~, start := 460690, stop := 460728 ), 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 := 460829, root := ~, start := 460481, stop := 460828 ), rec( content := "\n \ ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], name := "Item", next := 460845, root := ~, start := 460464, stop := 460844 ) ], count := [ "A", 4, 0, 10 ], name := "Row", next := 460858, root := ~, start := 460302, stop := 460857 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 16 ], name := "HorLine", next := 460875, root := ~, start := 460865, stop := 460874 ), 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 := 460940, root := ~, start := 460914, stop := 460939 ), rec( content := "\n \ ", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 17 ], name := "Item", next := 460956, root := ~, start := 460897, stop := 460955 ), 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 := 461022, root := ~, start := 460982, stop := 461021 ), rec( content := "\n \ ", count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 19 ], name := "Item", next := 461038, root := ~, start := 460965, stop := 461037 ), 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 := 461167, root := ~, start := 461141, stop := 46116\ 6 ), 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 := 461248, root := ~, start := 461208, stop := 461247\ ), 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 := 461302, root := ~, start := 461262, stop := 4613\ 01 ), 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 := 461402, root := ~, start := 461064, stop := 461401 ), rec( content := "\n \ ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], name := "Item", next := 461418, root := ~, start := 461047, stop := 461417 ) ], count := [ "A", 4, 0, 16 ], name := "Row", next := 461431, root := ~, start := 460883, stop := 461430 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 22 ], name := "HorLine", next := 461448, root := ~, start := 461438, stop := 461447 ), 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 := 461506, root := ~, start := 461487, stop := 461505 ), rec( content := "\n \ ", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 23 ], name := "Item", next := 461522, root := ~, start := 461470, stop := 461521 ), 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 := 461581, root := ~, start := 461548, stop := 461580 ), rec( content := "\n \ ", count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 25 ], name := "Item", next := 461597, root := ~, start := 461531, stop := 461596 ), 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 := 461729, root := ~, start := 461710, stop := 46172\ 8 ), 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 := 461803, root := ~, start := 461770, stop := 461802 ), 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 := 461867, root := ~, start := 461835, stop := 461866 ), 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 := 462091, root := ~, start := 461623, stop := 462090 ), rec( content := "\n \ ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], name := "Item", next := 462107, root := ~, start := 461606, stop := 462106 ) ], count := [ "A", 4, 0, 22 ], name := "Row", next := 462120, root := ~, start := 461456, stop := 462119 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 28 ], name := "HorLine", next := 462137, root := ~, start := 462127, stop := 462136 ), 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 := 462194, root := ~, start := 462176, stop := 462193 ), rec( content := "\n \ ", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 29 ], name := "Item", next := 462210, root := ~, start := 462159, stop := 462209 ), 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 := 462329, root := ~, start := 462238, stop := 462328 ), rec( content := "\n \ ", count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 31 ], name := "Item", next := 462345, root := ~, start := 462219, stop := 462344 ), 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 := 4624\ 19, root := ~, start := 462401, stop := 462418 ), 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 := 462551, root := ~, start := 462460, stop := 4625\ 50 ), 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 := 462583, root := ~, start := 462371, stop := 462582 ), rec( content := "\n \ ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], name := "Item", next := 462599, root := ~, start := 462354, stop := 462598 ) ], count := [ "A", 4, 0, 28 ], name := "Row", next := 462612, root := ~, start := 462145, stop := 462611 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 34 ], name := "HorLine", next := 462629, root := ~, start := 462619, stop := 462628 ), 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 := 462681, root := ~, start := 462668, stop := 462680 ), rec( content := "\n \ ", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 35 ], name := "Item", next := 462697, root := ~, start := 462651, stop := 462696 ), 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 := 462750, root := ~, start := 462723, stop := 462749 ), rec( content := "\n \ ", count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 37 ], name := "Item", next := 462766, root := ~, start := 462706, stop := 462765 ), 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 := 4628\ 35, root := ~, start := 462822, stop := 462834 ), 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 := 462903, root := ~, start := 462876, stop := 462902 ), 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 := 462935, root := ~, start := 462792, stop := 462934 ), rec( content := "\n \ ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 39 ], name := "Item", next := 462951, root := ~, start := 462775, stop := 462950 ) ], count := [ "A", 4, 0, 34 ], name := "Row", next := 462964, root := ~, start := 462637, stop := 462963 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 40 ], name := "HorLine", next := 462981, root := ~, start := 462971, stop := 462980 ), 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 := 463041, root := ~, start := 463020, stop := 463040 ), rec( content := "\n \ ", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 41 ], name := "Item", next := 463057, root := ~, start := 463003, stop := 463056 ), 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 := 463118, root := ~, start := 463083, stop := 463117 ), rec( content := "\n \ ", count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 43 ], name := "Item", next := 463134, root := ~, start := 463066, stop := 463133 ), 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 := 4632\ 11, root := ~, start := 463190, stop := 463210 ), 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 := 463287, root := ~, start := 463252, stop := 463286 ), 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 := 463319, root := ~, start := 463160, stop := 463318 ), rec( content := "\n \ ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], name := "Item", next := 463335, root := ~, start := 463143, stop := 463334 ) ], count := [ "A", 4, 0, 40 ], name := "Row", next := 463348, root := ~, start := 462989, stop := 463347 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 46 ], name := "HorLine", next := 463365, root := ~, start := 463355, stop := 463364 ) ], count := [ "A", 4, 0, 4 ], mathmode := "Text", name := "Table", next := 463379, root := ~, start := 459997, stop := 463378 ), rec( content := "\n\n ", count := [ "A", 4, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 1 ], mathmode := "Text", name := "Section", next := 463393, root := ~, start := 459759, stop := 463392 ), 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 := 463484, root := ~, start := 463411, stop := 463483 ), 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 := 463622, root := ~, start := 463512, stop := 463621 ), rec( content := "\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "Alt", next := 463633, root := ~, start := 463489, stop := 463632 ), 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 := 463679, root := ~, start := 463669, stop := 463678 ), 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 := 463749, root := ~, start := 463701, stop := 463748 ), 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 := 463809, root := ~, start := 463758, stop := 463808 ), 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 := 463890, root := ~, start := 463835, stop := 463889 ), rec( content := "\n \ ", count := [ "A", 5, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 9 ], name := "Item", next := 463906, root := ~, start := 463818, stop := 463905 ) ], count := [ "A", 5, 0, 4 ], name := "Row", next := 463919, root := ~, start := 463687, stop := 463918 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 10 ], name := "HorLine", next := 463936, root := ~, start := 463926, stop := 463935 ), 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 := 463997, root := ~, start := 463975, stop := 463996 ), rec( content := "\n \ ", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 11 ], name := "Item", next := 464013, root := ~, start := 463958, stop := 464012 ), 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 := 464070, root := ~, start := 464039, stop := 464069 ), rec( content := "\n \ ", count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 13 ], name := "Item", next := 464086, root := ~, start := 464022, stop := 464085 ), 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 := 464173, root := ~, start := 464142, stop := 4641\ 72 ), 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 := 4642\ 40, root := ~, start := 464232, stop := 464239 ), 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 := 464371, root := ~, start := 464112, stop := 464370 ), rec( content := "\n \ ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], name := "Item", next := 464387, root := ~, start := 464095, stop := 464386 ) ], count := [ "A", 5, 0, 10 ], name := "Row", next := 464400, root := ~, start := 463944, stop := 464399 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 16 ], name := "HorLine", next := 464417, root := ~, start := 464407, stop := 464416 ), 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 := 464481, root := ~, start := 464456, stop := 464480 ), rec( content := "\n \ ", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 17 ], name := "Item", next := 464497, root := ~, start := 464439, stop := 464496 ), 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 := 464543, root := ~, start := 464506, stop := 464542 ), 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 := 464702, root := ~, start := 464569, stop := 464701 ), rec( content := "\n \ ", count := [ "A", 5, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], name := "Item", next := 464718, root := ~, start := 464552, stop := 464717 ) ], count := [ "A", 5, 0, 16 ], name := "Row", next := 464731, root := ~, start := 464425, stop := 464730 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 22 ], name := "HorLine", next := 464748, root := ~, start := 464738, stop := 464747 ), 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 := 464807, root := ~, start := 464787, stop := 464806 ), rec( content := "\n \ ", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 23 ], name := "Item", next := 464823, root := ~, start := 464770, stop := 464822 ), 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 := 464907, root := ~, start := 464849, stop := 464906 ), rec( content := "\n \ ", count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 25 ], name := "Item", next := 464923, root := ~, start := 464832, stop := 464922 ), 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 := 4649\ 99, root := ~, start := 464979, stop := 464998 ), 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 := 465098, root := ~, start := 465040, stop := 4650\ 97 ), 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 := 465142, root := ~, start := 464949, stop := 465141 ), rec( content := "\n \ ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], name := "Item", next := 465158, root := ~, start := 464932, stop := 465157 ) ], count := [ "A", 5, 0, 22 ], name := "Row", next := 465171, root := ~, start := 464756, stop := 465170 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 28 ], name := "HorLine", next := 465188, root := ~, start := 465178, stop := 465187 ), 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 := 465249, root := ~, start := 465227, stop := 465248 ), rec( content := "\n \ ", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 29 ], name := "Item", next := 465265, root := ~, start := 465210, stop := 465264 ), 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 := 465316, root := ~, start := 465291, stop := 465315 ), rec( content := "\n \ ", count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 31 ], name := "Item", next := 465332, root := ~, start := 465274, stop := 465331 ), 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 := 465413, root := ~, start := 465388, stop := 4654\ 12 ), 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 := 465479, root := ~, start := 465457, stop := 46547\ 8 ), 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 := 465546, root := ~, start := 465530, stop := 465545\ ), 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 := 4655\ 65, root := ~, start := 465554, stop := 465564 ), 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 := 465629, root := ~, start := 465358, stop := 465628 ), rec( content := "\n \ ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], name := "Item", next := 465645, root := ~, start := 465341, stop := 465644 ) ], count := [ "A", 5, 0, 28 ], name := "Row", next := 465658, root := ~, start := 465196, stop := 465657 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 34 ], name := "HorLine", next := 465675, root := ~, start := 465665, stop := 465674 ), 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 := 465731, root := ~, start := 465714, stop := 465730 ), rec( content := "\n \ ", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 35 ], name := "Item", next := 465747, root := ~, start := 465697, stop := 465746 ), 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 := 465793, root := ~, start := 465756, stop := 465792 ), 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 := 465952, root := ~, start := 465819, stop := 465951 ), rec( content := "\n \ ", count := [ "A", 5, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], name := "Item", next := 465968, root := ~, start := 465802, stop := 465967 ) ], count := [ "A", 5, 0, 34 ], name := "Row", next := 465981, root := ~, start := 465683, stop := 465980 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 40 ], name := "HorLine", next := 465998, root := ~, start := 465988, stop := 465997 ), 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 := 466053, root := ~, start := 466037, stop := 466052 ), rec( content := "\n \ ", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 41 ], name := "Item", next := 466069, root := ~, start := 466020, stop := 466068 ), 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 := 466130, root := ~, start := 466095, stop := 466129 ), rec( content := "\n \ ", count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 43 ], name := "Item", next := 466146, root := ~, start := 466078, stop := 466145 ), 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 := 4662\ 18, root := ~, start := 466202, stop := 466217 ), 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 := 466294, root := ~, start := 466259, stop := 466293 ), 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 := 466326, root := ~, start := 466172, stop := 466325 ), rec( content := "\n \ ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], name := "Item", next := 466342, root := ~, start := 466155, stop := 466341 ) ], count := [ "A", 5, 0, 40 ], name := "Row", next := 466355, root := ~, start := 466006, stop := 466354 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 46 ], name := "HorLine", next := 466372, root := ~, start := 466362, stop := 466371 ), 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 := 466431, root := ~, start := 466411, stop := 466430 ), rec( content := "\n \ ", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 47 ], name := "Item", next := 466447, root := ~, start := 466394, stop := 466446 ), 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 := 466493, root := ~, start := 466456, stop := 466492 ), 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 := 466652, root := ~, start := 466519, stop := 466651 ), rec( content := "\n \ ", count := [ "A", 5, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], name := "Item", next := 466668, root := ~, start := 466502, stop := 466667 ) ], count := [ "A", 5, 0, 46 ], name := "Row", next := 466681, root := ~, start := 466380, stop := 466680 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 52 ], name := "HorLine", next := 466698, root := ~, start := 466688, stop := 466697 ), 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 := 466759, root := ~, start := 466737, stop := 466758 ), rec( content := "\n \ ", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 53 ], name := "Item", next := 466775, root := ~, start := 466720, stop := 466774 ), 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 := 466838, root := ~, start := 466801, stop := 466837 ), rec( content := "\n \ ", count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 55 ], name := "Item", next := 466854, root := ~, start := 466784, stop := 466853 ), 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 := 4669\ 32, root := ~, start := 466910, stop := 466931 ), 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 := 467010, root := ~, start := 466973, stop := 467009 ), 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 := 467042, root := ~, start := 466880, stop := 467041 ), rec( content := "\n \ ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], name := "Item", next := 467058, root := ~, start := 466863, stop := 467057 ) ], count := [ "A", 5, 0, 52 ], name := "Row", next := 467071, root := ~, start := 466706, stop := 467070 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 58 ], name := "HorLine", next := 467088, root := ~, start := 467078, stop := 467087 ), 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 := 467147, root := ~, start := 467127, stop := 467146 ), rec( content := "\n \ ", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 59 ], name := "Item", next := 467163, root := ~, start := 467110, stop := 467162 ), 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 := 467218, root := ~, start := 467189, stop := 467217 ), rec( content := "\n \ ", count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 61 ], name := "Item", next := 467234, root := ~, start := 467172, stop := 467233 ), 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 := 467319, root := ~, start := 467290, stop := 4673\ 18 ), 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 := 4673\ 72, root := ~, start := 467364, stop := 467371 ), 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 := 467692, root := ~, start := 467260, stop := 467691 ), rec( content := "\n \ ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], name := "Item", next := 467708, root := ~, start := 467243, stop := 467707 ) ], count := [ "A", 5, 0, 58 ], name := "Row", next := 467721, root := ~, start := 467096, stop := 467720 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 64 ], name := "HorLine", next := 467738, root := ~, start := 467728, stop := 467737 ), 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 := 467799, root := ~, start := 467777, stop := 467798 ), rec( content := "\n \ ", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 65 ], name := "Item", next := 467815, root := ~, start := 467760, stop := 467814 ), 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 := 467877, root := ~, start := 467841, stop := 467876 ), rec( content := "\n \ ", count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 67 ], name := "Item", next := 467893, root := ~, start := 467824, stop := 467892 ), 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 := 4679\ 71, root := ~, start := 467949, stop := 467970 ), 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 := 468048, root := ~, start := 468012, stop := 468047 ), 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 := 468080, root := ~, start := 467919, stop := 468079 ), rec( content := "\n \ ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], name := "Item", next := 468096, root := ~, start := 467902, stop := 468095 ) ], count := [ "A", 5, 0, 64 ], name := "Row", next := 468109, root := ~, start := 467746, stop := 468108 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 70 ], name := "HorLine", next := 468126, root := ~, start := 468116, stop := 468125 ), 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 := 468186, root := ~, start := 468165, stop := 468185 ), rec( content := "\n \ ", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 71 ], name := "Item", next := 468202, root := ~, start := 468148, stop := 468201 ), 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 := 468248, root := ~, start := 468211, stop := 468247 ), 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 := 468407, root := ~, start := 468274, stop := 468406 ), rec( content := "\n \ ", count := [ "A", 5, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], name := "Item", next := 468423, root := ~, start := 468257, stop := 468422 ) ], count := [ "A", 5, 0, 70 ], name := "Row", next := 468436, root := ~, start := 468134, stop := 468435 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 76 ], name := "HorLine", next := 468453, root := ~, start := 468443, stop := 468452 ), 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 := 468530, root := ~, start := 468492, stop := 468529 ), rec( content := "\n \ ", count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 77 ], name := "Item", next := 468546, root := ~, start := 468475, stop := 468545 ), 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 := 468592, root := ~, start := 468555, stop := 468591 ), 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 := 468780, root := ~, start := 468618, stop := 468779 ), rec( content := "\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], name := "Item", next := 468796, root := ~, start := 468601, stop := 468795 ) ], count := [ "A", 5, 0, 76 ], name := "Row", next := 468809, root := ~, start := 468461, stop := 468808 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 82 ], name := "HorLine", next := 468826, root := ~, start := 468816, stop := 468825 ), 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 := 468900, root := ~, start := 468865, stop := 468899 ), rec( content := "\n \ ", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 83 ], name := "Item", next := 468916, root := ~, start := 468848, stop := 468915 ), 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 := 468962, root := ~, start := 468925, stop := 468961 ), 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 := 469150, root := ~, start := 468988, stop := 469149 ), rec( content := "\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], name := "Item", next := 469166, root := ~, start := 468971, stop := 469165 ) ], count := [ "A", 5, 0, 82 ], name := "Row", next := 469179, root := ~, start := 468834, stop := 469178 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 88 ], name := "HorLine", next := 469196, root := ~, start := 469186, stop := 469195 ), 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 := 469255, root := ~, start := 469235, stop := 469254 ), rec( content := "\n \ ", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 89 ], name := "Item", next := 469271, root := ~, start := 469218, stop := 469270 ), 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 := 469317, root := ~, start := 469280, stop := 469316 ), 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 := 469512, root := ~, start := 469487, stop := 469511 ), 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 := 4695\ 67, root := ~, start := 469558, stop := 469566 ), 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 := 469628, root := ~, start := 469599, stop := 46962\ 7 ), rec( HTML := true, content := ".\n ", count := [ "A", 5, 0, 93 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], name := "Alt", next := 469646, root := ~, start := 469343, stop := 469645 ), rec( content := "\n \ ", count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], name := "Item", next := 469662, root := ~, start := 469326, stop := 469661 ) ], count := [ "A", 5, 0, 88 ], name := "Row", next := 469675, root := ~, start := 469204, stop := 469674 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 94 ], name := "HorLine", next := 469692, root := ~, start := 469682, stop := 469691 ) ], count := [ "A", 5, 0, 4 ], mathmode := "Text", name := "Table", next := 469706, root := ~, start := 463639, stop := 469705 ), rec( content := "\n\n ", count := [ "A", 5, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 1 ], mathmode := "Text", name := "Section", next := 469720, root := ~, start := 463397, stop := 469719 ), 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 := 469808, root := ~, start := 469738, stop := 469807 ), 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 := 469946, root := ~, start := 469836, stop := 469945 ), rec( content := "\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "Alt", next := 469957, root := ~, start := 469813, stop := 469956 ), 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 := 470003, root := ~, start := 469993, stop := 470002 ), 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 := 470073, root := ~, start := 470025, stop := 470072 ), 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 := 470133, root := ~, start := 470082, stop := 470132 ), 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 := 470214, root := ~, start := 470159, stop := 470213 ), rec( content := "\n \ ", count := [ "A", 6, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 9 ], name := "Item", next := 470230, root := ~, start := 470142, stop := 470229 ) ], count := [ "A", 6, 0, 4 ], name := "Row", next := 470243, root := ~, start := 470011, stop := 470242 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 10 ], name := "HorLine", next := 470260, root := ~, start := 470250, stop := 470259 ), 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 := 470321, root := ~, start := 470299, stop := 470320 ), rec( content := "\n \ ", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 11 ], name := "Item", next := 470337, root := ~, start := 470282, stop := 470336 ), 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 := 470421, root := ~, start := 470363, stop := 470420 ), rec( content := "\n \ ", count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 13 ], name := "Item", next := 470437, root := ~, start := 470346, stop := 470436 ), 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 := 4705\ 15, root := ~, start := 470493, stop := 470514 ), 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 := 470614, root := ~, start := 470544, stop := 470613 ), 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 := 470742, root := ~, start := 470463, stop := 470741 ), rec( content := "\n \ ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], name := "Item", next := 470758, root := ~, start := 470446, stop := 470757 ) ], count := [ "A", 6, 0, 10 ], name := "Row", next := 470771, root := ~, start := 470268, stop := 470770 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 16 ], name := "HorLine", next := 470788, root := ~, start := 470778, stop := 470787 ), 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 := 470851, root := ~, start := 470827, stop := 470850 ), rec( content := "\n \ ", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 17 ], name := "Item", next := 470867, root := ~, start := 470810, stop := 470866 ), 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 := 470921, root := ~, start := 470893, stop := 470920 ), rec( content := "\n \ ", count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 19 ], name := "Item", next := 470937, root := ~, start := 470876, stop := 470936 ), 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 := 471038, root := ~, start := 471014, stop := 47103\ 7 ), 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 := 471105, root := ~, start := 471077, stop := 4711\ 04 ), 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 := 471145, root := ~, start := 471110, stop := 471144 ), 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 := 471189, root := ~, start := 470963, stop := 471188 ), rec( content := "\n \ ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], name := "Item", next := 471205, root := ~, start := 470946, stop := 471204 ) ], count := [ "A", 6, 0, 16 ], name := "Row", next := 471218, root := ~, start := 470796, stop := 471217 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 22 ], name := "HorLine", next := 471235, root := ~, start := 471225, stop := 471234 ), 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 := 471309, root := ~, start := 471274, stop := 471308 ), rec( content := "\n \ ", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 23 ], name := "Item", next := 471325, root := ~, start := 471257, stop := 471324 ), 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 := 471379, root := ~, start := 471351, stop := 471378 ), rec( content := "\n \ ", count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 25 ], name := "Item", next := 471395, root := ~, start := 471334, stop := 471394 ), 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 := 471515, root := ~, start := 471480, stop := 47151\ 4 ), 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 := 471582, root := ~, start := 471554, stop := 4715\ 81 ), 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 := 471636, root := ~, start := 471587, stop := 471635 ), 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 := 471792, root := ~, start := 471421, stop := 471791 ), rec( content := "\n \ ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], name := "Item", next := 471808, root := ~, start := 471404, stop := 471807 ) ], count := [ "A", 6, 0, 22 ], name := "Row", next := 471821, root := ~, start := 471243, stop := 471820 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 28 ], name := "HorLine", next := 471838, root := ~, start := 471828, stop := 471837 ) ], count := [ "A", 6, 0, 4 ], mathmode := "Text", name := "Table", next := 471852, root := ~, start := 469963, stop := 471851 ), rec( content := "\n ", count := [ "A", 6, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 1 ], mathmode := "Text", name := "Section", next := 471865, root := ~, start := 469724, stop := 471864 ), 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 := 471968, root := ~, start := 471883, stop := 471967 ), 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 := 472106, root := ~, start := 471996, stop := 472105 ), rec( content := "\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "Alt", next := 472117, root := ~, start := 471973, stop := 472116 ), 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 := 472163, root := ~, start := 472153, stop := 472162 ), 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 := 472233, root := ~, start := 472185, stop := 472232 ), 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 := 472293, root := ~, start := 472242, stop := 472292 ), 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 := 472374, root := ~, start := 472319, stop := 472373 ), rec( content := "\n \ ", count := [ "A", 7, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 9 ], name := "Item", next := 472390, root := ~, start := 472302, stop := 472389 ) ], count := [ "A", 7, 0, 4 ], name := "Row", next := 472403, root := ~, start := 472171, stop := 472402 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 10 ], name := "HorLine", next := 472420, root := ~, start := 472410, stop := 472419 ), 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 := 472479, root := ~, start := 472459, stop := 472478 ), rec( content := "\n \ ", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 11 ], name := "Item", next := 472495, root := ~, start := 472442, stop := 472494 ), 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 := 472574, root := ~, start := 472521, stop := 472573 ), rec( content := "\n \ ", count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 13 ], name := "Item", next := 472590, root := ~, start := 472504, stop := 472589 ), 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 := 4726\ 66, root := ~, start := 472646, stop := 472665 ), 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 := 472760, root := ~, start := 472707, stop := 4727\ 59 ), 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 := 472792, root := ~, start := 472616, stop := 472791 ), rec( content := "\n \ ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], name := "Item", next := 472808, root := ~, start := 472599, stop := 472807 ) ], count := [ "A", 7, 0, 10 ], name := "Row", next := 472821, root := ~, start := 472428, stop := 472820 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 16 ], name := "HorLine", next := 472838, root := ~, start := 472828, stop := 472837 ), 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 := 472900, root := ~, start := 472877, stop := 472899 ), rec( content := "\n \ ", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 17 ], name := "Item", next := 472916, root := ~, start := 472860, stop := 472915 ), 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 := 472996, root := ~, start := 472942, stop := 472995 ), rec( content := "\n \ ", count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 19 ], name := "Item", next := 473012, root := ~, start := 472925, stop := 473011 ), 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 := 4730\ 91, root := ~, start := 473068, stop := 473090 ), 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 := 473186, root := ~, start := 473120, stop := 4731\ 85 ), 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 := 473255, root := ~, start := 473232, stop := 47325\ 4 ), 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 := 473311, root := ~, start := 473288, stop := 47331\ 0 ), 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 := 473438, root := ~, start := 473038, stop := 473437 ), rec( content := "\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], name := "Item", next := 473454, root := ~, start := 473021, stop := 473453 ) ], count := [ "A", 7, 0, 16 ], name := "Row", next := 473467, root := ~, start := 472846, stop := 473466 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 22 ], name := "HorLine", next := 473484, root := ~, start := 473474, stop := 473483 ), 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 := 473547, root := ~, start := 473523, stop := 473546 ), rec( content := "\n \ ", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 23 ], name := "Item", next := 473563, root := ~, start := 473506, stop := 473562 ), 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 := 473643, root := ~, start := 473589, stop := 473642 ), rec( content := "\n \ ", count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 25 ], name := "Item", next := 473659, root := ~, start := 473572, stop := 473658 ), 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 := 4737\ 39, root := ~, start := 473715, stop := 473738 ), 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 := 473834, root := ~, start := 473768, stop := 4738\ 33 ), 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 := 473904, root := ~, start := 473880, stop := 47390\ 3 ), 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 := 473960, root := ~, start := 473937, stop := 47395\ 9 ), 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 := 474087, root := ~, start := 473685, stop := 474086 ), rec( content := "\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], name := "Item", next := 474103, root := ~, start := 473668, stop := 474102 ) ], count := [ "A", 7, 0, 22 ], name := "Row", next := 474116, root := ~, start := 473492, stop := 474115 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 28 ], name := "HorLine", next := 474133, root := ~, start := 474123, stop := 474132 ), 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 := 474215, root := ~, start := 474172, stop := 474214 ), rec( content := "\n \ ", count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 29 ], name := "Item", next := 474231, root := ~, start := 474155, stop := 474230 ), 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 := 474277, root := ~, start := 474240, stop := 474276 ), 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 := 474465, root := ~, start := 474303, stop := 474464 ), rec( content := "\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], name := "Item", next := 474481, root := ~, start := 474286, stop := 474480 ) ], count := [ "A", 7, 0, 28 ], name := "Row", next := 474494, root := ~, start := 474141, stop := 474493 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 34 ], name := "HorLine", next := 474511, root := ~, start := 474501, stop := 474510 ) ], count := [ "A", 7, 0, 4 ], mathmode := "Text", name := "Table", next := 474525, root := ~, start := 472123, stop := 474524 ), rec( content := "\n ", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 1 ], mathmode := "Text", name := "Section", next := 474538, root := ~, start := 471869, stop := 474537 ), rec( content := "\n", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 1 ], mathmode := "Text", name := "Appendix", next := 474550, root := ~, start := 440899, stop := 474549 ), 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 := 474630, root := ~, start := 474593, stop := 474629 ), rec( attributes := rec( ), content := 0, count := [ "Ind", 0, 0, 1 ], mathmode := "Text", name := "TheIndex", next := 474642, root := ~, start := 474631, stop := 474641 ) ], count := [ 0, 0, 0, 1 ], name := "Book", next := 474651, root := ~, start := 1962, stop := 474650 ) ], 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://digraphs.github.io/Digraphs/.\n \n\n \ \n unzip and untar the file, this should create a directory \ 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 I\ nside 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. ty\ pe gap at the command line).\n \n\n \n ty\ pe LoadPackage(\"digraphs\");\n \n \n\n If 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 &Di\ graphs; package is written in &GAP; and C code and requires the\n &IO; pack\ age. The &IO; package is used to read and write transformations,\n partial\ permutations, and bipartitions to a file. \n\n \n\n \n The Grape package\n\n Th\ e &Grape; package must be available for the following operations to be\n \ available:\n\n \n with a digraph\ argument\n with a digraph argumen\ t\n with a &Grape; graph argument<\ /Item>\n \n\n If &Grape; is not available, then &Digraphs; ca\ n be used as normal with\n the exception that the functions above will no\ t 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 should \ be compiled. This component contains certain low-level\n functions required\ by &Digraphs;.\n \n\n It is not possible to use the &Digraphs; pack\ age without compiling it. \n\n To compile the kernel component inside th\ e pkg/&ARCHIVENAME;\n directory, type\n \n\n \n\n If you installed the package in anoth\ er 'pkg' directory than the standard 'pkg'\n directory in your &GAP; instal\ lation, then you have to do two things. Firstly\n during compilation you ha\ ve to use the option '--with-gaproot=PATH' of the\n 'configure' script wher\ e 'PATH' is a path to the main GAP root directory (if\n not given the defau\ lt '../..' is assumed). \n\n If you installed &GAP; on several architect\ ures, you must execute the\n configure/make step for each of the architectu\ res. You can either do this\n immediately after configuring and compiling G\ AP itself on this architecture, or\n alternatively set the environment\n \ variable 'CONFIGNAME' to the name of the configuration you used when compilin\ g\n GAP before running './configure'. Note however that your compiler choi\ ce 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 \n\n \n \n\n \n Rebuilding the do\ cumentation\n The &Digraphs; package comes complete with pdf, htm\ l, and text versions of\n the documentation. However, you might find it nec\ essary, at some point, to\n rebuild the documentation. To rebuild the docum\ entation, please use the\n function .\n\n \ \n \n Nothing\ \n \n This function should be called with no argume\ nt to compile the &Digraphs;\n documentation.\n \n\n\n \n\n \n \n\n \ \n Testing your installation\n\n In this section we d\ escribe how to test that &Digraphs; is working as\n intended. To test that\ &Digraphs; is installed correctly use\n or for more extensive tests use\n . <\ P/>\n\n If something goes wrong, then please review the instructions in Sec\ tion and ensure that &Digraphs; ha\ s been\n properly installed. If you continue having problems, please use t\ he https://github.com/digraphs/Digraphs/iss\ ues\n to report the issues you are having.\n\n \n \n true or false\ .\n \n This function can be called without argu\ ments to test your installation of\n &Digraphs; is working correctly. 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 called 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 quickly 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 series of\n \ very demanding tests, which measure the performance of a variety of\n func\ tions on large examples. These tests take a long time to\n complete, at le\ ast several minutes. \n\n For these tests to complete, the digraphs lib\ rary digraphs-lib must\n be downloaded and placed in the digraphs\ directory in a subfolder\n named digraphs-lib.\n This librar\ y can be found on the &Digraphs; website.\n \n\n\n\ \n \n\n \n \n\n\n\n \n Creating digraphs\n\n In this chapt\ er we describe how to create digraphs.\n\n Creating dig\ raphs\n \n \n \n Every digraph in &Digraphs; belongs to the categ\ ory\n IsDigraph. Some basic attributes and operations for digraphs\ \n are , , and\n\ .\n \n\n\n <\ ManSection>\n \n \n IsMutableDigraph is a synonym for a\ nd . A mutable digraph may be\ changed\n in-place by methods in the &Digraphs; package, and is not attrib\ ute-storing\n – see .\n \n\n A mutable digraph may be converted into an immutable attr\ ibute-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 and\n at\ tribute-storing. In particular, they lie in\n . \n\n A mutable digraph may be converted to \ an immutable\n digraph that lies in the category IsImmutableDigraph \ by calling\n on the digraph.\ \n\n The operation can be used\n \ to construct a mutable copy of an immutable digraph. It is however not\n po\ ssible to convert an immutable digraph into a mutable digraph in-place.\n \n\n\n \n \n \n IsCayleyDigraph is a subcate\ gory of IsDigraph. Digraphs that\n are Cayley digraphs of a group an\ d that are constructed by the operation\n are\ constructed in this category, and are\n always immutable.\n \n\n\n \n \n \n IsDigraphWithAdjacencyFun\ ction is a subcategory of IsDigraph.\n Digraphs that are crea\ ted using an adjacency function are constructed\n in this category.\n \ \n\n\n \n \n \n \n The t\ ype of all digraphs is DigraphByOutNeighboursType.\n The family of a\ ll digraphs is DigraphFamily.\n \n\n\n <\ ManSection>\n \n \n \n A digraph.\n \n If the optional first argume\ nt filt is present, then this should\n specify the category or repre\ sentation the digraph being created will\n belong to. For example, if fi\ lt is ,\n then the digraph being create\ d will be mutable, if filt is , \ then the digraph will be immutable. \n If the optional first argument fi\ lt is not present, then is used by\ default.\n \n for a list (i.e. an adjacency list)\ \n \n if obj is a list of lists of positive integers \ in\n the range from 1 to Length(obj), then this fu\ nction\n returns the digraph with vertices\n E ^ 0 = [\ 1 .. Length(obj)],\n and edges corresponding to the 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[i]\ ; the source of this edge is\n i and the range is j. If <\ C>j occurs in\n obj[i] with multiplicity k, th\ en 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 digraph with ve\ rtices\n E ^ 0 = [1 .. Length(obj)], and\n \ Length(source) edges. For each i in\n [1 .. Le\ ngth(source)] there exists an edge with source\n vertex s\ ource[i] and range vertex range[i].\n See and .\n \n\n The ver\ tices 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 ra\ nge are lists of equal length consisting of positive integers\n in \ the list [1 .. obj], then this function\n returns a digra\ ph with vertices\n E ^ 0 = [1 .. obj], and\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 li\ st and func is a function taking 2\n arguments that are elements\ of list, and func returns\n true or false,\ 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]) re\ turns true.\n \n\n for a group, a list, and two \ functions\n \n The arguments will be G, list, act,\ adj. \n\n Let G be a group acting on the objects in \ list via\n the action act, and let adj be a function \ taking\n two objects from list as arguments and returning tru\ e or\n false. The function adj will describe the adja\ cency\n between objects from list, which is invariant under 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 The action \ of the group G on the objects in list is stored\n in the \ attribute , and is used to speed up\n opera\ tions like .\n \n\n for \ a Grape package graph\n \n if obj is a &Grape;\ package graph (i.e. a record for which the\n function IsGraph r\ eturns true), then this function returns\n a digraph isomorphic \ to obj.\n \n\n for a binary relation\n \ \n if obj is a binary relation on the points [1 .. n\ ] for\n some posititve integer n, then this function returns\ the digraph\n defined by obj. Specifically, this function retur\ ns a digraph\n which has n vertices, and which has an edge with \ source i\n and range j if and only if [i,j] is a p\ air in\n the binary relation obj.\n \n \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> function(x,\ y) return Intersection(x, y) = []; end);;\ngap> Digraph(Petersen);\n\ngap> gr := Digraph([1 .. 10], ReturnTru\ e);\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, and\n \ it is actually the point graph of the van Lint-Schrijver partial\n geom\ etry, . The algebraic description\n is taken from t\ he seminal paper of Calderbank and Kantor .\n f := GF(3 ^ 4);\nGF(3^4)\ngap> gamma := First(f, x -> Ord\ er(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;\nfunction( \ x, y ) ... end\ngap> digraph := Digraph(AsList(f), adj);\n\ngap> group := Group(Z(3));;\ngap> act := \\*;\n<\ Operation \"*\">\ngap> digraph := Digraph(group, List(f), act, adj);\n\n]]>\n \n\n\n \n \n A digraph.\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 list is the adjacency m\ atrix 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 v\ ertices which has the edge [i,j] if and only if\n list[i][\ j] is true. \n\n DigraphByAdjacency\ Matrix([\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 := Digr\ aphByAdjacencyMatrix(IsMutableDigraph, \n> [[true, false, true],\n> [false, f\ alse, true],\n> [false, true, false]]);\n\n]]>\n \n\n\n \n \ \n A digrap\ h.\n \n If the optional first argument filt \ is present, then this should\n specify the category or representation the d\ igraph being created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutab\ le, if filt is , then the digrap\ h will be immutable. \n If the optional first argument filt is not p\ resent, then is used by default. \n\ \n If list is list of pairs of positive integers, then this function\ \n returns the digraph with the minimum number of vertices m such th\ at\n its list equal list. \n\n If the optional second argument\ n is a positive integer with\n n >= m (with m \ defined as above),\n then this function returns the digraph with n v\ ertices 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(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]]>\ \n \n\n\n \n \n \n An immutable digraph.\n \ \n \n If G is a permutation group, edges\ is an edge or list of\n edges, and n is a non-negative integer \ such that G fixes\n [1 .. n] setwise, then\n this o\ peration returns an immutable digraph with n vertices and the union\n \ of the orbits of the edges in edges under the action of the\n pe\ rmutation group G. An edge in this context is simply a pair of\n po\ sitive integers.\n \n\n If the optional third argument n is n\ ot present, then the largest\n moved point of the permutation group G is used by default. \n\n digraph := EdgeOrbit\ sDigraph(Group((1, 3), (1, 2)(3, 4)),\n> [[1, \ 2], [4, 5]], 5);\n\ngap> OutNeigh\ bours(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 should\n s\ pecify 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 list is a list of lists of posi\ tive integers list the range\n [1 .. Length(list)], then this f\ unction returns the\n digraph with vertices E^0=[1 .. Length(lis\ t)], and\n edges corresponding to the entries of list. More pr\ ecisely, there is an\n edge with source vertex i and range vertex j\ if i is list\n list[j]. \n\n If i occur\ s list list[j] with multiplicity k,\n then there are \ k multiple edges from i to j. \n\n See .\n\n 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(IsMutableDigraph, \n> \ [[2, 3, 2], [1], [1, 2, 3]]);\n\ \n]]>\n \n\n\n \n \n An immutable digraph.\n \n Let G be any group and let gens b\ e a list of elements of\n G. This operation returns an immutable dig\ raph that corresponds to\n the Cayley graph of G with respect\n <\ A>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 there 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 created b\ y this operation belongs to the category , the group G can be recovered from the\n digraph using , and the generators\n gens can be obtaine\ d using . \n\n Note that this f\ unction can only return an immutable digraph.\n\n \ G := DihedralGroup(8);\n\ngap> CayleyDig\ raph(G);\n\ngap> G := DihedralGro\ up(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> CayleyDigraph(G);\n\ngap> digraph := CayleyDigraph(G, [(\ )]);\n\ngap> GroupOfCayleyDigraph(\ digraph) = G;\ntrue\ngap> GeneratorsOfCayleyDigraph(digraph);\n[ () ]]]>\n \n\n\n \n\n C\ hanging representations\n \n \n A binary relation.\n \n If digraph is a digraph with a positive number of vertices\ \n n, and no multiple edges, then this operation returns a binary\n \ relation on the points [1..n].\n The pair [i,j] is in the \ binary relation if and only if\n [i,j] is an edge in digraph.\ \n \n D := Digraph([[3, 2], [1, 2], [2], [3, 4]\ ]);\n\ngap> AsBinaryRelation(D);\n\ Binary Relation on 4 points\n]]>\n \n\n\ \n \n \n \ A digraph, or fail.\n \n If the op\ tional first argument filt is present, then this should\n specify th\ e category or representation the digraph being created will\n belong to. Fo\ r example, if filt is ,\n then the d\ igraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optional f\ irst argument filt is not present, then is used by default. \n\n If trans is a transformation, a\ nd n is a non-negative integer\n such that the restriction of tra\ ns to [1 .. n] defines\n a transformation of [1 .. \ n], then AsDigraph\n returns the functional digraph with \ n vertices defined by\n trans. See .\n \n\n Specifically, the digraph returned by AsDigraph has n edges:\n for each vertex x in [1 .. n]\ , there is a unique edge\n with source x; this edge has range x^<\ A>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 AsDi\ graph(trans, n)\n returns fail.\n\n f := Transformation([4, 3, 3, 1, 7, 9, 10, 4, 2, 3]);\nTransform\ ation( [ 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 := BinaryRelat\ ionOnPoints(\n> [[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\nBinary Relation\ on 5 points\ngap> D := AsDigraph(b);\n\n]]>\n \n\n\n \n \ \n A &Grape; package graph.\n \n If digraph is a mutable or immutable digr\ aph without multiple edges, then this operation\n returns a &Grape; package\ graph that is isomorphic to digraph. \n\n If digraph is \ a multidigraph, then since &Grape; does not support\n multiple edges, the m\ ultiple edges will be reduced to a single edge in the\n result. In order wo\ rds, for a multidigraph this operation will return the\n same as Graph(D\ igraphRemoveAllMultipleEdges(digraph)).\n\n Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function\ (x, y) return Intersection(x, y) = []; end);;\ngap> Display(Petersen);\nrec(\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> Digraph(Pe\ tersen);\n\ngap> Graph(last) = P\ etersen;\ntrue]]>\n \n\n\n \n \n A &Grape; package g\ raph.\n \n If digraph is a digraph, then thi\ s method returns the same as\n , except that if dig\ raph is immutable, then the\n result will be stored as a mutable attrib\ ute of digraph.\n In this latter case, when AsGraph(digrap\ h) is called\n subsequently, the same &GAP; object will be retur\ ned as before.\n\n D := Digraph([[1, 2], [3], []])\ ;\n\ngap> G := AsGraph(D);\nrec( a\ djacencies := [ [ 1, 2 ], [ 3 ], [ ] ], group := Group(()), \n isGraph := tr\ ue, names := [ 1 .. 3 ], order := 3, \n representatives := [ 1, 2, 3 ], schre\ ierVector := [ -1, -2, -3 ] )\n]]>\n \n\ \n\n \n \n <\ Returns>A transformation, or fail\n \n If digraph is a functional digraph, then AsTransformation\n return\ s the transformation which is defined by digraph. See . Otherwise,\n AsTransformation(digraph\ ) returns fail. \n\n If digraph is a function\ al digraph with n vertices, then\n AsTransformation(digrap\ h) will return the\n transformation f of degree at most <\ M>n where for each 1\n \\leq i \\leq n, i ^ f is equal \ 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\ngap> AsTransformation(D);\nfail]]>\n \n\n\n \n\n \ New digraphs from old\n \n \n \n \n \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 synonym for \ DigraphCopySameMutability.\n D := CycleDigraph(\ 10);\n\ngap> DigraphCopy(D) = D;\ntr\ ue\ngap> IsIdenticalObj(DigraphCopy(D), D);\nfalse\ngap> DigraphMutableCopy(D)\ ;\n\n]]>\n \n\n\n \n \n \n \n \n A digraph.\n \n Each of these operations returns \ either the original argument\n digraph, or a new copy of digraph<\ /A> of the appropriate\n mutability, according to the mutability of digr\ aph.\n C := CycleDigraph(10);\n\ngap> D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ng\ ap> C = D;\ntrue\ngap> D := DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntr\ ue\ngap> D := DigraphMutableCopyIfMutable(C);\n\ngap> IsMutableDigraph(D);\nfalse\ngap> D := DigraphMutableCopyIfI\ mmutable(C);\n\ngap> IsMutableDigr\ aph(D);\ntrue\ngap> C := CycleDigraph(IsMutableDigraph, 10);\n\ngap> D := DigraphImmutableCopyIfImmutable(C);\n<\ mutable digraph with 10 vertices, 10 edges>\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 := DigraphMutableCopyIfMutable(C);\n\ngap> IsMutableDigraph(D);\ntrue\ngap> D := DigraphMutab\ leCopyIfImmutable(C);\n\ngap> IsId\ enticalObj(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 constr\ ucted from digraph by retaining precisely those vertices in\n ver\ ts, and those edges whose source and range vertices are both\n containe\ d in verts. \n\n The vertices of the induced subdigraph are\n \ [1..Length(verts)] but the original vertex labels can be\n accessed\ via . \n\n If digraph belong\ s to , then\n digraph is modified in\ place. If digraph belongs to \ , a new immutable digraph containing the\n appropriate\n vertices and ed\ ges 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 := Di\ graphMutableCopy(D);\n\ngap> n\ ew := 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 the su\ bdigraph of\n digraph induced by the set of non-isolated vertices,\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 digraph<\ /A> is preserved, as are\n the labels of the remaining vertices and edges;\ \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 digraph is immutable, then a new immutable digraph is returned.\n Otherwise, the\ isolated vertices of the mutable digraph digraph are\n removed in-p\ lace.\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 If <\ A>digraph is a digraph, then MaximalSymmetricSubdigraph\n return\ s a symmetric digraph without multiple edges which has the same\n vertex se\ t as digraph, and whose edge list is formed from\n digraph by\ ignoring the multiplicity of edges, and by ignoring\n edges [u,v] f\ or which there does not exist an edge [v,u].\n \n\n The digra\ ph returned by MaximalSymmetricSubdigraphWithoutLoops is\n the same,\ except that loops are removed. \n \n If digraph is immutable,\ then a new immutable digraph is returned.\n Otherwise, the mutable digraph\ digraph is changed in-place into such\n a digraph described above.<\ P/>\n\n See , , and\n for more information.\n D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsSymmetricDigraph(D) and IsM\ ultiDigraph(D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ \ 3, 1 ] ]\ngap> S := MaximalSymmetricSubdigraph(D);\n\ngap> IsSymmetricDigraph(S) and not IsMultiDigrap\ h(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 retai\ ning the vertices of digraph, discarding any duplicate edges,\n and \ 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 wor\ ds, for every\n symmetric pair of edges [i,j] and [j,i] in digraph, where \n i and j are distinct, it discards the \ the edge\n [\\max(i,j),\\min(i,j)].\n \n\n If digraph 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 IsMultiDigraph(D\ );\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ngap\ > D := MaximalAntiSymmetricSubdigraph(D);\n\ngap> IsAntiSymmetricDigraph(D) and not IsMultiDigraph\ (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 UndirectedSpanningForest<\ /C> returns an undirected spanning forest of\n digraph, otherwise th\ is 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 un\ directed spanning tree of digraph, otherwise this attribute\n return\ s fail. See for the\n def\ inition of an undirected spanning tree. \n\n If digraph is immuta\ ble, then an immutable digraph is returned.\n Otherwise, the mutable digrap\ h digraph is changed in-place into an\n undirected spanning tree of \ digraph. \n\n Note that for an immutable digraph that has known u\ ndirected spanning tree,\n the attribute UndirectedSpanningTree retu\ rns the same digraph as\n the attribute UndirectedSpanningForest.\n\ \n D := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]\ ]);\n\ngap> UndirectedSpannin\ gTree(D);\nfail\ngap> forest := UndirectedSpanningForest(D);\n\ngap> OutNeighbours(forest);\n[ [ 2 ], \ [ 1 ], [ 4 ], [ 3 ] ]\ngap> IsUndirectedSpanningForest(D, forest);\ntrue\ngap>\ DigraphConnectedComponents(forest).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> Digra\ phConnectedComponents(MaximalSymmetricSubdigraph(D)).comps;\n[ [ 1, 2 ], [ 3, \ 4 ] ]\ngap> UndirectedSpanningForest(MaximalSymmetricSubdigraph(D))\n> = fores\ t;\ntrue\ngap> D := CompleteDigraph(4);\n\ngap> tree := UndirectedSpanningTree(D);\n\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap\ > tree = UndirectedSpanningForest(D);\ntrue\ngap> UndirectedSpanningForest(Emp\ tyDigraph(0));\nfail\ngap> D := PetersenGraph(IsMutableDigraph);\n\ngap> UndirectedSpanningTree(D);\n\ngap> D;\n\n]]>\n \n\n\n \ \n \n A digraph.\ \n \n If digraph is a digraph, and p \ is a partition of the vertices\n of digraph, then this operation ret\ urns a digraph constructed by\n amalgamating all vertices of digraph\ which lie in the same part of\n p.\n \n\n A partition of \ the vertices of digraph is a list of non-empty\n disjoint lists, suc\ h that the union of all the sub-lists is equal\n to vertex set of digrap\ h. 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 i\ n 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 quotient\ .\n In particular, this means that the quotient of a digraph has no multipl\ e 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 b\ elongs to , a new immutable digraph w\ ith 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 ] ]\nga\ p> p := [[1], [2, 4], [3]];\n[ [ 1 ], [ 2, 4 ], [ 3 ] ]\ngap> quo := QuotientD\ igraph(D, p);\n\ngap> DigraphVerti\ ces(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 digr\ aph.\n \n The reverse of a digraph is the digraph f\ ormed by reversing the orientation\n of each of its edges, i.e. for every e\ dge [i, j] of a digraph, the\n reverse contains the corresponding ed\ ge [j, i]. \n\n DigraphReverse returns the reverse of the \ digraph digraph. If\n digraph is immutable, then a new immuta\ ble digraph is returned. Otherwise,\n the mutable digraph digraph is\ changed in-place into its reverse.\n 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 := C\ ycleDigraph(IsMutableDigraph, 3);\n\ \ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> DigraphReverse(D);\n\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 th\ ere is no edge from i to j in digraph.\n The dual is sometimes called the complement. \n\n DigraphDual re\ turns the dual of the digraph digraph. If\n digraph is an imm\ utable digraph, then a new immutable digraph is\n returned. Otherwise, the \ mutable digraph digraph is changed in-place\n into its dual.\n D := Digraph([[2, 3], [], [4, 6], [5], [],\n> [7, 8, 9]\ , [], [], []]);\n\ngap> DigraphDua\ l(D);\n\ngap> D := CycleDigraph(I\ sMutableDigraph, 3);\n\ngap> Digraph\ Dual(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\n digraph itself is return\ ed if no vertices are added\n (i.e. m=0 or labels is e\ mpty), otherwise\n the result is a new immutable digraph.\n \n\n \ D := CompleteDigraph(3);\n\ngap> new := DigraphAddVertices(D, 3);\n\ngap> DigraphVertices(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 := CompleteBipartit\ eDigraph(IsMutableDigraph, 2, 3);\n\ \ngap> new := DigraphAddVertices(D, 4);\n\ngap> D = new;\ntrue\n]]>\n \n\n\ \n \n \n \n A digraph.\n \n If edge is a pair of ver\ tices of digraph, or src and\n ran are vertices of \ digraph,\n then this operation returns a digraph constructed from di\ graph\n by adding a new edge with source edge[1] [src<\ /A>] and range\n edge[2] [ran].\n \n \n I\ f digraph belongs to , then the edge\n \ is added directly to digraph. If digraph belongs to , then an immutable copy of digraph wit\ h the\n additional edge is returned. \n\n \ D1 := Digraph([[2], [3], []]);\n\n\ gap> 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> D3 := 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 d\ igraph.\n \n \n This operation returns a new digra\ ph 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 o\ f the of\n digraph. If edge is \ already an edge in digraph, then\n digraph is returned unchan\ ged. 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(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 edges is\ a (possibly empty) list of pairs of vertices of\n digraph, then thi\ s operation returns a digraph constructed from\n digraph by adding t\ he edges specified by edges. More\n precisely, for every edge in edges, a new edge\n will be added with source edge[1] an\ d range edges[2].\n \n\n If an edge is included in edges with multiplicity k,\n then it will be added k times.\n\n \ If digraph belongs to , then the edg\ es\n are added directly to digraph. If digraph belongs to , then the result is returned as an immu\ table\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;;\ngap> D := func(1024);\n\ngap> new := DigraphAddEdges(D,\n> [[1\ 023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);\n\ngap> D = new;\nfalse\ngap> D2 := DigraphMutableC\ opy(func(1024));\n\ngap> new :\ = DigraphAddEdges(D2,\n> [[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);\ \n\ngap> D2 = new;\ntrue\ \n]]>\n \n\n\n \n \n A digraph.\n \n If v is a vertex of digraph, then thi\ s operation returns a\n digraph constructed from digraph by removing\ vertex v,\n along with any edge whose source or range vertex is \ v. \n\n If digraph has n vertices, then the vertices o\ f the returned\n digraph are [1..n-1], but the original labels can b\ e\n accessed via . \n\n If dig\ raph belongs to , then the\n vertex is \ removed directly from digraph. If digraph 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> 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 := CycleDig\ raph(IsMutableDigraph, 5);\n\ngap> n\ ew := DigraphRemoveVertex(D, 1);\n\n\ gap> DigraphVertexLabels(D);\n[ 2, 3, 4, 5 ]\ngap> D = new;\ntrue\n]]>\n \n\n\n \n \n A digraph.\n \ \n If verts is a (possibly empty) duplicate-free list \ of vertices of\n digraph, then this operation returns a digraph cons\ tructed from\n digraph by removing every vertex in verts, alo\ ng with any edge\n whose source or range vertex is in verts. \n\n\ If digraph has n vertices, then the vertices of the new\n \ digraph are [1 .. n-Length(verts)], but the\n original label\ s can be accessed via . \n\n If d\ igraph belongs to , then the\n vertices\ are removed directly from digraph. If digraph belongs\n to <\ Ref Filt=\"IsImmutableDigraph\"/>, an immutable copy of digraph\n wi\ thout the vertices is returned. \n\n D := Digra\ ph([[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]]>\n \n\n\n \n \n \n A di\ graph.\n \n If digraph is a digraph with no \ multiple edges and edge is a\n pair of vertices of digraph, o\ r src and ran are\n vertices of digraph, then this ope\ ration returns a digraph\n constructed from digraph by removing the \ edge specified by\n edge or [src, ran]. \n\n If d\ igraph belongs to , then the edge i\ s removed directly from\n digraph. If digraph belongs to , an immutable copy of digraph wit\ hout\n the edge is returned.\n \n\n Note that if digraph b\ elongs to ,\n then a new copy of digra\ ph will be returned even if edge or \n [src, ran] does no\ t define an edge of digraph. \n\n D := C\ ycleDigraph(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);;\ngap> IsIdenticalObj(new, D);\nt\ rue]]>\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 digraph and with the orbit of t\ he edge edge (under the action\n of the \ of digraph) removed.\n If edge is not an edge in digraph<\ /A>, then digraph is\n returned unchanged. The argument digraph must be an immutable\n digraph.\n \n\n An edge is simply a pai\ r of vertices of digraph.\n gr1 := CayleyDi\ graph(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(gr\ 2);\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 following holds:\n \n \ \n digraph is a digraph with no multiple edges, and\n \ edges is a list of pairs of vertices of digraph, or\n \ \n digraph is a digraph and edges is an empty\ list \n \n\n then this operation returns a digraph constr\ ucted from digraph\n by removing all of the edges specified by ed\ ges\n (see ).\n \n\n If digraph belongs to , then the edge\n is removed directly from digraph. If <\ A>digraph belongs to , the edge i\ s removed from an immutable copy of\n digraph and this new digraph i\ s returned. \n\n Note that if edges is empty, then this operation\ \n will always return digraph rather than a copy. Also, if any eleme\ nt\n of edges is invalid (i.e. does not define an edge of digraph\ )\n then that element will simply be ignored.\n\n 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]]>\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 digraph by removing every loo\ p. A loop\n is an edge with equal source and range. \n\n If digrap\ h is immutable, then a new immutable digraph 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> DigraphRemov\ eLoops(D);\n\ngap> D := Digraph(Is\ MutableDigraph, [[1, 2], [1]]);\n\ng\ ap> DigraphRemoveLoops(D);\n\ngap> D\ ;\n\n]]>\n \ \n\n\n \n \n \n A digraph.\n \n If d\ igraph is a digraph, then this operation returns a\n digraph construct\ ed from digraph by removing all multiple edges.\n The result is the \ largest subdigraph of digraph which does not\n contain multiple edge\ s. \n\n If digraph is immutable, then a new immutable digraph is \ returned.\n Otherwise, the multiple edges of the mutable digraph digraph\ are\n removed in-place.\n 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> DigraphRemoveAllMultipleEdges(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n \n \n A digraph.\n \n If digraph is a digraph without multiple edges, and edges is a\n l\ ist of pairs of vertices of digraph (the entries of each pair\n corr\ esponding to the source and the range of an edge, respectively),\n then \ DigraphReverseEdges returns a digraph constructed from\n digraph\ by reversing the orientation of every edge specified by\n edges. If\ only one edge is to be reversed, then\n DigraphReverseEdge can be u\ sed instead. In this case, the second\n argument should just be a single ve\ rtex-pair, or the second and third\n arguments should be the source and ran\ ge of an edge respectively. \n\n Note that even though digraph c\ annot have multiple edges, the\n output may have multiple edges. \n\n \ If digraph belongs to , then the edge\ s\n are reversed in digraph. If digraph belongs to , an immutable copy of digraph with the\ \n specified edges reversed is returned. \n \ 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 := Dig\ raphReverseEdge(D, [7, 19]);\n\ng\ ap> DigraphEdges(new);\n[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 9, 1 ], [\ 11, 2 ], \n [ 19, 7 ], [ 21, 1 ] ]\ngap> D2 := DigraphMutableCopy(new);;\nga\ p> new := DigraphReverseEdges(D2, [[19, 7]]);;\ngap> D2 = new;\ntrue\ngap> D =\ new;\ntrue\n]]>\n \n\n\n \n \n \n A digraph.\ \n \n In the first form, if D1, D2, etc. are di\ graphs, then\n DigraphDisjointUnion returns their disjoint union.\n\ \n In the second form, if list is a non-empty list of digraphs, then\ \n DigraphDisjointUnion returns the disjoint union of the digraphs\n\ contained in the list. \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 \n\n More specifically, for\ a collection of digraphs D1, D2,\n ..., the disjoint \ union with have\n DigraphNrVertices(D1) +\n \ DigraphNrVertices(D2) + ... vertices.\n \ The edges of D1 will remain unchanged, whilst the edges of the\n \ ith digraph, D[i], will be changed so that they\n belong \ to the vertices of the disjoint union corresponding to\n D[i]\ . In particular, the edges of D[i] will\n have their source \ and range increased by\n DigraphNrVertices(D1) + ... +\n DigraphNrVertices(D[i-1]).\n\n\ \n\n Note that previously set w\ ill be lost. \n\n If the first digraph D1 [list[1]] belong\ s to , then D1 [list[1]] is\ modified in\n place to contain the appropriate vertices and edges. If d\ igraph 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 := Complet\ eDigraph(3);\n\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]]>\n \n\n\n \n \n \ \n A digraph.\n \n In the first form, i\ f D1, D2, etc. are digraphs, then\n DigraphEdgeUnion r\ eturns their edge union.\n\n In the second form, if list is a non-em\ pty list of digraphs, then\n DigraphEdgeUnion returns the edge union\ of the digraphs contained\n in the list. \n\n The vertex set of th\ e edge union of a collection of digraphs is the\n union of the verte\ x sets, whilst the edge list of the edge union is\n the concatenation of the edge lists. The number of vertices of the\n edge union is equal t\ o the maximum number of vertices of one of the\n digraphs, whilst th\ e number of edges of the edge union will equal the\n sum of the numb\ er of edges of each digraph. \n\n Note that previously set will be lost.\n \n\n If the first digraph D1 [list[1]] belongs to , the\ n D1 [list[1]] is modified in\n place to contain the appropri\ ate vertices and edges. If digraph belongs\n to , a new immutable digraph containing the\n appropriate vertic\ es and edges is returned.\n 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 := Digraph\ EdgeUnion(D1, D2);\n\ngap> Ou\ tNeighbours(union);\n[ [ 2, 2, 3 ], [ 1, 2 ], [ 1 ] ]\ngap> union = DigraphByE\ dges(\n> Concatenation(DigraphEdges(D1), DigraphEdges(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 list is a non-empty list of \ digraphs,\n then DigraphJoin returns the join of the digraphs contai\ ned in the\n list. \n\n The join of a collection 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 bet\ ween\n vertices corresponding to digraphs D[i] and\n D<\ /A>[j] in the collection; the join is created by including\n all suc\ h edges. \n\n For example, the join of two empty digraphs is a complete \ bipartite\n digraph. \n\n Note that previously set will be lost.\n \n\n If the first digraph D1 [list[1]] belongs to , then D\ 1 [list[1]] is modified in\n place to contain the appropriate ve\ rtices 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> IsCompleteDigraph(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 gr1, gr2, etc. are \ digraphs, then\n DigraphCartesianProduct returns a digraph isomorphi\ c to their\n Cartesian product.\n \n\n In the second form, if \ list is a non-empty list of digraphs,\n then DigraphCartesianProduct\ returns a digraph isomorphic to the \n Cartesian product of the digrap\ hs contained in the list.\n \n \n Mathematically, the Cartesian prod\ uct of two digraphs G, H\n is a digraph with vertex set Ca\ rtesian(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 returned by\n DigraphCar\ tesianProduct has vertex set\n [1 .. DigraphNrVertices(G)*DigraphNr\ Vertices(H)] 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 regained by using\n . In addition,\n are pr\ eserved: if vertex pair\n [u,u'] was encoded as i then the ve\ rtex label of i\n will be the pair of vertex 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 i\ s associative, the Cartesian product of a collection \n of digraphs gr1<\ /A>, gr2, ... is computed in the obvious \n fashion. \n \n\n 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], [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 DigraphDirectProd\ uct returns a digraph isomorphic to their direct\n product.\n \n\ \n In the second form, if list is a non-empty list of digraphs, then\ \n DigraphDirectProduct returns a digraph isomorphic to the direct\n\ product of the digraphs contained in the list.\n \n\n Mathematic\ ally, 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 ther\ e is an edge from u to v in\n G and an edge from u'\ to v' in H. \n \n\n Due to technical reasons, th\ e digraph D returned by\n DigraphDirectProduct has vertex set\ \n [1 .. DigraphNrVertices(G)*DigraphNrVertices(H)] instead, and th\ e\n exact method of encoding pairs of vertices into integers is implementat\ ion\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 of i\n will be the pa\ ir of vertex labels of u and u' i.e.\n DigraphVertexLabel(\ D,i) = [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H,u')].\n \ \n\n As the direct product is associative, the direct product of a collecti\ on \n of digraphs gr1, gr2, ... is computed in the obv\ ious \n fashion. \n \n\n gr := ChainDigraph(\ 4);\n\ngap> gr2 := CycleDigraph(3);\n\ \ngap> gr3 := DigraphDirectProduct(gr\ , gr2);\n\ngap> IsIsomorphicDigra\ ph(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 transformations.\n \n If digraph is a Cartesian product digraph, \n digra\ ph = DigraphCartesianProduct(gr_1, gr_2, ... ), then\n DigraphCartes\ ianProductProjections returns a list proj\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 endomorphism of digraph. I\ f\n gr1, gr2, ... are all loopless digraphs, then the induced \n \ subdigraph of digraph on the image of proj[i] is \n isomorphi\ c to gr_i.\n \n\n Currently this attribute is only set upon c\ reating an immutable \n digraph via DigraphCartesianProduct and ther\ e is no way \n of calculating it for other digraphs.\n \n\n For m\ ore information see\n \n\n D := DigraphCartesianProd\ uct(ChainDigraph(3), CycleDigraph(4),\n> Digraph([[2], [2]]));;\ngap> HasDigra\ phCartesianProductProjections(D);\ntrue\ngap> proj := DigraphCartesianProductP\ rojections(D);; Length(proj);\n3\ngap> IsIdempotent(proj[2]);\ntrue\ngap> Rank\ OfTransformation(proj[3]);\n2\ngap> S := ImageSetOfTransformation(proj[2]);;\n\ gap> IsIsomorphicDigraph(CycleDigraph(4), InducedSubdigraph(D, S));\ntrue\n]]>\ \n \n\n\n \n \n A list of tr\ ansformations.\n \n If digraph is a direct p\ roduct digraph, \n digraph = DigraphDirectProduct(gr_1, gr_2, ... ),\ then\n DigraphDirectProductProjections returns a list proj\n\ such that proj[i] is the projection onto the i-th\n coord\ inate of the product.\n \n\n A projection is an idempotent endomorph\ ism 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 attribute is only set upon creati\ ng an immutable\n digraph via DigraphDirectProduct and there is no w\ ay of \n calculating it for other digraphs.\n \n\n For more infor\ mation, see\n \n\n D := DigraphDirectProduct(ChainDigra\ ph(3), CycleDigraph(4),\n> Digraph([[2], [2]]));;\ngap> HasDigraphDirectProduc\ tProjections(D);\ntrue\ngap> proj := DigraphDirectProductProjections(D);; Leng\ th(proj);\n3\ngap> IsIdempotent(proj[2]);\ntrue\ngap> RankOfTransformation(pro\ j[3]);\n2\ngap> P := DigraphRemoveAllMultipleEdges(\n> ReducedDigraph(OnDigrap\ hs(D, proj[2])));; \ngap> IsIsomorphicDigraph(CycleDigraph(4), P);\ntrue\n]]><\ /Example>\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 ver\ tex v to a vertex u if and only\n if the terminal vertex of t\ he edge associated with v is the start\n vertex of the edge associat\ ed with u. \n\n Note that the returned digraph is always a new im\ mutable digraph, and the\n argument digraph is never modified.\n \ LineDigraph(CompleteDigraph(3));\n\ngap> LineDigraph(ChainDigraph(3));\n\n]]>\n \n\ \n\n \n \ \n \n A digra\ ph.\n \n Given a symmetric digraph digraph, \ the operation returns the\n symmetric digraph obtained by associating a ver\ tex with each edge of\n digraph, ignoring directions and multiplicit\ es, and adding an edge\n between two vertices if and only if the correspond\ ing edges have a vertex in\n common. \n\n Note that the returned digr\ aph is always a new immutable digraph, and the\n argument digraph is\ never modified.\n LineUndirectedDigraph(CompleteD\ igraph(3));\n\ngap> LineUndirected\ Digraph(DigraphSymmetricClosure(ChainDigraph(3)));\n\n]]>\n \n\n\n \n \n A digra\ ph.\n \n Let digraph be a digraph with verte\ x set V. This function\n returns the double digraph of digraph. The vertex set of the double\n digraph is the orginal vertex set togethe\ r 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 constructed as\n described above is\ returned.\n gamma := Digraph([[2], [3], [1]]);\n<\ immutable digraph with 3 vertices, 3 edges>\ngap> DoubleDigraph(gamma);\n\n]]>\n \n\n\n \n \n A digraph.\n \n Let digr\ aph be a digraph with vertex set V. This function\n returns the \ bipartite double digraph of digraph. The vertex set of\n the double \ digraph is the original vertex set together with a duplicate. The\n edges a\ re [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 th\ e orignal edges are not included in the resulting digraph. \n\n If di\ graph is mutable, then digraph is modified in-place. If\n dig\ raph is immutable, then a new immutable digraph constructed as\n descri\ bed above is returned.\n gamma := Digraph([[2], [3\ ], [1]]);\n\ngap> BipartiteDoubleD\ igraph(gamma);\n\n]]>\n \ \n\n\n \n \n \n A digraph.\n \n For a digraph \ digraph this operation returns a new digraph constructed\n from d\ igraph, such that a loop is added for every vertex which did not\n have\ a loop in digraph. \n\n If digraph is immutable, then a n\ ew immutable digraph is returned.\n Otherwise, the loops are added to the l\ oopless vertices of the mutable digraph\n digraph 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 <\ Oper Name=\"DistanceDigraph\" Arg=\"digraph, list\"\n Label=\"for digra\ ph and list\"/>\n A digraph.\n \n The fir\ st argument is a digraph, the second argument is a non-negative\n integer o\ r a list of positive integers. This operation returns a digraph on\n the sa\ me set of vertices as digraph, with two vertices being adjacent\n if\ and only if the distance between them in digraph equals i\n \ or is a number in list. See . \n\n If digraph is \ mutable, then digraph is modified in-place. If\n digraph is i\ mmutable, then a new immutable digraph constructed as\n described above is \ returned.\n digraph := DigraphFromSparse6String(\n\ > \":]n?AL`BC_DEbEF`GIaGHdIJeGKcKL_@McDHfILaBJfHMjKM\");\n\ngap> DistanceDigraph(digraph, 1);\n\ngap> DistanceDigraph(digraph, [1, 2]);\n\n]]>\n \n<\ /ManSection>\n\n \n \n A digraph.\n \n Given a symmet\ ric loopless digraph with no multiple edges digraph,\n the k-closure of digraph is defined to be the unique\n smallest sym\ metric loopless digraph C with no multiple edges on the\n vertices o\ f digraph that contains all the edges of digraph\n and satsif\ ies the property that the sum of the degrees of every two\n non-adjacenct v\ ertices in C is less than k. See , , \ , and . \n\n The operation DigraphC\ losure 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> IsCompleteDigraph(closure);\ntrue\ \n]]>\n \n\n\n \n \n \n A digraph.\n \n\ If digraph is a symmetric digraph, then\n DigraphMycielskian<\ /C> returns the Mycielskian of digraph. \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 information, see h\ ttps://en.wikipedia.org/wiki/Mycielskian. \n\n If digraph i\ s immutable, then a new immutable digraph is returned.\n Otherwise, the mut\ able digraph digraph is changed in-place into\n its Mycielskian. \ \n\n D := CycleDigraph(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 := CompleteBipartiteDigraph(IsMut\ able, 2, 3);\n\ngap> DigraphMyciels\ kian(D);\n\ngap> D;\n\n]]>\n \n\n\n \n\n Random digraphs\n \n \n A \ digraph.\n \n If the optional first argument fil\ t 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 defaul\ t.\n\n If n is a positive integer, then this function returns a \ random\n digraph with n vertices and without multiple edges. The res\ ult\n may or may not have loops. \n\n If the optional second argument\ p is a float with value\n 0 \\leq 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 prob\ ability will be assumed\n (chosen with uniform probability).\n RandomDigraph(1000);\n\ngap> RandomDigraph(10000, 0.023);\n\ngap> RandomDigraph(IsMutableDigraph, 1000, 1 / 2);\n\n]]>\n \n\ \n\n \n \n A digraph.\n \n If n i\ s a positive integer, then this function returns a random\n digraph with n vertices. If the optional second argument\n m is a positive i\ nteger, then the digraph will have m\n edges. If m is not spe\ cified, then the number of edges will be\n chosen randomly (with uniform pr\ obability) from the range\n [1 .. {n \\choose 2}].\n \ \n\n The method used by this function chooses each edge from the set \ of\n all possible edges with uniform probability. No effort is made to avoi\ d\n creating multiple edges, so it is possible (but not guaranteed) that th\ e\n result will have multiple edges. The result may or may not have loops.<\ P/>\n RandomMultiDigraph(1000);\n\ngap> RandomMultiDigraph(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 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 non-negative inte\ ger, this function returns a random\n tournament with n vertices. Se\ e . \n RandomTournamen\ t(10);\n\ngap> RandomTournament(IsMutab\ leDigraph, 10);\n\n]]>\ \n \n\n\n \n \n A digraph.\n \n If \ the optional first argument filt is present, then this should\n spec\ ify 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 opt\ ional first argument filt is not present, then is used by default. \n\n If n is a positive int\ eger, 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 Rando\ mLattice(10);\n\ngap> Ra\ ndomLattice(IsMutableDigraph, 10);\n\n]]>\n \n\n\n \n\n <\ Heading>Standard examples\n \n \n A digraph.\n \n\ If n is a positive integer, this function 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 e\ dge with source i and range i + 1. \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 The of a chain represents a\n total order.\n ChainDigraph(42);\n\ngap> ChainD\ igraph(IsMutableDigraph, 10);\n\n]]\ >\n \n\n\n \n \n A digraph.\n <\ Description>\n If n is a non-negative integer, this function returns\ the complete\n digraph with n vertices. See . \n\n If the optional first argument filt is present,\ then this should\n specify the category or representation the digraph bein\ g created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if f\ ilt is , then the digraph will be i\ mmutable. \n If the optional first argument filt is not present, the\ n 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 digra\ ph is a digraph whose vertices\n can be partitioned into two non-empty vert\ ex sets, such there exists\n a unique edge with source i and range <\ C>j if and only if\n i and j lie in different vertex sets\ . \n\n If m and n are positive integers, this function\n \ returns the complete bipartite digraph with vertex sets of sizes\n m<\ /A> (containing the vertices [1 .. m]) and\n n (containing th\ e vertices [m + 1 .. m + n]).\n \n\n If the optional first ar\ gument filt is present, then this should\n specify the category or r\ epresentation the digraph being created will\n belong to. For example, if <\ A>filt is ,\n then the digraph being cr\ eated will be mutable, if filt is , then the digraph will be immutable. \n If the optional first argument <\ A>filt is not present, then is u\ sed 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 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 CompleteMultipartiteDigraph([5, 4, 2]);\n\ngap> CompleteMultipartiteDigraph(\ IsMutableDigraph, [5, 4, 2]);\n\n]\ ]>\n \n\n\n \n \n A digraph.\n \n If n is a positive integer, 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 dire\ cted\n edge with source i and range i + 1. In addition, there\ is\n an edge with source n and range 1. \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 CycleDigraph(1);\n\ngap> CycleDigraph(123);\n\ngap> CycleDigraph(IsMutableDigraph, 10);\n\n]]>\n \n\n\n \n \n \n A digraph.<\ /Returns>\n \n If n is a non-negative integer, this fu\ nction returns the\n empty or null digraph with n vert\ ices. An empty\n digraph is one with no edges.\n \n \n If th\ e optional first argument filt is present, then this should\n specif\ y the category or representation the digraph being created will\n belong to\ . For example, if filt is ,\n then t\ he digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the option\ al 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> EmptyDigraph(IsMutableDigraph, 10);\n]]>\n \n\n\n \n \n A digraph.\n \n If n and k<\ /A> are non-negative integers, then this operation\n returns a symmetric di\ graph which corresponds to the undirected Johnson\n graph J(n, k)\ . \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 The Jo\ hnson graph J(n, k) has vertices given by all the\n k-sub\ sets of the range [1 .. n], and two vertices are\n connecte\ d 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> J\ ohnsonDigraph(1, 0);\n\ngap> JohnsonDig\ raph(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 graph 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 Juli\ us\n Petersen, who in 1898 constructed it to be the smallest bridgeless cub\ ic\n graph with no three-edge-coloring. \n \n If the optional \ first argument filt is present, then this should\n specify the categ\ ory or representation the digraph being created will\n belong to. For examp\ le, 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 ar\ gument filt is not present, then is used by default. \n\n See also .\n\n ChromaticNumber(PetersenGraph());\n3\ngap\ > PetersenGraph(IsMutableDigraph);\n\n]]>\n \n\n\n \n \n A digra\ ph.\n \n If n is a positive integer and k\ is a non-negative\n integer less than n / 2, then this o\ peration returns\n the generalised Petersen graph GPG(n, k\ ). \n\n From\n https://en.wikipedia.org/wiki/Generalized_Pe\ tersen_graph:\n \n\n The generalized Petersen graphs are a \ family of cubic\n graphs formed by connecting the vertices of a regular pol\ ygon to the\n corresponding vertices of a star polygon. They include the Pe\ tersen graph\n and generalize one of the ways of constructing the Petersen \ graph. The\n generalized Petersen graph family was introduced in 1950 by H.\ S. M.\n Coxeter and was given its name in 1969 by Mark Watkins.\n <\ P/>\n\n If the optional first argument filt is present, then this sh\ ould\n specify the category or representation the digraph being created wil\ l\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt is <\ Ref\n Filt=\"IsImmutableDigraph\"/>, then the digraph will be immutable. \n\ If the optional first argument filt is not present, then is used by default. \n\n See also .\n\n GeneralisedPetersenGraph(\ 7, 2);\n\ngap> General\ isedPetersenGraph(40, 1);\n\ngap> D := GeneralisedPetersenGraph(5, 2);\n\ngap> IsIsomorphicDigraph(D, PetersenGraph());\n\ true\ngap> GeneralisedPetersenGraph(IsMutableDigraph, 9, 4);\n\n]]>\n \n\n\ \n \n\n\n \n Operators\n\n Operators for digraphs<\ /Heading>\n \n digraph1 = digraph2\n \n = (for digraphs)\n \ returns true if digraph1 and digraph2 have the same\n\ vertices, and DigraphEdges(digraph1) =\n DigraphEdges\ (digraph2), up to some re-ordering of the edge\n lists. \ \n\n Note that this operator does not compare the vertex labels\n \ of digraph1 and digraph2.\n \n\n digr\ aph1 < digraph2\n \n < (for digraphs)\n\n This operator returns true\ if one of the following holds:\n\n \n \n \ The number n_1 of vertices in digraph1 is less than\n \ the number n_2 of vertices in digraph2;\n \n \ \n n_1 = n_2, and the number m_1 of edge\ s in\n digraph1 is less than the number m_2 of edges i\ n\n digraph2;\n \n \n \ n_1 = n_2, m_1 = m_2, and\n DigraphEdges(digraph\ 1) is less than\n DigraphEdges(digraph2) after\ having both of these sets\n have been sorted with respect to the le\ xicographical order.\n \n \n \n \n\n \n \n <\ Returns>true or false.\n \n If sup\ er and sub are digraphs, then this operation returns\n true if sub is a subdigraph of super, and false\n if it \ is not. \n\n A digraph sub is a subdigraph of a digraph <\ A>super if\n sub and super share the same number of verti\ ces, and the\n collection of edges of super (including repeats) cont\ ains the\n collection of edges of sub (including repeats). \n\n\ In other words, sub is a subdigraph of super if and only if\ \n DigraphNrVertices(sub) = DigraphNrVertices(super), \ and\n for each pair of vertices i and j, there are at least a\ s many\n edges of the form [i, j] in super as there are in sub.\n \n\n g := Digraph([[2, 3], [1],\ [2, 3]]);\n\ngap> h := Digraph([[\ 2, 3], [], [2]]);\n\ngap> IsSubdig\ raph(g, h);\ntrue\ngap> IsSubdigraph(h, g);\nfalse\ngap> IsSubdigraph(Complete\ Digraph(4), CycleDigraph(4));\ntrue\ngap> IsSubdigraph(CycleDigraph(4), ChainD\ igraph(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]]>\n \n\n\n \ \n \n \n true o\ r false.\n \n The operation IsUndirectedS\ panningTree returns true if the\n digraph sub is an undir\ ected spanning tree of the digraph\n super, and the operation IsU\ ndirectedSpanningForest returns\n true if the digraph sub\ is an undirected spanning forest of\n the digraph super. \n\n \ An undirected spanning tree of a digraph super is a subdigraph\ \n of super that is an undirected tree (see \n and ). Note that a digraph whose <\ Ref\n Oper=\"MaximalSymmetricSubdigraph\"/> is not connected has no undir\ ected\n spanning trees (see ). \n\n \ An undirected spanning forest of a digraph super is a\n sub\ digraph of super that is an undirected forest (see and ), and is not\n conta\ ined in any larger such subdigraph of super. Equivalently, an\n und\ irected spanning forest is a subdigraph of super whose connected\n c\ omponents coincide with those of the of super (see ). \n\n Note that an undirected spanning tree is an undirected spannin\ g forest that\n is connected.\n\n D := Complete\ Digraph(4);\n\ngap> tree := Digrap\ h([[3], [4], [1, 4], [2, 3]]);\n\n\ gap> IsSubdigraph(D, tree) and IsUndirectedTree(tree);\ntrue\ngap> IsUndirecte\ dSpanningTree(D, tree);\ntrue\ngap> forest := EmptyDigraph(4);\n\ngap> IsSubdigraph(D, forest) and IsUndirectedFore\ st(forest);\ntrue\ngap> IsUndirectedSpanningForest(D, forest);\nfalse\ngap> Is\ Subdigraph(tree, forest);\ntrue\ngap> D := DigraphDisjointUnion(CycleDigraph(2\ ), CycleDigraph(2));\n\ngap> IsUnd\ irectedTree(D);\nfalse\ngap> IsUndirectedForest(D) and IsUndirectedSpanningFor\ est(D, D);\ntrue]]>\n \n\n\n \ \n\n \n Attributes and operations\n Ve\ rtices and edges\n \n \n A list of integers.\n \ \n Returns the vertices of the digraph digraph. \n\n Note tha\ t the vertices of a digraph are always a range of\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> DigraphVertic\ es(gr);\n[ 1 .. 6 ]\ngap> DigraphVertices(RandomDigraph(100));\n[ 1 .. 100 ]\n\ gap> 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 digraph. \n\n gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\",\ \"b\", \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphNrVertices(gr);\n3\ngap> gr := Digrap\ h([1, 2, 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> \ [2, 7, 5, 3, 7]);\n\ngap> Digraph\ NrVertices(gr);\n6\ngap> DigraphNrVertices(RandomDigraph(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 re\ turns a list of edges of the digraph digraph,\n where each edge is a\ pair of elements of of\n the form [sour\ ce,range].\n \n\n The entries of DigraphEdges(digraph<\ /A>) are in one-to-one\n corresponence with the edges of digraph<\ /A>. 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 firs\ t component (i.e. by the source of each edge), but they are not\n necessari\ ly then sorted by the second component.\n gr := Di\ graphFromDiSparse6String(\".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(IsMutableDigraph, 3);\n\ \ngap> DigraphEdges(D);\n[ [ 1, 2 ],\ [ 2, 3 ], [ 3, 1 ] ]\n]]>\n \n\n\n <\ ManSection>\n \n An \ integer.\n \n This function returns the number of e\ dges of the digraph digraph.\n gr := Digrap\ h([\n> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]]);;\ngap> Digraph\ NrEdges(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 return\ s a list of the sinks of the digraph\n digraph.\n A sink of a dig\ raph is a vertex with out-degree zero.\n See .\n gr := Digraph([[3, 5, 2, 2], [3], [], [5, \ 2, 5, 3], []]);\n\ngap> Digra\ phSinks(gr);\n[ 3, 5 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSinks(D);\n[ ]\n]]>\ \n \n\n\n \n \n A list of vertices.\n \n This function returns an immutable list of the sources of the dig\ raph\n digraph.\n A source of a digraph is a vertex with in-degre\ e zero.\n See .\n 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]]>\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 dig\ raph ordered\n so that every edge's source appears no earlier in the li\ st than its range.\n If the digraph digraph contains directed cycles\ of length greater\n than 1, then this 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 attribute has complexity O(m+n)<\ /M> where\n m is the number of edges (counting multiple edges as one\ ) and\n n is the number of vertices in the digraph. \n D := Digraph([\n> [2, 3], [], [4, 6], [5], [], [7, 8, 9], [\ ], [], []]);\n\ngap> DigraphTopolo\ gicalSort(D);\n[ 2, 5, 4, 7, 8, 9, 6, 3, 1 ]\ngap> D := Digraph(IsMutableDigra\ ph, [[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 ope\ ration 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 ca\ n be changed an arbitrary number of times. If no\n label has been set for t\ he vertex i, then the default value is\n i. \n\n If digraph is a digraph created from a record with a component\n verti\ ces, then the labels of the vertices are set to the value of\n this com\ ponent. \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 := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphVertexLabel(D, 3);\n3\ \ngap> D := Digraph([\"a\", \"b\", \"c\"], [], []);\n\ngap> DigraphVertexLabel(D, 2);\n\"b\"\ngap> SetDigraphVertex\ Label(D, 2, \"d\");\ngap> DigraphVertexLabel(D, 2);\n\"d\"\ngap> D := InducedS\ ubdigraph(D, [1, 2]);\n\ngap> Digraph\ VertexLabel(D, 2);\n\"d\"\ngap> D := Digraph(IsMutableDigraph, [\"e\", \"f\", \ \"g\"], [], []);\n\ngap> DigraphVertexL\ abel(D, 1);\n\"e\"\ngap> SetDigraphVertexLabel(D, 1, \"h\");\ngap> DigraphVert\ exLabel(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 digraph is a digraph, then D\ igraphVertexLabels returns a\n copy of the labels of the vertices in digraph.\n SetDigraphVertexLabels can be used to set the labels\ of the vertices\n in digraph to the list of\n arbitrary &GAP; ob\ jects list. \n\n The label of a vertex can be changed an arbitra\ ry number of times. If no\n label has been set for the vertex i, the\ n the default value is\n i. \n\n If digraph is a digra\ ph created from a record with a component\n vertices, then the label\ s of the vertices are set to the value of\n this component. \n\n Indu\ ced subdigraphs, and other operations which create new digraphs from\n old \ ones, inherit their labels from their parents.\n\n \ D := DigraphFromDigraph6String(\"&DHUEe_\");\n\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\ng\ ap> DigraphVertexLabels(D);\n[ \"a\", \"c\" ]\ngap> D := Digraph(IsMutableDigr\ aph, [\"e\", \"f\", \"g\"], [], []);\n\ \ngap> SetDigraphVertexLabels(D, [\"h\", \"i\", \"j\"]);\ngap> DigraphVertexLa\ bels(D);\n[ \"h\", \"i\", \"j\" ]\ngap> InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"h\", \"j\"\ ]\n]]>\n \n\n\n \n \n \n \n If digra\ ph is a digraph without multiple edges, then the first\n operation retu\ rns the label of the edge from vertex i to vertex\n j. The s\ econd operation can be used to set the label of the edge\n between vertex <\ A>i and vertex j to the arbitrary &GAP; 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 the default value is 1. <\ P/>\n\n Induced subdigraphs, and some other operations which create new dig\ raphs from\n old ones, inherit their edge labels 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 := InducedSubdig\ raph(D, [2, 5]);\n\ngap> DigraphEd\ geLabel(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[ 17\ 29 ]\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 digraph without multiple edges\ , then\n DigraphEdgeLabels returns a copy of the labels of the edges\ \n in digraph as a list of lists labels such that\n lab\ els[i][j] is the label on the edge from vertex i\n to vertex \ OutNeighbours(digraph)[i][j].\n\n SetDigraphEdgeLabels can be us\ ed to set the labels of the edges in\n digraph without multiple edge\ s to the list labels of lists of\n arbitrary &GAP; objects such that\ list[i][j] is the label on the edge\n from vertex i to the v\ ertex OutNeighbours(digraph>[i][j].\n\n Alternatively SetDigraphE\ dgeLabels can be called with binary function\n func that as its \ second argument that when passed two vertices i\n and j retur\ ns 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 the default value is 1. <\ P/>\n\n Induced subdigraphs, and some other operations which create new dig\ raphs\n from old ones, inherit their labels from their parents.\n\n 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> DigraphEdgeLabels(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 ] ]\ngap> OutNeighbours(D);\n[ [ 3 ], [ 1, 3 ], [ 1 ] ]\ngap> D := Co\ mpleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> DigraphEdgeLabels(D);\n[ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1\ ], [ 1, 1 ], [ 1, 1 ] ]\ngap> SetDigraphEdgeLabel(D, 2, 4, \"a\");\ngap> Digr\ aphEdgeLabels(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 DigraphInEdges returns the list of all e\ dges of digraph\n which have vertex 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 return\ s the list of all edges of digraph\n which have vertex as the\ ir source.\n D := Digraph([[2, 2], [3, 3], [4, 4],\ [1, 1]]);\n\ngap> DigraphOut\ Edges(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 and only if the list\n list<\ /A> specifies an edge in the digraph digraph. Specifically,\n this \ operation returns true if list is a pair of positive\n intege\ rs where list[1] is the source and list[2]\n is\ the range of an edge in digraph, and false otherwise. \n\n \ The second form simply returns true if [u, v] i\ s\n an edge in digraph, and false otherwise.\n\n D := Digraph([[2, 2], [6], [], [3], [], [1]]);\n\ngap> IsDigraphEdge(D, [1, 1]);\nfalse\ngap> I\ sDigraphEdge(D, [1, 2]);\ntrue\ngap> IsDigraphEdge(D, [1, 8]);\nfalse\n]]>\n \n\n\n \n \n \n \n\ \n true\ or false.\n \n If digraph is a d\ igraph and list is a list of pairs of\n vertices of digraph, \ then IsMatching returns true if\n list is a matching o\ f digraph. The operation\n IsMaximalMatching returns true\ if list is a maximal \n matching, IsMaximumMatching retu\ rns true if list is a\n maximum matching and IsPerfectMatc\ hing 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 M of a digraph digr\ aph is a subset of the\n edges of digraph, i.e. DigraphEdges(\ digraph), such\n that no pair of distinct edges in M are \ incident to the same vertex\n of digraph. Note that this definition\ allows a matching to contain\n loops. See . The matching M is\n maximal if it is contained in no larg\ er matching of the digraph,\n is maximum if it has the greatest card\ inality among all matchings and\n is perfect if every vertex of the \ digraph is incident to an edge in\n the matching. Every maximum or perfect\ matching is maximal. Note, however,\n that not every perfect matching of d\ igraphs 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> IsMaximalMatching(D, edges);\nfals\ e\ngap> edges := [[2, 1], [3, 4]];;\ngap> IsMaximalMatching(D, edges);\ntrue\n\ gap> 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> IsMaximumMat\ ching(D, edges);\ntrue\n]]>\n \n\n\n \ \n \n A list of pairs of vertices.\n \n This functi\ on returns a maximal matching of the digraph digraph.\n \n\n \ For the definition of a maximal matching, see . \n\n D := DigraphFromDiSparse6String(\".IeAoXCJ\ U@|SHAe?d\");\n\ngap> M := Digra\ phMaximalMatching(D);; IsMaximalMatching(D, M);\ntrue\ngap> D := RandomDigraph\ (100);;\ngap> IsMaximalMatching(D, DigraphMaximalMatching(D));\ntrue\ngap> D :\ = GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> IsMaximalMatching(D, DigraphMaximalMatching(D));\ntr\ ue\n]]>\n \n\n\n \n \n A list of pair\ s of vertices.\n \n This function returns a maximum\ matching of the digraph digraph.\n \n\n For the definition o\ f a maximum matching, see .\n If digrap\ h is bipartite (see ), then \n the al\ gorithm used has complexity O(m*sqrt(n)). Otherwise for general \n g\ raphs the complexity is O(m*n*log(n)). Here n is the number of v\ ertices \n and m is the number of edges.\n\n D := DigraphFromDigraph6String(\"&I@EA_A?AdDp[_c??OO\");\n\ngap> M := DigraphMaximumMatching(D);; IsMaxim\ alMatching(D, M);\ntrue\ngap> Length(M);\n5\ngap> D := Digraph([[5, 6, 7, 8], \ [6, 7, 8], [7, 8], [8], \n> [], [], [], []]);;\ngap> M := Dig\ raphMaximumMatching(D);\n[ [ 1, 5 ], [ 2, 6 ], [ 3, 7 ], [ 4, 8 ] ]\ngap> D :=\ GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> M := DigraphMaximumMatching(D);;\ngap> IsMaximalMatch\ ing(D, M);\ntrue\ngap> Length(M);\n9\n]]>\n \n\n\n \n\n Neighbours and degree\ \n \n \n \n A square m\ atrix of non-negative integers.\n \n This function \ returns the adjacency matrix mat of the 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 range 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 f\ unction AdjacencyMatrixMutableCopy returns a copy\n of AdjacencyM\ atrix 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\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 ] ]\nga\ p> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> Display(AdjacencyMatrix(D));\n[ [ 0, 1, 0 ],\n [ 0, 0, 1\ ],\n [ 1, 0, 0 ] ]\n]]>\n \n\n\n \ \n \n \ A polynomial with integer coefficients.\n \n \ This function returns the characteristic polynomial of the digraph\n digraph. That is it returns the characteristic polynomial\n of the a\ djacency 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> CharacteristicPolynomi\ al(D);\nx_1^3-1\n]]>\n \n\n\n \n \n \n A square \ matrix of booleans.\n \n If digraph is a dig\ raph with a positive number of vertices\n n, then BooleanAdjacenc\ yMatrix(digraph)\n returns the boolean adjacency matrix <\ C>mat of digraph. The\n value of the matrix entry mat[j][i]<\ /C> is true if and only if\n there exists an edge in digraph \ with source j and range\n i. If digraph has no vertic\ es, then the empty list is\n returned. \n\n Note that the boolean ad\ jacency matrix loses information about multiple\n edges. \n\n The a\ ttribute BooleanAdjacencyMatrix returns an immutable list of\n immut\ able lists, whereas the function\n BooleanAdjacencyMatrixMutableCopy\ returns a copy of the\n BooleanAdjacencyMatrix that is a mutable li\ st 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 ] ]\ngap> 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 ] ]\ngap> BooleanAdjacencyMatrix(EmptyDig\ raph(0));\n[ ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> PrintArray(BooleanAdjacencyMatrix(D));\n[ [ \ false, true, false ],\n [ false, false, true ],\n [ true, false, \ false ] ]\n]]>\n \n\n\n \ \n \n A fu\ nction.\n \n If digraph is a digraph, then <\ C>DigraphAdjacencyFunction returns\n a function which takes two integer\ parameters x, y and returns\n true if there exists an edge f\ rom vertex x to vertex y\n in digraph and false\ if not.\n\n digraph := Digraph([[1, 2], [3], []])\ ;\n\ngap> foo := DigraphAdjacencyF\ unction(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 := Digrap\ h([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \"b\"],\n> \ [\"b\", \"a\", \"a\"]);\n\ngap> foo := DigraphAdjacencyFunction(gr);\nfunction( u, v ) ... end\ng\ ap> foo(1, 2);\ntrue\ngap> foo(3, 2);\nfalse\ngap> foo(3, 1);\nfalse\ngap> D :\ = CycleDigraph(IsMutableDigraph, 3);\n\ngap> foo := DigraphAdjacencyFunction(D);\nfunction( u, v ) ... end\ngap> f\ oo(1, 2);\ntrue\ngap> foo(2, 1);\nfalse\n]]>\n \n\n\n \n \ \n \n A list of posit\ ive integers.\n \n DigraphRange and Digra\ phSource return the range and source of\n the digraph digraph. M\ ore precisely, position i in\n DigraphRange(digraph) i\ s 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 ]\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 := CycleDi\ graph(IsMutableDigraph, 3);\n\ngap> \ DigraphRange(D);\n[ 2, 3, 1 ]\ngap> DigraphSource(D);\n[ 1, 2, 3 ]\n]]>\n \n\n\n \n \n \ \n \n \n The adjacencies of a d\ igraph.\n \n This function returns the list out<\ /C> of out-neighbours of each vertex\n of the digraph digraph.\n \ More specifically, a vertex j appears in out[i] each time\n t\ here exists an edge with source i and range j in\n digraph\ . \n\n The function OutNeighbours returns an immutable list \ of\n lists, whereas the function OutNeighboursMutableCopy returns a \ copy\n of OutNeighbours which is a mutable list of mutable lists. \n\n 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\ \ngap> OutNeighbours(gr);\n[ [ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> OutNeighb\ oursMutableCopy(gr);\n[ [ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> D := CycleDigr\ aph(IsMutableDigraph, 3);\n\ngap> Ou\ tNeighbours(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 inn of in-neighbours of each ve\ rtex\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 digraph. \n\n The function <\ C>InNeighbours returns an immutable list of\n lists, whereas the functi\ on InNeighboursMutableCopy returns a copy\n of InNeighbours w\ hich is a mutable list of mutable lists. \n\n Note that each entry of <\ C>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 := Di\ graph([[1, 2, 3], [2, 1], [3]]);\n\ \ngap> InNeighbours(gr);\n[ [ 1, 2 ], [ 1, 2 ], [ 1, 3 ] ]\ngap> gr := Digraph\ ByAdjacencyMatrix([\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\ ]]>\n \n\n\n \n \n \n \n A lis\ t of non-negative integers.\n \n\n Given a digraph \ digraph with n vertices, the function\n OutDegrees ret\ urns an immutable list out of length n, such that\n for a ver\ tex i in digraph, the value of out[i] is the\n out-deg\ ree of vertex i.\n See . \n\n \ The function OutDegreeSequence returns the same list,\n after it h\ as been sorted into non-increasing order. \n\n The function OutDegre\ eSet returns the same list, sorted into\n increasing order with duplica\ te entries removed. \n\n D := Digraph([[1, 3, \ 2, 2], [], [2, 1], []]);\n\ng\ ap> OutDegrees(D);\n[ 4, 0, 2, 0 ]\ngap> OutDegreeSequence(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 \n \n A \ list of non-negative integers.\n \n\n Given a digra\ ph digraph with n vertices, the function\n InDegrees r\ eturns an immutable list inn of length n, such that\n for a v\ ertex i in digraph, the value of inn[i] is\n the in-de\ gree of vertex i.\n See . \n\n \ The function InDegreeSequence returns the same list,\n after it ha\ s been sorted into non-increasing order. \n\n The function InDegreeS\ et 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-n\ egative integer.\n \n This operation returns the ou\ t-degree of the vertex vertex in the\n digraph digraph.\n \ The out-degree of vertex is the number of edges in digraph\n \ whose source is vertex.\n \n D := Di\ graph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1, 2, 2, 1, 2, 2]]);\n\ngap> OutDegreeOfVertex(D, 1);\n3\n\ gap> OutDegreeOfVertex(D, 2);\n2\ngap> OutDegreeOfVertex(D, 3);\n4\ngap> OutDe\ greeOfVertex(D, 4);\n7\n]]>\n \n\n\n \ \n \ \n \n \ A list of vertices.\n \n\n This operation returns t\ he list out of vertices of the digraph\n digraph.\n A vert\ ex i appears in the list out each time there exists an\n edge\ with source vertex and range i in digraph; in\n part\ icular, this means that out may contain duplicates. \n <\ ![CDATA[\ngap> D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2,\ 1, 2, 2]]);\n\ngap> OutNeig\ hboursOfVertex(D, 1);\n[ 2, 2, 3 ]\ngap> OutNeighboursOfVertex(D, 3);\n[ 2, 2,\ 3 ]\n]]>\n \n\n\n \n \n A non-nega\ tive integer.\n \n This operation returns the in-de\ gree of the vertex vertex in the\n digraph digraph.\n The \ in-degree of vertex is the number of 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> InDegreeOfVertex(D, 1);\n5\ngap> InDegreeOfVe\ rtex(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 v\ ertices.\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 there exists an\n edge with sour\ ce i and range vertex in digraph; in\n particular, thi\ s means that inn may contain duplicates. \n\n D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2\ ]]);\n\ngap> InNeighboursOfV\ ertex(D, 1);\n[ 2, 4, 4, 4 ]\ngap> InNeighboursOfVertex(D, 2);\n[ 1, 1, 3, 3, \ 4, 4, 4, 4 ]\n]]>\n \n\n\n \n \n A list of vert\ ices.\n \n If digraph is a digraph, then \ DigraphLoops returns the list\n consisting 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> Digraph\ Loops(D);\n[ 3, 4 ]\ngap> D := Digraph(IsMutableDigraph, [[1], [1]]);\n\ngap> DigraphLoops(D);\n[ 1 ]\n]]>\n \n\n\n \n \n \n A po\ sitive integer or fail\n \n If the first a\ rgument is a partial order digraph \ then these operations return the meet, or\n the join, of the two input vert\ ices. If the meet (or join) is does not\n exist then fail is returne\ d. The meet (or join) is guaranteed to\n exist when the first argument sati\ sfies\n (or\n ) - see the documentation for these\n properties fo\ r the definition of the meet (or the join).\n\n D \ := Digraph([[1], [1, 2], [1, 3], [1, 2, 3, 4]]);\n\ngap> PartialOrderDigraphMeetOfVertices(D, 2, 3);\n4\ngap> Pa\ rtialOrderDigraphJoinOfVertices(D, 2, 3);\n1\ngap> PartialOrderDigraphMeetOfVe\ rtices(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);\nfail\ngap>\ PartialOrderDigraphJoinOfVertices(D, 3, 4);\nfail\ngap> PartialOrderDigraphMe\ etOfVertices(D, 1, 2);\nfail\ngap> PartialOrderDigraphJoinOfVertices(D, 1, 2);\ \nfail\n]]>\n \n\n\n \n \ \n A square matrix of n\ on-negative integers.\n \n This function returns th\ e outdegree matrix mat of the digraph\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 <\ A>digraph has no vertices, then the empty list is returned. \n\n Se\ e for more information.\n 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[ ]\n\ gap> 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 retu\ rns the outdegree Laplacian matrix mat of the\n digraph digraph. The outdegree Laplacian matrix is defined 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> 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 := Cy\ cleDigraph(IsMutableDigraph, 3);\n\n\ gap> Display(LaplacianMatrix(D));\n[ [ 1, -1, 0 ],\n [ 0, 1, -1 ],\ \n [ -1, 0, 1 ] ]\n]]>\n \n\n\n <\ /Section>\n\n Reachability and connectivity\n \ \n \n A\ n integer or fail.\n \n This function return\ s the diameter of the digraph digraph.\n \n\n If a digraph digraph is strongly connected and has at least 1\n vertex, then the diameter is the maximum shortest distance between\n any pair of distin\ ct vertices. Otherwise then the diameter of\n digraph is undefined, \ and this function returns the value\n fail. \n\n See . \n D := Digr\ aph([[2], [3], [4, 5], [5], [1, 2, 3, 4, 5]]);\n\ngap> DigraphDiameter(D);\n3\ngap> D := ChainDigraph(2);\n\ngap> DigraphDiameter(D);\nfail\ngap> Is\ StronglyConnectedDigraph(D);\nfalse\ngap> D := GeneralisedPetersenGraph(IsMuta\ bleDigraph, 6, 2);\n\ngap> Digraph\ Diameter(D);\n4\n]]>\n \n\n\n \n \n \n \ \n An integer or fa\ il\n \n If there is a directed path in the digr\ aph digraph between vertex\n 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 s\ ection for the definit\ ion of a directed path. \n\n If the 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 at distance 0 from\n itself . \n\ \n If the third form is used, then list1 and list2 are both l\ ists\n of vertices. The shortest directed path between list1 and\n \ list2 is then the length of the shortest directed path which starts\n\ with a vertex in list1 and terminates at a vertex in list2, \ if\n such directed path exists. If list1 and list2 have non-e\ mpty\n intersection, the operation returns 0. \n D := Digraph([[2], [3], [1, 4], [1, 3], [5]]);\n\ngap> DigraphShortestDistance(D, 1, 3);\n2\ngap> Di\ graphShortestDistance(D, [3, 3]);\n0\ngap> DigraphShortestDistance(D, 5, 2);\n\ fail\ngap> DigraphShortestDistance(D, [1, 2], [4, 5]);\n2\ngap> DigraphShortes\ tDistance(D, [1, 3], [3, 5]);\n0\n]]>\n \n\n\n \n \n A square matrix.\n \n If digra\ ph is a digraph with n vertices, then this\n function returns an\ n \\times n matrix mat, where each entry is\n either a non-n\ egative integer, or fail. If n = 0, then an\n empty list is \ returned. \n\n If there is a directed path from vertex i to vert\ ex j, then\n the value of mat[i][j] is the length of the shor\ test 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 dista\ nce from every vertex to\n itself is 0, i.e. mat[i][i] = 0 fo\ r all vertices i.\n \n\n The method used in this function is \ a version of the Floyd-Warshall\n algorithm, and has complexity O(n^3).\n\n D := Digraph([[1, 2], [3], [1, 2], [4]]);\ \n\ngap> mat := DigraphShortestDis\ tances(D);;\ngap> PrintArray(mat);\n[ [ 0, 1, 2, fail ],\n [ \ 2, 0, 1, fail ],\n [ 1, 1, 0, fail ],\n [ fail, fai\ l, fail, 0 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphShortestDistances(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 digraph and v is a vertex in d\ igraph,\n then this operation returns the length of the longest directe\ d walk in\n digraph which begins at vertex v. See section for the definitions of d\ irected\n walk, directed cycle, and loop.\n \n\n \n \n If there exists a directed walk starting at vertex v\n \ which traverses a loop or a directed cycle,\n then we consider the\ re to be a walk of infinite length from v\n (realised by repeate\ dly traversing the loop/directed cycle),\n and so the result is infi\ nity.\n To disallow walks using loops, try using\n : \n\n DigraphLongestDistanceFromVertex(\ DigraphRemoveLoops(digraph,v)).\n \n\n \n Otherwise, if all directed walks\n starting at vertex v<\ /A> have finite length,\n then the length of the longest such walk is r\ eturned.\n \n \n\n Note that the result is 0 i\ f and only if v is a sink of\n digraph. See .\n\n D := Digraph([[2], [3, 4], [], [5], \ [], [6]]);\n\ngap> DigraphLongestD\ istanceFromVertex(D, 1);\n3\ngap> DigraphLongestDistanceFromVertex(D, 3);\n0\n\ gap> 3 in DigraphSinks(D);\ntrue\ngap> DigraphLongestDistanceFromVertex(D, 6);\ \ninfinity\n]]>\n \n\n\n \ \n \n \n A list of vertices\n \n This o\ peration returns the list of all vertices in digraph digraph\n such \ that the shortest distance to a vertex vertex is\n distance o\ r is in the list distances. \n\n digraph should be a digr\ aph, vertex should be a positive\n integer, distance should b\ e a non-negative integer, and\n distances should be a list of non-ne\ gative integers.\n\n D := Digraph([[2], [3], [1, 4\ ], [1, 3]]);\n\ngap> DigraphDistan\ ceSet(D, 2, [1, 2]);\n[ 3, 1, 4 ]\ngap> DigraphDistanceSet(D, 3, 1);\n[ 1, 4 ]\ \ngap> DigraphDistanceSet(D, 2, 0);\n[ 2 ]\n]]>\n \n<\ /ManSection>\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 length of its shortest simple circuit.\n\ See section for the defin\ itions\n of simple circuit, directed cycle, and loop.\n \n\n If <\ A>digraph has no directed cycles, then this function will return\n i\ nfinity. If digraph contains a loop, then this function will\n \ return 1.\n \n\n In the worst case, the method used in this f\ unction is a version of the\n Floyd-Warshall algorithm, and has complexity \ O(n ^ 3), where\n n is the number of vertices in di\ graph. 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 := Digraph([[1], [1]]);\n\ngap> DigraphGirth(D);\n1\ngap> D := Digraph([[2, 3\ ], [3], [4], []]);\n\ngap> Digraph\ Girth(D);\ninfinity\ngap> D := Digraph([[2, 3], [3], [4], [1]]);\n\ngap> DigraphGirth(D);\n3\ngap> D := Generali\ sedPetersenGraph(IsMutableDigraph, 6, 2);\n\ngap> DigraphGirth(D);\n2\n]]>\n \n\n\n \n \ \n An integer, or infinity.\n \n T\ his attribute returns the odd girth of the digraph digraph.\n \ The odd girth of a digraph is the length of its shortest simple\n c\ ircuit of odd length. See Section for the definitions of simple circuit, directed\n cycle, and loo\ p.\n \n\n If digraph has no directed cycles of odd length, th\ en this function\n will return infinity, 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);\n\ infinity\ngap> D := CycleDigraph(IsMutableDigraph, 4);\n\ngap> DigraphDisjointUnion(D, CycleDigraph(5));\n\ngap> DigraphOddGirth(D);\n5\n]]>\ \n \n\n\n \n \n An integer or infinity.<\ /Returns>\n \n If digraph is a symmetric digraph, then\ this function returns the\n girth of digraph when treated as an und\ irected graph (i.e. each pair\n of edges [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 the length of its shortest simple\n \ cycle, i.e. the shortest non-trivial path 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 function will return\n infinity<\ /K>. \n D := Digraph([[2, 4], [1, 3], [2, 4], \ [1, 3]]);\n\ngap> DigraphUndirecte\ dGirth(D);\n4\ngap> D := Digraph([[2], [1, 3], [2]]);\n\ngap> DigraphUndirectedGirth(D);\ninfinity\ngap> D := Di\ graph([[1], [], [4], [3]]);\n\ngap\ > DigraphUndirectedGirth(D);\n1\ngap> D := GeneralisedPetersenGraph(IsMutableD\ igraph, 9, 2);\n\ngap> DigraphUndi\ rectedGirth(D);\n5\n]]>\n \n\n\n \n \n \n A recor\ d.\n \n This function returns the record wcc\ corresponding to the weakly\n connected components of the digraph digra\ ph. Two vertices of\n digraph are in the same weakly connected \ component whenever they are\n equal, or there exists a directed path (ignor\ ing the orientation of edges)\n between them. More formally, two vertices \ are in the same weakly connected\n component of digraph if and only \ if they are in the same strongly\n connected component (see )\n of the of digraph. \n\n The set of weakly connected compone\ nts is a partition of\n the vertex set of digraph. \n\n The r\ ecord wcc has 2 components: comps and id.\n The compon\ ent comps is a list of the weakly connected components\n of digr\ aph (each of which is a list of vertices).\n The component id is\ a list such that the vertex i is an\n element of the weakly connect\ ed 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 Digrap\ hNrConnectedComponents(digraph) is simply a shortcut\n for Le\ ngth(DigraphConnectedComponents(digraph).comps),\n and is no mor\ e efficient.\n\n gr := Digraph([[2], [3, 1], []]);\ \n\ngap> DigraphConnectedComponent\ s(gr);\nrec( comps := [ [ 1, 2, 3 ] ], id := [ 1, 1, 1 ] )\ngap> gr := Digraph\ ([[1], [1, 2], []]);\n\ngap> Digra\ phConnectedComponents(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<\ immutable cycle digraph with 3 vertices>\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 verti\ ces.\n \n If vertex is a vertex in the digra\ ph digraph, then\n this operation 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> DigraphConnectedComponent(D, 4);\n[ 4 ]\n]]>\n \n\n\n \n \n \n A record.\n \ \n This function returns the record scc corresponding \ to the strongly\n connected components of the digraph digraph. Two \ vertices of\n digraph are in the same strongly connected component w\ henever they\n are equal, or there is a directed path from each vertex to t\ he other. The\n set of strongly connected 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 components\n of digraph (each of wh\ ich is a list of vertices).\n The component id is a list such that t\ he vertex i is an\n element of the strongly connected component c\ omps[id[i]]. \n\n The method used in this function is a non-recursi\ ve version of Gabow's\n Algorithm and has complexity \ O(m+n) where\n m is the number of edges (counting multiple ed\ ges as one) and\n n is the number of vertices in the digraph. \n\ \n DigraphNrStronglyConnectedComponents(digraph) is simply a\ \n shortcut for\n Length(DigraphStronglyConnectedComponents(digrap\ h).comps),\n and is no more efficient.\n\n 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> DigraphStronglyConnectedComponent\ s(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\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]]>\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> DigraphS\ tronglyConnectedComponent(D, 3);\n[ 1, 3 ]\ngap> DigraphStronglyConnectedCompo\ nent(D, 2);\n[ 2 ]\ngap> DigraphStronglyConnectedComponent(D, 4);\n[ 4 ]\n]]><\ /Example>\n \n\n\n \n \n A pair of lists of verti\ ces, or fail.\n \n If digraph is a bi\ partite digraph, i.e. if it satisfies , then DigraphBicomponents returns a\n pair of bicomponents of d\ igraph. Otherwise,\n DigraphBicomponents returns fail. \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 se\ ts. The\n parts of this partition are called bicomponents of digr\ aph.\n Equivalently, a pair of bicomponents of digraph consists \ of the\n color-classes of a 2-coloring of digraph. \n\n For a\ bipartite digraph with at least 3 vertices, there is a unique pair of\n bi\ components of bipartite if and only if the digraph is connected. See for more information.\n D := CycleDigraph(3);\n\ngap> \ DigraphBicomponents(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 ] ]\ng\ ap> D := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 2 ], [ 3, 4, 5\ ] ]\n]]>\n \n\n\n \n \n A list of vertic\ es.\n \n A connected digraph is biconnected \ if it is still connected (in the\n sense of ) when any vertex is removed. \n If the digraph digraph is not b\ iconnected but is connected, then any\n vertex v of digraph w\ hose removal makes the resulting digraph\n disconnected is called an art\ iculation point. \n\n ArticulationPoints returns a list of th\ e articulation points of\n digraph, if any, and, in particular, retu\ rns the empty list if\n digraph is not connected. \n\n Multip\ le edges are ignored by this method. \n\n The method used in this opera\ tion has complexity O(m+n) where\n m is the number of edges a\ nd 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 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 \n\n See also and \n .\n ArticulationPoints(CycleDigraph(5));\n[ ]\ngap> D := Digrap\ h([[2, 7], [3, 5], [4], [2], [6], [1], []]);;\ngap> ArticulationPoints(D);\n[ \ 2, 1 ]\ngap> ArticulationPoints(ChainDigraph(5));\n[ 4, 3, 2 ]\ngap> Articulat\ ionPoints(NullDigraph(5));\n[ ]\ngap> D := ChainDigraph(IsMutableDigraph, 4);\ \n\ngap> ArticulationPoints(D);\n[ 3\ , 2 ]\n]]>\n \n\n\n \n <\ Attr Name=\"Bridges\" Arg=\"D\"/>\n A (possibly empty) list of edges\ .\n \n A connected 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 any edge [u, v] of D whose\ removal\n makes the resulting digraph disconnected is called a bridge. \n\n Bridges returns a list of the bridges of D, if an\ y, and, in\n particular, returns the empty list if D is not connecte\ d. \n\n Multiple edges are ignored by this method. \n\n The meth\ od 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 bridge, then a node incident to the bridg\ e is an\n articulation point (see ) if an\ d only if\n it has degree at least 2. It follows that if D h\ as a\n bridge and at least 3 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 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 := Digraph([[2], [3], [4], [2]]);\n\ngap> Bridges(D);\n[ [ 1, 2 ] ]\ngap> Brid\ ges(ChainDigraph(2));\n[ [ 1, 2 ] ]\ngap> ArticulationPoints(ChainDigraph(2));\ \n[ ]\n]]>\n \n\n\n \n \ \n \n A digraph or fail.\n \n A strong orientation of a connected symmetric digraph D (if\n it exists) is a strongly connected 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 s\ tates\n that a digraph admits a strong 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<\ /A> is symmetric and D admits a strong orientation. If D\n is\ symmetric but does not admit a strong orientation, then fail is\n r\ eturned. If D is not symmetric, then an error is given. \n \n\n \ If D is immutable, StrongOrientation(D) returns an\n \ immutable digraph, and if D is mutable, then\n StrongOrientatio\ n(D) returns a mutable digraph. \n\n The method used in this\ operation has complexity O(m+n) where\n m is the number of e\ dges and n is the number of vertices in\n the digraph.\n StrongOrientation(DigraphSymmetricClosure(CycleDigraph(5))) \n> =\ CycleDigraph(5);\ntrue\ngap> D := DigraphSymmetricClosure(Digraph(\n> [[2, 7]\ , [3, 5], [4], [2], [6], [1], []]));;\ngap> IsBridgelessDigraph(D);\nfalse\nga\ p> StrongOrientation(D);\nfail\ngap> StrongOrientation(NullDigraph(0));\n\ngap> StrongOrientation(DigraphDisjointUn\ ion(CompleteDigraph(3), \n> Complete\ Digraph(3)));\nfail\n]]>\n \n\n\n \n \n An int\ eger.\n \n This function returns the period of the \ digraph digraph. \n\n If a digraph digraph has at least on\ e directed cycle, then the period\n is the greatest positive integer which \ divides the lengths of all directed\n cycles of digraph. If digr\ aph has no directed cycles, then\n this function returns 0. See\ section for the defin\ ition of a directed cycle. \n\n A digraph with a period of 1 is \ said to be aperiodic. See\n . \n 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\ngap> DigraphPeriod(D);\n1\ngap> D := ChainDigraph(2)\ ;\n\ngap> DigraphPeriod(D);\n0\ngap> \ IsAcyclicDigraph(D);\ntrue\ngap> D := GeneralisedPetersenGraph(IsMutableDigrap\ h, 5, 2);\n\ngap> DigraphPeriod(D)\ ;\n1\n]]>\n \n\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 ope\ ration DigraphFloydWarshall is customised 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 func is where the work to calcul\ ate the desired outcome must be\n performed. \n\n This method initia\ lises the matrix mat by setting entry\n mat[i][j] to equal edge if there is an edge with source\n i and range j, an\ d by setting entry mat[i][j] to equal\n nopath otherwise.\n \ The final part of DigraphFloydWarshall then calls the function\n <\ A>func inside three nested for loops, over the vertices of\n digraph\ : \n\n \n\n The mat\ rix mat is then returned as the result. An example of using\n Dig\ raphFloydWarshall to calculate the shortest (non-zero) distances\n betw\ een the vertices of a digraph is shown below: \n\n 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]]>\ \n \n\n\n \n \n true or false.\n \n This operation returns true if there exists a \ non-trivial directed\n walk from vertex u to vertex v in the \ digraph digraph,\n and false if there does not exist such a d\ irected walk. See section\n for the definition of a\n non-trivial directed walk.\n \n\n Th\ e method for IsReachable has worst case complexity of O(m +\n n\ ) where m is the number of edges and n the number of\n ve\ rtices in digraph.\n\n D := Digraph([[2], [3], \ [2, 3]]);\n\ngap> IsReachable(D, 1\ , 3);\ntrue\ngap> IsReachable(D, 2, 1);\nfalse\ngap> IsReachable(D, 3, 3);\ntr\ ue\ngap> IsReachable(D, 1, 1);\nfalse\n]]>\n \n\n\n \n \n A pair of lists, or fail.\n \n\ If there exists a non-trivial directed path (or a non-trivial cycle, in th\ e\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). Otherwise, 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-zero length n-1,\n (v_\ 1, e_1, v_2, e_2, ..., e_{n-1}, v_n),\n is represented by a pair of lis\ ts [v,a] as follows:\n\n \n \n v is t\ he 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 OutNeighboursOfVertex(digraph,\ v_i)\n corresponding to the edge e_i. 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 \n \n\n The method for DigraphPath has worst case complexity of O(\ m +\n n) where m is the number of edges and n the numbe\ r of\n vertices in digraph.\n\n D := Digraph\ ([[2], [3], [2, 3]]);\n\ngap> Digr\ aphPath(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 the vertex\n u to the ver\ tex v, if such a path exists. If u =\n v, then \ the shortest non-trivial cycle is returned, again, if it\n exists. Otherwis\ e, this operation returns fail. See Section\n for the definition of a directed\n path and a dire\ cted cycle.\n \n\n See for details on th\ e output.\n\n The method for DigraphShortestPath has worst case comp\ lexity of\n O(m + n) where m is the number of edges and n<\ /M> 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> Digra\ phShortestPath(D, 5, 5);\nfail\ngap> DigraphShortestPath(D, 1, 1);\n[ [ 1, 1 ]\ , [ 1 ] ]\n]]>\n \n\n\n \ \n \n An iter\ ator.\n \n If digraph is a digraph or a list\ of adjacencies which defines a\n digraph - see - then this operation returns an\n iterator of the non-trivial directed \ paths (or directed cycles, in the case\n that u = v) i\ n digraph from the vertex u\n to the vertex v.\n \n\n See for more information about the repes\ entation\n of a directed path or directed cycle which is used, and see for more information about itera\ tors.\n See Section for the d\ efinition of\n a directed path and a directed 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> Ne\ xtIterator(iter);\n[ [ 1, 4 ], [ 3 ] ]\ngap> NextIterator(iter);\n[ [ 1, 2, 5,\ 4 ], [ 4, 2, 1 ] ]\ngap> IsDoneIterator(iter);\ntrue\ngap> iter := IteratorOf\ Paths(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 si\ mple circuits in digraph. \n\n See section for the definition\n of a simple circuit, a\ nd related notions. Note that a loop is a simple\n circuit. \n\n For\ a digraph without multiple edges, a simple circuit is uniquely\n determine\ d by its subsequence of vertices. However this is not the case for\n a mult\ idigraph. The attribute DigraphAllSimpleCircuits ignores\n multiple\ edges, and identifies a simple circuit using only its subsequence\n of ver\ tices. For example, although the simple circuits (v, e, v) and\n \ (v, e', v) (for distinct edges e and e') are\n mathematic\ ally distinct, DigraphAllSimpleCircuits considers them to\n be the s\ ame. \n\n With this approach, a directed circuit of length n can\ be determined\n by a list of its first n vertices. Thus a simple ci\ rcuit (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, \\ldots, v_n], or any cyclic perm\ utation thereof. For\n each simple circuit of digraph,\n Digr\ aphAllSimpleCircuits(digraph) includes 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> DigraphAllSimpleCircuits(D);\n[ ]\ngap> D := Digraph\ ([[3], [1], [1]]);\n\ngap> Digraph\ AllSimpleCircuits(D);\n[ [ 1, 3 ] ]\ngap> D := Digraph([[1, 1]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1\ ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1, 2, 3 ] ]\n]]>\n \n\n\n \n \n A list of vertices, o\ r fail.\n \n If digraph is a digraph,\ then DigraphLongestSimpleCircuit\n returns the longest simple ci\ rcuit in digraph. See section\n for the definition of simple\n circuit, and the definition\ of length for a simple circuit. \n\n This attribute computes\n Di\ graphAllSimpleCircuits(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 describe\ d in . \n\n If digraph ha\ s no simple circuits, then this attribute returns\n fail. If dig\ raph has multiple simple circuits of maximal\n length, then this attrib\ ute returns one of them. \n\n D := Digraph([[],\ [3], [2, 4], [5, 4], [4]]);;\ngap> DigraphLongestSimpleCircuit(D);\n[ 4, 5 ]\ \ngap> D := ChainDigraph(10);;\ngap> DigraphLongestSimpleCircuit(D);\nfail\nga\ p> D := Digraph([[3], [1], [1, 4], [1, 1]]);;\ngap> DigraphLongestSimpleCircui\ t(D);\n[ 1, 3, 4 ]\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 4, 1)\ ;\n\ngap> DigraphLongestSimpleCircu\ it(D);\n[ 1, 2, 3, 4, 8, 7, 6, 5 ]\n]]>\n \n\n\n \n \n A list.\n \n This operation return\ s a list list such that list[i] is the\n list of vertices who\ se minimum distance from the vertex vertex in\n digraph is i - 1. Vertex vertex is\n assumed to be at distance 0 fr\ om itself.\n D := CompleteDigraph(4);;\ngap> Digra\ phLayers(D, 1);\n[ [ 1 ], [ 2, 3, 4 ] ]\n]]>\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 degeneracy of digraph. \n\n\ The degeneracy of a digraph is the least integer k such\n that e\ very induced of digraph contains a vertex whose number of\n neighbou\ rs (excluding itself) is at most k. Note that this means\n that loop\ s are ignored. \n\n If digraph is not symmetric or has multiple e\ dges then this\n attribute returns fail.\n 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 := GeneralisedP\ etersenGraph(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 DigraphDegenera\ cy(digraph) is a non-negative integer\n k - see - then this attribute\n returns a degenerac\ y ordering of the vertices of the vertices of\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(digra\ ph) returns fail,\n then this attribute returns fail.\n D := DigraphSymmetricClosure(ChainDigraph(5)\ );;\ngap> DigraphDegeneracyOrdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap> D := Complete\ Digraph(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> DigraphDegeneracyOrdering(D);\n[ 1, 6, 5, 2, 4,\ 3 ]\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 3, 1);\n\ngap> DigraphDegeneracyOrdering(D);\n[ 6, 5, \ 4, 1, 3, 2 ]\n]]>\n \n\n\n \n \n A list or <\ K>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 cycle of length n. If digrap\ h is a digraph that contains a\n Hamiltonian path, then this function r\ eturns one, described in the form\n used by . Note if digraph has\n 0 or 1 vertices, then Hamiltonia\ nPath returns [] or\n [1], respectively. \n\n The m\ ethod used in this attribute has the same worst 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\ngap> HamiltonianPath(D);\nfail\ngap>\ D := Digraph([[2], [3], [1]]);\n\ \ngap> HamiltonianPath(D);\n[ 1, 2, 3 ]\ngap> D := GeneralisedPetersenGraph(Is\ MutableDigraph, 5, 2);\n\ngap> Ham\ iltonianPath(D);\nfail\n]]>\n \n\n\n \ \n \n A\ n integer.\n \n Returns the number of spanning tree\ s of the symmetric digraph\n digraph.\n NrSpanningTrees wi\ ll 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> NrSpan\ ningTrees(D);\n125\ngap> D := DigraphSymmetricClosure(CycleDigraph(24));;\ngap\ > NrSpanningTrees(D);\n24\ngap> NrSpanningTrees(EmptyDigraph(0));\n0\ngap> D :\ = GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> NrSpanningTrees(D);\n1134225\n]]>\n \n\n\n \n \n \n <\ Returns>Two lists.\n \n If digraph is a digrap\ h and source and target are\n vertices of digraph, then\ DigraphDijkstra calculates the\n length of the shortest path from \ source to target and returns\n two lists. Each element of the firs\ t list is the distance of the\n corresponding element from source. If\ a vertex was not visited in\n the process of calculating the shortest distan\ ce to target or if\n there is no path connecting that vertex with s\ ource, then\n the corresponding distance is infinity. Each elemen\ t of the\n second list gives the previous vertex in the shortest path\n from\ source to the corresponding vertex. For\n source and for any \ vertices that remained unvisited this\n will be -1. \n\n If th\ e optional second argument target is not present, then\n DigraphD\ ijkstra returns the shortest path from source to\n every vertex \ that is reachable from source.\n\n mat := [[0\ , 1, 1], [0, 0, 1], [0, 0, 0]];\n[ [ 0, 1, 1 ], [ 0, 0, 1 ], [ 0, 0, 0 ] ]\nga\ p> 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> DigraphD\ ijkstra(D, 1, 2);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\n]]>\n\n\n\n \n\n Cayley graphs of group\ s\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 GroupOfCayleyDi\ graph returns G.\n \n\n If digraph is a Cayley gra\ ph of a semigroup S and\n digraph belongs to the category , then\n SemigroupOfCayleyDigraph return\ s S.\n \n\n See also .\n G := DihedralGroup(IsPermGroup, 8);\nGroup([ \ (1,2,3,4), (2,4) ])\ngap> digraph := CayleyDigraph(G);\n\ngap> GroupOfCayleyDigraph(digraph) = G;\ntrue\n]]>\n \n\n\n \n \n A list of generators.\ \n \n If digraph is an immutable Cayley grap\ h of a group or semigroup with\n respect to a set of generators gens\ and digraph belongs to\n the category , then\n GeneratorsOfCayleyDigraph return the list of generators <\ C>gens\n over which digraph is defined.\n \n\n See als\ o \n or .\n G := DihedralGroup(IsPermGroup, 8);\ \nGroup([ (1,2,3,4), (2,4) ])\ngap> digraph := CayleyDigraph(G);\n\ngap> GeneratorsOfCayleyDigraph(digraph) = G\ eneratorsOfGroup(G);\ntrue\ngap> digraph := CayleyDigraph(G, [()]);\n\ngap> GeneratorsOfCayleyDigraph(digraph) =\ [()];\ntrue]]>\n \n\n\n \n\n\ Associated semigroups\n \n <\ Oper Name=\"AsSemigroup\" Arg=\"filt, digraph\"\n Label=\"for a filter an\ d a digraph\"/>\n \n A semilattice of partial perms.\n \n The o\ peration AsSemigroup requires that filt be equal to\n .\n If digraph i\ s a\n or \n then AsSemigroup returns a semigroup of partial perms \ which is\n isomorphic to the semigroup whose elements are the vertices of\ \n digraph with the binary operation\n .\n If digraph\n satisfies but not\n then\n AsSemigroup returns a semigroup of partial per\ ms which is\n isomorphic to the semigroup whose elements are the vertices\ of\n digraph with the binary operation\n .\n \n\n The operation AsMonoid\ behaves similarly to AsSemigroup\n except that filt ma\ y also be equal to\n \ ,\n digraph must satisfy , and the\ \n output satisfies . \n\n \ The output of both of these operations is guaranteed to be of minimal\n \ degree (see ).\ \n Furthermore the \ of\n the output is guaranteed to be the 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\ngap> ForAll(Elements(S), IsIdempotent);\ntrue\ngap> IsInver\ seSemigroup(S);\ntrue\ngap> Size(S);\n5\ngap> di := Digraph([[1], [1, 2], [1, \ 2, 3]]);\n\ngap> M := AsMonoid(IsP\ artialPermMonoid, di);\n\ngap\ > Size(M);\n3]]>\n \n \n\n \n \n A Clif\ ford semigroup of partial perms. \n \n The oper\ ation AsSemigroup requires that filt be equal to\n .\n If Y is a or\n ,\n gps is a list of groups corresponding to each vertex, \ and\n homs is a list containing for each edge (i, j) in\n \ the transitive reduction of digraph a triple [i, j, hom]\n \ where hom is a group homomorphism from gps[i] to\n g\ ps[j], and the diagram of homomorphisms commutes, then\n AsSemigro\ up returns a semigroup of partial perms which is\n isomorphic to the \ strong semilattice of groups S[Y; gps; homs].\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 := 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]]>\n \n \n\n \n\n Planarity\n \n \n A list or fail\ .\n \n KuratowskiPlanarSubdigraph return\ s the immutable list of\n lists of out-neighbours of a (not necessarily ind\ uced) subdigraph of the\n digraph digraph that witnesses the fact th\ at digraph is not\n planar, or fail if digraph is plan\ ar. In other words,\n KuratowskiPlanarSubdigraph returns the out-ne\ ighbours of a\n subdigraph of digraph that is homeomorphic to the co\ mplete graph\n with 5 vertices, or to the complete bipartite graph w\ ith vertex sets\n of sizes 3 and 3. \n \n The direc\ tions 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 the reference implementation in\n \ &edge-addition-planarity-suite; by John 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> Kura\ towskiPlanarSubdigraph(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(Is\ MutableDigraph, [[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(IsM\ utableDigraph, [[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> KuratowskiPlanarSubdigra\ ph(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 Kuratowsk\ iOuterPlanarSubdigraph returns the immutable list of\n immutable lists \ of out-neighbours of a (not necessarily induced)\n subdigraph of the digrap\ h digraph that witnesses the fact\n that digraph is not outer\ planar, or fail if\n digraph is outer planar. In other word\ s,\n KuratowskiOuterPlanarSubdigraph returns the out-neighbours of a\ \n subdigraph of digraph that is homeomorphic to the complete graph\ \n with 4 vertices, or to the complete bipartite graph with vertex s\ ets\n of sizes 2 and 3. \n \n The directions and mu\ ltiplicities of any edges in digraph are\n ignored when considering \ whether or not digraph is outer planar. \n \n\n See also \n \ ,\n , and \n .\n \n\ \n This method uses the reference implementation in\n &edge-addition-pla\ narity-suite; by John 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> KuratowskiOuterPlanarSu\ bdigraph(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> IsOuterPlanarDigrap\ h(D);\nfalse\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ ], [ ], [ ], [ \ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ], \n [ ] ]\ngap> D := Digraph(Is\ MutableDigraph, [[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> KuratowskiOuterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], [ 9, 8\ , 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\nga\ p> 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> IsOuterPlanarDigraph(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 digr\ aph is a planar digraph, then \n PlanarEmbedding returns the imm\ utable list of lists of\n out-neighbours of a subdigraph of digraph \ such that each vertex's\n neighbours are given in clockwise order. If di\ graph is not planar,\n then fail is returned. \n \n Th\ e directions and multiplicities of any edges in digraph are\n ignore\ d by PlanarEmbedding.\n \n\n See also \n .\n \n\n This method uses the reference implementation\ in\n &edge-addition-planarity-suite; by John Boyer of the algorithms descr\ ibed\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\ng\ ap> 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, 1\ 0], [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> Plana\ rEmbedding(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\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, 1\ 0],\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> PlanarEmb\ edding(D);\nfail\n]]>\n \n\n\n \n \n A \ list or fail.\n \n If digraph is an o\ uter planar digraph, then\n OuterPlanarEmbedding returns the immutab\ le 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 digrap\ h is not\n outer planar, then fail is returned. \n \n \ The directions and multiplicities of any edges in digraph are\n igno\ red by OuterPlanarEmbedding.\n \n\n See also .\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> 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> OuterPlanar\ Embedding(D);\nfail\ngap> OuterPlanarEmbedding(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 := 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> Oute\ rPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(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 r\ eturn the immutable list of lists of\n out-neighbours of a subdigraph of th\ e digraph digraph which is\n homeomorphic to one of the following:\n\ the complete bipartite graph with vertex sets of sizes 2 and\n <\ C>3; the complete bipartite graph with vertex 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 impleme\ ntation in\n &edge-addition-planarity-suite; by John Boyer of the algorithm\ s described\n in .\n\n D := Dig\ raph([[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> Subdigr\ aphHomeomorphicToK23(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> Subdigraph\ HomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 10, 5\ ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], \ [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\ngap> SubdigraphHomeomorphi\ cToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomorphicToK33(CompleteDigrap\ h(5));\nfail\ngap> SubdigraphHomeomorphicToK33(CompleteBipartiteDigraph(3, 3))\ ;\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ], [ ] ]\ngap> Subdigrap\ hHomeomorphicToK4(CompleteDigraph(3));\nfail\ngap> D := Digraph(IsMutableDigra\ ph, [[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> SubdigraphH\ omeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 \ ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> 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> SubdigraphHomeomorphicToK23(D);\n[ \ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11\ ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\ngap> SubdigraphH\ omeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomorphicToK33(Comp\ leteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(CompleteBipartiteDigr\ aph(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 Edge properties\n \n \n tr\ ue 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 ar\ gument digraph is mutable, then the return value of\n this property \ is recomputed every time it is called. \n\n D := Digra\ ph([[1, 2], [2]]);\n\ngap> Digraph\ Edges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 2 ] ]\ngap> DigraphHasLoops(D);\ntrue\ng\ ap> 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(IsMutableDigraph, 4);\n\ \ngap> DigraphHasLoops(D);\nfalse]]\ >\n \n\n\n \n \n \n true or false.\ \n \n This property is true if the digraph digraph<\ /A>\n is antisymmetric, and false if it is not.\n \n\n A d\ igraph is antisymmetric if whenever there is an edge with source\n <\ C>u and range v, and an edge with source v and range\n u, then the vertices u and v are equal.\n \n \n\ If the argument digraph is mutable, then the return value of\n t\ his property is recomputed every time it is called. \n\n 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> IsAntisymmetricDigr\ aph(gr2);\ntrue\ngap> DigraphEdges(gr2);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 3 ], [ 2,\ 3 ], [ 3, 1 ] ]\n]]>\n \n\n\n \n \n t\ rue or false.\n \n This property is tru\ e if the digraph digraph is bipartite, 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 distinct sets.\n Equivalently, a digraph\ is bipartite if and only if it is 2-colorable; see\n . \n\n See also .\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 D := ChainDigraph(4);\n\ngap> IsBipartiteDigraph(D);\ntrue\ngap> D := Cy\ cleDigraph(3);\n\ngap> IsBipartiteDig\ raph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ \ngap> IsBipartiteDigraph(D);\ntrue\ ]]>\n \n\n\n \n \n true or\ false.\n \n Returns true if the digr\ aph digraph is a complete bipartite\n digraph, and false if i\ t is not. \n\n A digraph is a complete bipartite digraph if it i\ s bipartite, see\n , and there exists a u\ nique edge with\n source i and range j if and only if i and j lie\n in different bicomponents of digraph, see . \n\n Equivalently, a bipartite \ digraph with bicomponents of size m and\n n is complete preci\ sely when it has 2mn edges, none of which\n are multiple edges. \ \n\n See also .\n \n \n\ If the argument digraph is mutable, then the return value of\n t\ his property is recomputed every time it is called. \n\n D := CycleDigraph(2);\n\ngap> \ IsCompleteBipartiteDigraph(D);\ntrue\ngap> D := CycleDigraph(4);\n\ngap> IsBipartiteDigraph(D);\ntrue\ngap> IsCompl\ eteBipartiteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDi\ graph, 5, 4);\n\ngap> IsCompleteBip\ artiteDigraph(D);\ntrue]]>\n \n\n\n <\ ManSection>\n \n \ true or false.\n \n Returns true if the digraph digraph is complete, and\n false if it is n\ ot. \n\n A digraph is complete if it has no loops, and for all\n\ distinct vertices i and j,\n there is exactly one \ edge with source i and range j.\n\n Equivalently, a digraph w\ ith n vertices is complete precisely when\n it has n(n - 1) e\ dges, no loops, and no multiple edges.\n \n \n If the argument \ digraph is mutable, then the return value of\n this property is reco\ mputed every time it is called. \n\n D := Digraph(\ [[2, 3], [1, 3], [1, 2]]);\n\ngap>\ IsCompleteDigraph(D);\ntrue\ngap> D := Digraph([[2, 2], [1]]);\n\ngap> IsCompleteDigraph(D);\nfalse\ngap> \ D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteDigraph(D);\nfalse]]>\n \n\n\n \n \n true or false.\n \n This property returns true if digraph is a complete\n multipartite digraph, and false if not. \n\n \ A digraph is a complete multipartite digraph if and only if\n its \ vertices can be partitioned into at least two maximal independent sets,\n w\ here every possible edge between these independent sets occurs in the\n dig\ raph exactly once.\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 := CompleteMultipartiteDigraph(\ [2, 4, 6]);\n\ngap> IsCompleteMultipartiteDigraph(D);\ntrue\ngap> D := CompleteBipartite\ Digraph(IsMutableDigraph, 5, 4);\n\ \ngap> IsCompleteMultipartiteDigraph(D);\ntrue]]>\n \ \n\n\n \n \n \n true\ or false.\n \n Returns true if t\ he digraph digraph is empty, and\n false if it is not. A digr\ aph is empty if it has no\n edges. \n\n IsNullDigraph i\ s a synonym for IsEmptyDigraph.\n \n \n If the argument \ digraph is mutable, then the return value of\n this property is reco\ mputed every time it is called. \n\n D := Digraph(\ [[], []]);\n\ngap> IsEmptyDigraph(D);\ \ntrue\ngap> IsNullDigraph(D);\ntrue\ngap> D := Digraph([[], [1]]);\n\ngap> IsEmptyDigraph(D);\nfalse\ngap> IsNul\ lDigraph(D);\nfalse]]>\n \n\n\n \n \n <\ K>true or false.\n \n A digraph is an eq\ uivalence digraph if and only if the digraph satisifies\n all of ,\n and .\n A partial order digraph corresponds t\ o 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> IsEquivalenceDigra\ ph(D);\ntrue\n]]>\n \n\n\n \n \n true\ or false.\n \n This property is true\ if the digraph digraph is\n functional. \n\n A digraph i\ s functional if every vertex is the source of a\n unique edge.\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 gr1 := Digraph([[3], [2], [2], [1], [6], [5]]);\n\ngap> IsFunctionalDigraph(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> IsFunctionalDigra\ ph(gr3);\ntrue\n]]>\n \n\n\n \n \n true or false.\n \n A multidigraph is on\ e 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 pro\ perty is recomputed every time it is called. \n\n \ D := Digraph([\"a\", \"b\", \"c\"], [\"a\", \"b\", \"b\"], [\"b\", \"c\", \"a\ \"]);\n\ngap> IsMultiDigraph(D);\n\ false\ngap> D := DigraphFromDigraph6String(\"&Bug\");\n\ngap> IsDuplicateFree(DigraphEdges(D));\ntrue\ngap> IsMu\ ltiDigraph(D);\nfalse\ngap> D := Digraph([[1, 2, 3, 2], [2, 1], [3]]);\n\ngap> IsDuplicateFree(DigraphEdges\ (D));\nfalse\ngap> IsMultiDigraph(D);\ntrue\ngap> D := DigraphMutableCopy(D); \ \n\ngap> IsMultiDigraph(D);\ntr\ ue]]>\n \n\n\n \n \n true or fa\ lse.\n \n This property is true if the d\ igraph 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 is mutable, then the return value\ of\n this property is recomputed every time it is called. \n <\ ![CDATA[\ngap> D := Digraph([[1, 2], [2]]);\n\ngap> IsReflexiveDigraph(D);\ntrue\ngap> D := Digraph([[3, 1], [4,\ 2], [3], [2, 1]]);\n\ngap> IsRefl\ exiveDigraph(D);\nfalse\n]]>\n \n\n\n \ \n \n true or false.\n \n This property \ is true if the digraph digraph\n is symmetric, and false if it is not. \n\n A symmetric digraph is one where for each n\ on-loop edge, having\n source u and range v, there is a corre\ sponding edge with\n source v and range u. If there are n\ edges with\n source u and range v, then there must be pr\ ecisely n\n edges with source v and range u. In other\ words, a symmetric\n digraph has a symmetric adjacency matrix .\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 gr1 := Digraph([[2], [1, 3], [2, \ 3]]);\n\ngap> IsSymmetricDigraph(g\ r1);\ntrue\ngap> adj1 := AdjacencyMatrix(gr1);;\ngap> Display(adj1);\n[ [ 0, \ 1, 0 ],\n [ 1, 0, 1 ],\n [ 0, 1, 1 ] ]\ngap> adj1 = TransposedMat(ad\ j1);\ntrue\ngap> gr1 = DigraphReverse(gr1);\ntrue\ngap> gr2 := Digraph([[2, 3]\ , [1, 3], [2, 3]]);\n\ngap> IsSymm\ etricDigraph(gr2);\nfalse\ngap> adj2 := AdjacencyMatrix(gr2);;\ngap> Display(a\ dj2);\n[ [ 0, 1, 1 ],\n [ 1, 0, 1 ],\n [ 0, 1, 1 ] ]\ngap> adj2 = T\ ransposedMat(adj2);\nfalse\n]]>\n \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 (und\ irected) graph.\n Specifically, a tournament is a digraph which has a uniqu\ e directed edge\n (of some orientation) between any pair of distinct vertic\ es, and no loops.\n \n \n If the argument digraph is mut\ able, then the return value of\n this property is recomputed every time it \ is called. \n D := Digraph([[2, 3, 4], [3, 4], [4]\ , []]);\n\ngap> IsTournament(D);\n\ true\ngap> D := Digraph([[2], [1], [3]]);\n\ngap> IsTournament(D);\nfalse\ngap> D := CycleDigraph(IsMutableDigra\ ph, 3);\n\ngap> IsTournament(D);\ntr\ ue\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 digraph\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, the methods used for this property\ use a version\n of the Floyd-Warshall algorithm, and have complexity O(\ n^3).\n\n However for digraphs which are topologically sortable\n [<\ Ref Attr=\"DigraphTopologicalSort\"/>], then methods with\n complexity O\ (m + n + m \\cdot n) will be used when appropriate.\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], [3]]);\n\ngap> IsTransitiveDigraph(D);\nfalse\ngap> gr2 := Digraph([[1, 2\ , 3], [3], [3]]);\n\ngap> IsTransi\ tiveDigraph(gr2);\ntrue\ngap> gr2 = DigraphTransitiveClosure(D);\ntrue\ngap> g\ r3 := Digraph([[1, 2, 2, 3], [3, 3], [3]]);\n\ngap> IsTransitiveDigraph(gr3);\ntrue\n]]>\n \n\n\n \n \n \n\ true or false.\n \n A digr\ aph 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 t\ he 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 propert\ y is recomputed every time it is called. \n\n D := Dig\ raph([[1], [2, 3], [2, 3]]);\n\nga\ p> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[1 .. 4], [1 .. 4], [1 .. 4\ ], [1 .. 4]]);\n\ngap> IsPreorder\ Digraph(D);\ntrue\ngap> D := Digraph([[2], [3], [4], [5], [1]]);\n\ngap> IsPreorderDigraph(D);\nfalse\ngap> D :=\ Digraph([[1], [1, 2], [2, 3]]);\n\ \ngap> IsQuasiorderDigraph(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\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 <\ A>digraph corresponds\n to the partial order relation \\leq defi\ ned by x \\leq y if and\n only if [x, y] is an edge of dig\ raph.\n \n \n If the argument digraph is mutable, th\ en the return value of\n this property is recomputed every time it is calle\ d. \n\n D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsPartialOrderDigraph(D);\ntrue\nga\ p> D := CycleDigraph(5);\n\ngap> IsPa\ rtialOrderDigraph(D);\nfalse\ngap> D := Digraph([[1, 1], [1, 1, 2], [3], [3, 3\ , 4, 4]]);\n\ngap> IsPartial\ OrderDigraph(D);\ntrue\n]]>\n \n\n\n \ \n \n <\ Prop Name=\"IsJoinSemilatticeDigraph\" Arg=\"digraph\"/>\n \n true or false.\n \n IsMeetSemilatticeDigraph returns true\ if the digraph\n digraph is a meet semilattice; IsJoinSemilattic\ eDigraph\n returns true if the digraph digraph is a join \ semilattice;\n and IsLatticeDigraph returns true if the digr\ aph\n digraph is both a meet and a join semilattice.\n \n\n \ For a partial order digraph the\n c\ orresponding 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 me\ et semilattice if it is a partial order and every\n pair of vertices ha\ s a greatest lower bound (meet) with respect to the\n aforementioned relati\ on. A join semilattice is a partial order where\n every pair of vert\ ices has a least upper bound (join) with respect to\n the relation.\n \n \n If the argument digraph is mutable, then the return val\ ue of\n this property is recomputed every time it is called. \n\n D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsMeetSemilatticeDigraph(D);\nfalse\ngap> IsJoin\ SemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ngap> D := Digr\ aph([[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> IsMeetSemilattic\ eDigraph(D);\ntrue\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDig\ raph(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);\ntrue\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 such that\n for every vertex\ v of digraph digraph there are exactly\n n edges term\ inating in v.\n\n See also and\n\ .\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 IsInRegularDigra\ ph(CompleteDigraph(4));\ntrue\ngap> IsInRegularDigraph(ChainDigraph(4));\nfals\ e\n]]>\n \n\n\n \n \n true or \ false.\n \n This property is true if the\ re is an integer n such that\n for every vertex v of digraph \ digraph there are exactly\n n 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 this property is recomputed every ti\ me it is called. \n IsOutRegularDigraph(CompleteDi\ graph(4));\ntrue\ngap> IsOutRegularDigraph(ChainDigraph(4));\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n This property is true if there is an intege\ r n such that\n for every vertex v of digraph digraph \ there are exactly\n n edges starting and terminating at v. In\ other words,\n the property is true if digraph is both in-re\ gular and\n and out-regular.\n\n See also and\n .\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 IsR\ egularDigraph(CompleteDigraph(4));\ntrue\ngap> IsRegularDigraph(ChainDigraph(4\ ));\nfalse\n]]>\n \n\n\n \ \n \n t\ rue 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 digraph\n and any two integer\ s i and j between 0 and the\n diameter of digraph, the number of vertices at distance i\n from u and distanc\ e j from v depends only on\n i, j, and the dist\ ance 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 exac\ tly b_i neighbors of v which are at\n distance i - 1 a\ way from u, and c_i neighbors of\n v which are at dist\ ance i + 1 away from u. This\n definition is used to check wh\ ether digraph is distance regular. \n\n In the case where digr\ aph is not symmetric or not connected, the\n property is false.\ \n \n \n If the argument digraph is mutable, then the re\ turn value of\n this property is recomputed every time it is called. \n\n \ D := DigraphSymmetricClosure(ChainDigraph(5));;\nga\ p> 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 prope\ rty is true if the digraph digraph is acyclic, and\n false\ if it is not. A digraph is acyclic if every directed\n cycle on\ the digraph is trivial. See Section for the definition of a directed cycle, and of a trivial\n d\ irected cycle.\n\n The method used in this operation 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 in the digraph. \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\n Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y\ )\n> return IsEmpty(Intersection(x, y));\n> end);;\ngap> D := Digraph(Peters\ en);\n\ngap> IsAcyclicDigraph(D)\ ;\nfalse\ngap> D := DigraphFromDiSparse6String(\n> \".b_OGCIDBaPGkULEbQHCeRIdr\ HcuZMfRyDAbPhTi|zF\");\n\ngap> I\ sAcyclicDigraph(D);\ntrue\ngap> IsAcyclicDigraph(ChainDigraph(10));\ntrue\ngap\ > D := CompleteDigraph(IsMutableDigraph, 4);\n\ngap> IsAcyclicDigraph(D);\nfalse\ngap> IsAcyclicDigraph(CycleDigr\ aph(10));\nfalse]]>\n \n\n\n \n \n true or false.\n \n IsChainDigraph retur\ ns true if the digraph digraph is\n isomorphic to the chain d\ igraph 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 directed tree, in which\n every v\ ertex has out degree at most one; see \n and \ .\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> IsChainDigraph(D)\ ;\nfalse\ngap> D := ChainDigraph(5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> D := DigraphReverse(D);\n\ngap> IsChainDigraph(D);\ntrue\ngap> D := Chain\ Digraph(IsMutableDigraph, 5);\n\ngap\ > IsChainDigraph(D);\ntrue\ngap> DigraphReverse(D);\n\ngap> IsChainDigraph(D);\ntrue]]>\n \n\n\n \n \n true or false.\n \n This property is true if the digraph digraph\n is we\ akly connected and false if it is not. A digraph\n digraph i\ s weakly connected if it is possible to travel\n from any vertex to \ any other vertex by traversing edges in either\n direction (possibly agains\ t the orientation of some of them). \n\n The method used in this functi\ on 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 vertice\ s of the digraph).\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([[2], [3], []]);;\ng\ ap> IsConnectedDigraph(D);\ntrue\ngap> D := Digraph([[1, 3], [4], [3], []]);;\ \ngap> IsConnectedDigraph(D);\nfalse\ngap> D := Digraph(IsMutableDigraph, [[2]\ , [3], []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> D := Digraph(IsMutableD\ igraph, [[1, 3], [4], [3], []]);;\ngap> IsConnectedDigraph(D);\nfalse]]>\n \n\n\n \n \n true or false.<\ /Returns>\n \n A connected digraph is biconnected if i\ t is still connected (in the\n sense of )\ 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 retur\ ns false if\n digraph is not connected. \n \n If \ the argument digraph is mutable, then the return value of\n this pro\ perty is recomputed every time it is called. \n\n Multiple edges are ig\ nored by this method. \n\n The method used in this operation has comple\ xity 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 See also , , and \n .\n \n \n IsBiconnecte\ dDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBiconnectedDigraph(Cy\ cleDigraph(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> I\ sBiconnectedDigraph(D);\ntrue]]>\n \n\n\ \n \n \n true or false.\n \n A connected di\ graph is bridgeless if it is still connected (in the\n sense of ) when any edge is removed. \n If D ha\ s at least 3 vertices, then implies \ IsBridgelessDigraph;\n see or for a more\n detailed explanation.\n \n\n IsBridgelessDigraph returns true if the digraph\n D is b\ ridgeless, and false if it is not. In\n particular, IsBridgelessD\ igraph 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 called. \n\n Multiple edges ar\ e ignored by this method. \n\n The method used in this operation has co\ mplexity 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 See also , , and \n .\n \n IsBridgelessDigr\ aph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBridgelessDigraph(CycleDig\ raph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4]]);;\n\ gap> IsBridgelessDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMuta\ bleDigraph, 5, 4);\n\ngap> IsBridge\ lessDigraph(D);\ntrue\ngap> D := Digraph([[2, 5], [1, 3, 4, 5], [2, 4], [2, 3]\ , [1, 2]]);\n\ngap> IsBridgelessD\ igraph(D);\ntrue\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> D := Digraph([[2]\ , [3], [4], [2]]);\n\ngap> IsBridg\ elessDigraph(D);\nfalse\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> IsBridgele\ ssDigraph(ChainDigraph(2));\nfalse\ngap> IsBiconnectedDigraph(ChainDigraph(2))\ ;\ntrue\n]]>\n \n\n\n \n \ \n tr\ ue or false.\n \n This property is tr\ ue if the digraph digraph is strongly\n connected and false if it is not. \n\n A digraph digraph 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 ope\ ration is based on Gabow's Algorithm and has comp\ lexity O(m+n), where m is\n the number of edges (counting mul\ tiple edges as one) and n is the\n number of vertices in the digraph\ .\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 := CycleDigraph(250000);\n\ngap> IsStronglyConnectedDigraph(D);\ntrue\ngap> D :=\ DigraphRemoveEdges(D, [[250000, 1]]);\n\ngap> IsStronglyConnectedDigraph(D);\nfalse\ngap> D := CycleD\ igraph(IsMutableDigraph, 250000);\n\ngap> IsStronglyConnectedDigraph(D);\ntrue\ngap> DigraphRemoveEdge(D\ , [250000, 1]);\n\ngap> Is\ StronglyConnectedDigraph(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n Th\ is property is true if the digraph digraph\n is aperiodic, i.\ e. if its is equal to 1.\n Otherwise, the p\ roperty is false.\n \n \n If the argument digraph\ is mutable, then the return value of\n this property is recomputed every t\ ime it is called. \n D := Digraph([[6], [1], [2], \ [3], [4, 4], [5]]);\n\ngap> I\ sAperiodicDigraph(D);\nfalse\ngap> D := Digraph([[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);\ntru\ e]]>\n \n\n\n \n \n true or false.\n \n Returns true if the digraph digr\ aph is a directed tree, and\n false 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 no\ t considered a directed\n tree as it has no source. \n\n See also .\n \n \n If the argument digrap\ h is mutable, then the return value of\n this property is recomputed ev\ ery time it is called. \n\n D := Digraph([[], [2]]\ );\n\ngap> IsDirectedTree(D);\nfals\ e\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 <\ C>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 sym\ metric digraph without loops, in which for\n any pair of distinct vertices \ u and v, there is exactly one\n directed path from u t\ o v. See and , and see\n Section for the definition of\n directed path. This definition implies tha\ t an undirected tree has\n no multiple edges. \n\n An undirected \ forest is a digraph, each of whose connected components\n is an undirec\ ted tree. In other words, an undirected forest is isomorphic to\n a disjoin\ t union of undirected trees. See and . In particular, every\n undirected tree is an undirected forest. \ \n\n Please note that the digraph with zero vertices is considered to b\ e neither\n an undirected tree nor an undirected forest.\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([[3], [3], [1, 2]]);\n\ngap> IsUndirectedTree(D);\ntrue\ngap> IsSymmetricDigraph(D) and no\ t DigraphHasLoops(D);\ntrue\ngap> D := Digraph([[3], [5], [1, 4], [3], [2]]);\ \n\ngap> IsConnectedDigraph(D);\nf\ alse\ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedForest(D);\ntrue\ngap\ > D := Digraph([[1, 2], [1], [2]]);\n\ngap> IsUndirectedTree(D) or IsUndirectedForest(D);\nfalse\ngap> IsSymmetr\ icDigraph(D) or not DigraphHasLoops(D);\nfalse]]>\n \ \n\n\n \n \n true or false.\n \n This property returns true if the digraph digraph is Eulerian.\n\ \n\n A connected digraph is called Eulerian if there exists \ a directed\n circuit on the digraph which includes every edge exactly once.\ See\n Section for the de\ finition of\n a directed circuit. Note that the empty digraph with at most \ one vertex is\n considered to be Eulerian.\n \n \n If the ar\ gument digraph is mutable, then the return value of\n this property \ is recomputed every time it is called. \n\n D := D\ igraph([[]]);\n\ngap> IsEulerianDigraph\ (D);\ntrue\ngap> D := Digraph([[2], []]);\n\ngap> IsEulerianDigraph(D);\nfalse\ngap> D := Digraph([[3], [], [2]])\ ;\n\ngap> IsEulerianDigraph(D);\nf\ alse\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, then this property returns\n \ true, and false if it is not. \n\n A digraph with n v\ ertices is Hamiltonian if it has a\n directed cycle of length n. See Section for the\ definition of a directed cycle.\n Note the empty digraphs on 0 and 1 verti\ ces 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 argument digraph is mutable, then \ the return value of\n this property is recomputed every time it is called. \ \n\n g := Digraph([[]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\ngap> g := Digraph([[2], \ [1]]);\n\ngap> IsHamiltonianDigrap\ h(g);\ntrue\ngap> g := Digraph([[3], [], [2]]);\n\ngap> IsHamiltonianDigraph(g);\nfalse\ngap> g := Digraph([[2],\ [3], [1]]);\n\ngap> IsHamiltonian\ Digraph(g);\ntrue\n]]>\n \n\n\n \n \n true\ or false.\n \n IsCycleDigraph re\ turns true if the digraph digraph is\n isomorphic to the cycl\ e digraph with the same number of vertices as\n digraph, and fals\ e if it is not; see . \n\n A digr\ aph is a cycle if and only if it is strongly connected and has\n the\ same number of edges as vertices. \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 := Digraph([[1, 3]\ , [2, 3], [3]]);\n\ngap> IsCycleDi\ graph(D);\nfalse\ngap> D := CycleDigraph(5);\n\ngap> IsCycleDigraph(D);\ntrue\ngap> D := OnDigraphs(D, (1, 2, 3));\ \n\ngap> D = CycleDigraph(5);\nfal\ se\ngap> IsCycleDigraph(D);\ntrue]]>\n \n\n\n \n \n Planarity\n \n \n true\ or false.\n \n A planar digraph is a\ digraph that can be embedded in the plane in\n such a way that its edges d\ o not intersect. A digraph is planar if and only\n if it does not have a su\ bdigraph that is homeomorphic to either the\n complete graph on 5 ve\ rtices or the complete bipartite graph with\n vertex sets of sizes 3\ and 3.\n \n \n IsPlanarDigraph returns true\ if the digraph digraph is\n planar and false if it is not. T\ he directions and multiplicities of\n any edges in digraph are ignor\ ed by IsPlanarDigraph.\n \n\n See also .\n \n\n This method uses the reference implementatio\ n in\n &edge-addition-planarity-suite; by John Boyer of the algorithms desc\ ribed\n in .\n\n IsPlanarDi\ graph(CompleteDigraph(4));\ntrue\ngap> IsPlanarDigraph(CompleteDigraph(5));\nf\ alse\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(2, 3));\ntrue\ngap> IsPlan\ arDigraph(CompleteBipartiteDigraph(3, 3));\nfalse\ngap> IsPlanarDigraph(Comple\ teDigraph(IsMutableDigraph, 4));\ntrue\ngap> IsPlanarDigraph(CompleteDigraph(I\ sMutableDigraph, 5));\nfalse\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(Is\ MutableDigraph, 2, 3));\ntrue\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(I\ sMutableDigraph, 3, 3));\nfalse\n]]>\n \n\n\n \n \ \n true or false.\n \n An <\ E>outer planar digraph is a digraph that can be embedded in the\n plane\ in such a way that its edges do not intersect, and all vertices\n belong t\ o 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 IsOuterPla\ narDigraph 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 th\ e reference implementation in\n &edge-addition-planarity-suite; by John Boy\ er of the algorithms described\n in .\n\n IsOuterPlanarDigraph(CompleteDigraph(4));\nfalse\ngap> IsOuterPlana\ rDigraph(CompleteDigraph(5));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipart\ iteDigraph(2, 3));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(\ 3, 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(10));\ntrue\ngap> IsOut\ erPlanarDigraph(CompleteDigraph(IsMutableDigraph, 4));\nfalse\ngap> IsOuterPla\ narDigraph(CompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsOuterPlanarDi\ graph(CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 2, 3));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipar\ titeDigraph(IsMutableDigraph,\n> \ 3, 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(IsMutableDigraph, 1\ 0));\ntrue\n\n]]>\n \n\n\n \n\ \n Homomorphisms and transformations\n \n \n true or false.\n \n This property returns tr\ ue if digraph is a core,\n and false if it is not.\n\ \n A digraph D is a core if and only if it has no proper\n \ subdigraphs A such that there exists 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 automorphism 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(CycleDig\ raph(6));\n\ngap> Digra\ phHomomorphism(D, CompleteDigraph(2));\nTransformation( [ 1, 2, 1, 2, 1, 2 ] )\ \ngap> IsDigraphCore(D);\nfalse\n]]>\n \n\n\n \n \n \ true or false.\n \n If di\ graph is a digraph without multiple edges, then\n IsEdgeTransitive returns true if digraph\n is edge transitive, and false\ otherwise. A digraph is\n edge transitive if its automorphism g\ roup acts\n transitively on its edges (via 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(Complete\ Digraph(2));\ntrue\ngap> IsEdgeTransitive(ChainDigraph(3));\nfalse\ngap> IsEdg\ eTransitive(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 <\ A>digraph is a digraph, then IsVertexTransitive returns\n tru\ e if digraph is vertex transitive, and false\n otherwise.\ A digraph is vertex transitive if its automorphism group\n acts tra\ nsitively 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(Comple\ teDigraph(2));\ntrue\ngap> IsVertexTransitive(ChainDigraph(3));\nfalse\n]]>\n \n\n\n \n\n \n\n <\ !-- properties -->\n Homomorphisms<\ /Heading>\n \n\n Acting on digraphs\n \n \n \n A digraph.\n \n If digraph is a digraph, and the second a\ rgument perm is a\n permutation of the vertices of digraph\ , then this operation\n returns a digraph constructed by relabelling th\ e vertices of\n digraph according to perm. Note that for an \ automorphism\n f of a digraph, we have 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 constructed by transforming the source a\ nd range of each edge according to\n trans. 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 contain multiple edges, even if digraph does not.\n If trans is mathematically a permutation, t\ hen the result coincides\n with OnDigraphs(digraph, AsPermutation\ (trans)).\n \n\n The \ of digraph will not be retained\n in the returned digraph. \n\n \ If digraph belongs to , then\n re\ labelling of the vertices is performed directly on digraph.\n If \ digraph belongs to , an immutable\n c\ opy of digraph with the vertices relabelled 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 := Tra\ nsformation([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]]>\n \n\n\n \n \n \n A digraph.\n \n If d\ igraph is a digraph, and pair is a pair consisting of a\n permut\ ation of the vertices and a permutation of the edges of\n digraph, t\ hen this operation returns a digraph\n constructed by relabelling the verti\ ces and edges of digraph\n according to perm[1] and perm[2\ ], respectively. \n\n In its second form, OnMultiDigraphs re\ turns a digraph with vertices\n and edges permuted by perm1 and p\ erm2, respectively. \n\n Note that OnDigraphs(digraph,\n \ perm)=OnMultiDigraphs(digraph, [perm, ()]) where perm i\ s\n a permutation of the vertices of digraph. If you are only intere\ sted\n in the action of a permutation on the vertices of a digraph, then yo\ u can\n use OnDigraphs instead of OnMultiDigraphs.\n\n If \ digraph belongs to , then\n relabell\ ing of the vertices is performed directly on digraph.\n If digrap\ h 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\ng\ ap> 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 canonical labellings\n From\ version 0.11.0 of &Digraphs; it is possible to use either &bliss; or\n &na\ uty; (via &NautyTracesInterface;) to calculate canonical labellings and\n a\ utomorphism groups of digraphs; see and for more details about &bliss; and &nauty;,\n respectively. \n\n\ \n \n \n Nothing.\n \n These functions can be used to specify whether &nauty; or &bliss; shoul\ d be\n used by default by &Digraphs;. If &NautyTracesInterface; is not avai\ lable,\n then these functions do nothing. Otherwise, by calling\n Dig\ raphsUseNauty subsequent computations will default to using\n &nauty; r\ ather than &bliss;, where possible.\n \n\n You can call these functi\ ons at any point in a &GAP; session, as many\n times as you like, it is gua\ ranteed that existing digraphs remain\n valid, and that comparison of exist\ ing digraphs and newly created digraphs\n via\n ,\n ,\n , and\n are also valid. \ \n\n It is also possible to compute the automorphism group of a specific di\ graph\n using both &nauty; and &bliss; using \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 digraph\n is an isomorphism\ from digraph to itself. See for more information\n about isomorphisms of digra\ phs. \n\n If digraph is not a multidigraph then the automorphism\ group is\n returned as a group of permutations on the set of vertices of\n\ digraph. \n\n If digraph is a multidigraph then the a\ utomorphism group is returned\n as the direct product of a group of permuta\ tions on the set of vertices of\n digraph with a group of permutatio\ ns on the set of edges of\n digraph. These groups can be accessed us\ ing on the\n returned group. \n\n By default, the \ automorphism group is found using &bliss; by Tommi Junttila\n and Petteri K\ aski. If &NautyTracesInterface; is available, then &nauty; by\n Brendan Mck\ ay and Adolfo Piperno can be used instead; see\n , \n , ,\n and .\n \n\n If the argument digraph is mutable, then the ret\ urn value of\n this attribute is recomputed every time it is called. \n\n \ 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]]>\n \n \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 &bliss; \ by Tommi Junttila\n and Petteri Kaski. \n\n The attribute and\n operation \ returns\n the value of either BlissAutomorphismGroup or . These groups are, of course, equal but\n \ their generating sets may differ. \n The attribute re\ turns the value of\n BlissAutomorphismGroup as it is not implemented\ for &nauty; The \n requirements for the optional arguments vertex_colou\ rs and \n edge_colours are documented in .\n\n \ See also , and .\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 \ G := BlissAutomorphismGroup(JohnsonDigraph(5, 2));;\n\ gap> 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 attribut\ e contains the group of\n automorphisms of digraph as calculated usi\ ng &nauty; by\n Brendan Mckay and Adolfo Piperno via &NautyTracesInterface;\ .\n\n \n\n T\ he attribute and\n \ operation returns\n the value of either NautyAutomorphismGr\ oup or .\n These groups are, of course, equal but their generating sets may di\ ffer. \n\n See also , and .\n \n\n If the argument digraph is mutable\ , then the return value of\n this attribute is recomputed every time it is \ called. \n\n 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) ])]]>\n \n \n\n \n \ \n A permutation group.\n \n\n This operation computes the automorphism group of\ a vertex-coloured digraph.\n A vertex-coloured digraph can be specified by\ its underlying digraph\n digraph and its colouring vert_colours<\ /A>. Let n be the\n number of vertices of digraph. The colour\ ing vert_colours may\n have one of the following two forms:\n\n <\ List>\n \n a list of n integers, where vert_colour\ s[i] is the\n colour of vertex i, using the colours <\ C>[1 .. m] for some\n m <= n; or\n \n \n a list of non-empty disjoint lists whose union is\n Di\ graphVertices(digraph), such that\n vert_colours[i\ ] is the list of all vertices with colour\n i.\n \n \n\n The automorphism group of a coloured digraph d\ igraph with\n colouring vert_colours is the group consisting of \ its automorphisms; an\n automorphism of digraph is an isomorp\ hism of coloured\n digraphs from digraph to itself. This group is e\ qual to the\n subgroup of AutomorphismGroup(digraph) consisti\ ng of those\n automorphisms that preserve the colouring specified by ver\ t_colours. See\n , and see \n for more information about isomorphisms of colo\ ured digraphs. \n\n If digraph is not a multidigraph then the au\ tomorphism group is\n returned as a group of permutations on the set of ver\ tices of\n digraph. \n\n If digraph is a multidigraph \ then the automorphism 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 a\ ccessed using on the\n returned group. \n\n By def\ ault, the automorphism group is found using &bliss; by Tommi Junttila\n and\ Petteri Kaski. If &NautyTracesInterface; is available, then &nauty; by\n B\ rendan Mckay and Adolfo Piperno can be used instead; see\n ,\n \ , ,\n \ and .\n\n cycle :\ = CycleDigraph(9);\n\ngap> G := Autom\ orphismGroup(cycle);;\ngap> IsCyclic(G) and Size(G) = 9;\ntrue\ngap> colours :\ = [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;\ngap> H := AutomorphismGroup(cycle, colo\ urs);;\ngap> Size(H);\n3\ngap> H = AutomorphismGroup(cycle, [1, 2, 3, 1, 2, 3,\ 1, 2, 3]);\ntrue\ngap> H = SubgroupByProperty(G, p -> OnTuplesSets(colours, p\ ) = colours);\ntrue\ngap> IsTrivial(AutomorphismGroup(cycle, [1, 1, 2, 2, 2, 2\ , 2, 2, 2]));\ntrue]]>\n \n\n\n \n \n <\ Returns>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 d\ igraph and colourings vert_colours, edge_colours.\n Let <\ C>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 integers,\n where ver\ t_colours[i] is the colour of vertex i,\n using the c\ olours [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 OutNeighbours(digraph), where\ \n edge_colours[i][j] is the colour of the edge\n \ OutNeighbours(digraph)[i][j], using the colours [1 .. k]\n \ for some k <= n;\n \n \n\n Giving vert\ _colours [edge_colours] as fail is\n equivalent to settin\ g all vertices [edges] to be the same colour. \n\n Unlike , it is\n possible to obtain th\ e automorphism group of an edge-coloured multidigraph\n (see ) when no two edges share the same\n source, range, and co\ lour.\n\n The automorphism group of a vertex/edge-coloured digraph\n\ digraph with colouring c is the group consisting of its\n \ vertex/edge-colour preserving automorphisms; an\n automorphism of <\ A>digraph is an isomorphism of vertex/edge-coloured\n digraphs from \ digraph to itself. This group is equal to the\n subgroup of Automor\ phismGroup(digraph) consisting of those\n automorphisms that pre\ serve the colouring specified by colours. See\n , and see \n for more infor\ mation about isomorphisms of coloured digraphs. \n\n If digraph \ is not a multidigraph then the automorphism group is\n returned as a group \ of permutations on the set of vertices of\n digraph. \n\n If \ digraph is a multidigraph then the automorphism 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 d\ igraph. These groups can be accessed using on the\n \ returned group. \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 i\ nstead; see\n ,\n , , and\n .\n\n cycle := CycleDigraph(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> Si\ ze(AutomorphismGroup(cycle));\n12\ngap> Size(AutomorphismGroup(cycle, vert_col\ ours));\n4\ngap> Size(AutomorphismGroup(cycle, fail, 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 dig\ raph is a canonical\n representative map if the following two cond\ itions 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 of a digraph G (under\ \\rho) is an\n isomorphism of G onto its canonical repres\ entative,\n \\rho(G). See \n for more information about isomorphisms of digraph\ s. \n\n BlissCanonicalLabelling returns a canonical labelling of\ the digraph\n digraph found using &bliss; by Tommi Junttila and Pet\ teri Kaski.\n NautyCanonicalLabelling returns a canonical labelling \ of the digraph\n digraph found using &nauty; by Brendan McKay and Ad\ olfo Piperno.\n Note that the canonical labellings returned by &bliss; and \ &nauty; are not\n usually the same (and may depend of the version used). \n\n BlissCanonicalLabelling can only be computed if digraph\ has\n no multiple edges; see . \n\n \ digraph1 := DigraphFromDiSparse6String(\".ImNS_AiB?qR\ N\");\n\ngap> BlissCanonicalLabel\ ling(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, Blis\ sCanonicalLabelling(digraph1)) =\n> OnDigraphs(digraph2, BlissCanonicalLabe\ lling(digraph2));\ntrue]]>\n \n\n\n <\ ManSection>\n \n \n <\ Returns>A permutation.\n \n A function \\rho\ that maps a coloured digraph to a coloured digraph is\n a canonical rep\ resentative 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 digraphs; and\n \n\ \n \\rho(G)=\\rho(H) if and 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 re\ presentative, \\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 vertices of\n digraph. T\ he colouring colours may have one of the following\n two forms: \ \n\n \n \n a list of n integers, where co\ lours[i] is the\n colour of vertex i, using the colou\ rs [1 .. m] for some\n m <= n; or\n \n \ \n a list of non-empty disjoint lists whose union is\n <\ C>DigraphVertices(digraph), such that\n colours[i]\ is the list of all vertices with colour\n i.\n \ \n \n\n If digraph and colours together form a colo\ ured digraph,\n BlissCanonicalLabelling returns a canonical labellin\ g of the digraph\n digraph found using &bliss; by Tommi Junttila and\ Petteri Kaski.\n Similarly, NautyCanonicalLabelling returns a canon\ ical labelling of\n the digraph digraph found using &nauty; by Brend\ an McKay and Adolfo\n Piperno. Note that the canonical labellings returned\ by &bliss; and\n &nauty; are not usually the same (and may depend of the v\ ersion used). \n\n BlissCanonicalLabelling can only be computed i\ f digraph has\n no multiple edges; see . The canonical\n labelling of digraph is given as a permutation o\ f its vertices.\n The canonical representative of digraph can be cre\ ated from\n digraph and its canonical labelling p by using th\ e\n operation :\n gap> OnDigraphs(digraph, p);\n\n The colouring of the ca\ nonical representative can easily be\n constructed. A vertex v (in <\ A>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 that 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(digraph), i -> colours[i / p]\ );\n\n On the other hand, if colours has the second form above\ , then the\n canonical representative has colouring:\n\n gap> OnTup\ lesSets(colours, p);\n\n \n\n digraph\ := DigraphFromDiSparse6String(\".ImNS_AiB?qRN\");\n\ngap> colours := [[1, 2, 8, 9, 10], [3, 4, 5, 6, 7]];;\nga\ p> 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> co\ lours := [1, 1, 1, 1, 2, 3, 1, 3, 2, 1];;\ngap> p := BlissCanonicalLabelling(d\ igraph, colours);\n(1,6,9,7)(3,4,5,8,10)\ngap> OnDigraphs(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 Bli\ ssCanonicalLabelling returns the canonical\n representative found by ap\ plying .\ The digraph returned is canonical in the sense\n that\n\n \n \ \n BlissCanonicalDigraph(digraph) and digraph are\n isomorphic as digraphs; and\n \n \n \ If gr is any digraph then BlissCanonicalDigraph(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 return value of\n this attrib\ ute is recomputed every time it is called. \n\n di\ graph := 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]]>\n \n\n\n \ \n \n A pe\ rmutation group.\n \n If digraph is immutabl\ e and was created knowing a subgroup of\n its automorphism group, then this\ group is stored in the attribute\n DigraphGroup. If digraph \ is mutable, or was not created\n knowing a subgroup of its automorphism gro\ up, then DigraphGroup\n returns the entire automorphism group of \ digraph. \n Note that if digraph is mutable, then the automor\ phism group is\n recomputed every time this function is called.\n \n\ \n Note that certain other constructor operations such as , , and , may not require a group as one of the arguments,\n \ but use the standard constructor method using a group, and hence set the\n \ DigraphGroup attribute for the resulting digraph.\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, gr\ oup, [1 .. n], \\^, adj);\n\ngap> Ha\ sDigraphGroup(D);\nfalse\ngap> DigraphGroup(D);\nGroup([ (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> H\ asDigraphGroup(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\ > HasDigraphGroup(D);\nfalse\ngap> HasAutomorphismGroup(D);\nfalse\ngap> Digra\ phGroup(D);\nGroup([ (2,3) ])\ngap> HasAutomorphismGroup(D);\ntrue\ngap> group\ := DihedralGroup(8);\n\ngap> D := Cayle\ yDigraph(group);\n\ngap> HasDigra\ phGroup(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) ]]]>\n \n\n\n \n \n \n An immutable list of lists of integers.\n \n \n DigraphOrbits returns the orbits of the actio\ n of the\n on the set of vertices of digrap\ h.\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 \n An immutable li\ st of integers.\n \n \n DigraphOrbitReps re\ turns a list of orbit representatives\n of the action of the on the set of vertices of\n digraph.\n\n \ D := CayleyDigraph(AlternatingGroup(4));\n\ngap> DigraphOrbitReps(D);\n[ 1 ]\ngap> D := Digrap\ hMutableCopy(D);\n\ngap> DigraphOr\ bitReps(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]]>\n \n\n\n \n \n \n An immutable list of integers.\n \n <\ Description>\n DigraphSchreierVector returns the so-called Schrei\ er vector\n of the action of the on the se\ t 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 and\n DigraphOrbitR\ eps(digraph)[-sch[i]] = i.\n \n\n sch[i] \ > 0:\n \n implies that i / gens[sch[i]]\ \n is one step closer to the root (or representative) of the tree, wher\ e\n gens is the generators of DigraphGroup(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(IsPermGroup, n);\nGroup([ (1\ ,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph, group, [1 .. n], \\^, adj);\n<\ mutable digraph with 4 vertices, 8 edges>\ngap> sch := DigraphSchreierVector(D\ );\n[ -1, 2, 2, 1 ]\ngap> D := CayleyDigraph(AlternatingGroup(4));\n\ngap> sch := DigraphSchreierVector(D);\n[\ -1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1 ]\ngap> DigraphOrbitReps(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 / ge\ ns[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 stabilizer of the vertex v\n\ under of the action of the on the set of\n \ vertices of digraph.\n\n D := DigraphFromD\ igraph6String(\"&GYHPQgWTIIPW\");\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> DigraphStabilizer(D, 2);\nG\ roup(())\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> DigraphStabilizer(D,\ 2);\nGroup(())\n]]>\n \n\n\n \n \n true or false.\n \n This operation returns true if there exists an isomorphism\ from the\n digraph digraph1 to the digraph digraph2. See for more info\ rmation\n about isomorphisms of digraphs. \n\n By default, an isomor\ phism is found using the canonical labellings of the\n digraphs obtained fr\ om &bliss; by Tommi Junttila\n and Petteri Kaski. If &NautyTracesInterface;\ is available, then &nauty; by\n Brendan Mckay and Adolfo Piperno can be us\ ed instead; see\n , and .\n\n digraph1 := CycleDigraph(4);\n<\ immutable cycle digraph with 4 vertices>\ngap> digraph2 := CycleDigraph(5);\n<\ immutable cycle digraph with 5 vertices>\ngap> IsIsomorphicDigraph(digraph1, d\ igraph2);\nfalse\ngap> digraph2 := DigraphReverse(digraph1);\n\ngap> IsIsomorphicDigraph(digraph1, digraph2);\nt\ rue\ngap> digraph1 := Digraph([[3], [], []]);\n\ngap> digraph2 := Digraph([[], [], [2]]);\n\ngap> IsIsomorphicDigraph(digraph1, digraph2);\ntrue]]><\ /Example>\n \n\n\n \n \n true or fal\ se.\n \n This operation tests for isomorphism o\ f coloured digraphs. A coloured\n digraph can be specified by its underlyi\ ng digraph digraph1 and its\n colouring colours1. Let n be the number of vertices of\n digraph1. The colouring colours\ 1 may have one of the following\n 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 som\ e\n m <= n; or\n \n \n a list o\ f non-empty disjoint lists whose union is\n DigraphVertices(digra\ ph), such that\n colours[i] is the list of all ve\ rtices with colour\n i.\n \n \n\n If digraph1 and digraph2 are digraphs without multiple edges,\n an\ d colours1 and colours2 are colourings of digraph1\n a\ nd digraph2, respectively, then this operation returns true\n \ if there exists an isomorphism between these two coloured digraphs. See\n \ for more information about isomorphisms of coloured 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 Ka\ ski. If &NautyTracesInterface; is available, then &nauty; by\n Brendan Mcka\ y and Adolfo Piperno can be used instead; see\n , and .\n\n d\ igraph1 := ChainDigraph(4);\n\ngap> d\ igraph2 := ChainDigraph(3);\n\ngap> I\ sIsomorphicDigraph(digraph1, digraph2,\n> [[1, 4], [2, 3]], [[1, 2], [3]]);\n\ false\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, 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 operati\ on returns an isomorphism between the digraphs digraph1\n and dig\ raph2 if one exists, else this operation returns fail.\n \n\ \n An isomorphism from a digraph digraph1 to a digraph\n <\ A>digraph2 is a bijection p from the vertices of\n digraph1 to the vertices of digraph2 with the following\n property: for al\ l 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 thi\ s operation returns one.\n The form of this isomorphism is a permutation p of the vertices of\n digraph1 such that \n\n OnDigr\ aphs(digraph1, p) = digraph2.\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 Kaski. If &NautyTracesInterface; is avail\ able, then &nauty; by\n Brendan Mckay and Adolfo Piperno can be used instea\ d; see\n , and .\n\n digraph1 := CycleDigraph(4);\n\ngap> digraph2 := CycleDigraph(5);\n\ngap> IsomorphismDigraphs(digraph1, digraph2);\ \nfail\ngap> digraph1 := CompleteBipartiteDigraph(10, 5);\n\ngap> digraph2 := Complete\ BipartiteDigraph(5, 10);\n\ngap> p := IsomorphismDigraphs(digraph1, digraph2);\n(1,6,1\ 1)(2,7,12)(3,8,13)(4,9,14)(5,10,15)\ngap> OnDigraphs(digraph1, p) = digraph2;\ \ntrue\n]]>\n \n\n\n \n \ \n A permut\ ation, 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 co\ lours[i] is the\n colour of vertex i, using the colou\ rs [1 .. m] for some\n m <= n; or\n \n \ \n a list of non-empty disjoint lists whose union is\n <\ C>DigraphVertices(digraph), such that\n colours[i]\ is the list of all vertices with colour\n i.\n \ \n \n\n An isomorphism between coloured digraphs is an iso\ morphism between\n the underlying digraphs that preserves 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 col\ ouring\n colours1) to the digraph digraph2 (with colouring\n \ colours2), and let p be the permutation 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 colours1[i] =\n c\ olours2[i ^ p] for all vertices i in digraph1.\n \ \n\n This operation returns such an isomorphism if one exists, else it retu\ rns\n fail. \n\n By default, an isomorphism is found using th\ e canonical labellings of the\n digraphs obtained from &bliss; by Tommi Jun\ ttila\n and Petteri Kaski. If &NautyTracesInterface; is available, then &na\ uty; by\n Brendan Mckay and Adolfo Piperno can be used instead; see\n , and .\n\n <\ Example> 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];;\nga\ p> colours2 := [1, 1, 1, 1];;\ngap> p := IsomorphismDigraphs(digraph1, digraph\ 2, colours1, colours2);\n(1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = digraph2;\ \ntrue\ngap> List(DigraphVertices(digraph1), i -> colours1[i ^ p]) = colours2;\ \ntrue\ngap> colours1 := [1, 1, 2, 2];;\ngap> colours2 := [2, 2, 1, 1];;\ngap>\ p := IsomorphismDigraphs(digraph1, digraph2, colours1, colours2);\n(1,4)(2,3)\ \ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\ngap> List(DigraphVertices(di\ graph1), i -> colours1[i ^ p]) = colours2;\ntrue\ngap> IsomorphismDigraphs(dig\ raph1, digraph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfail]]>\n \n\n\n \n \n An immutable list of lists.\n \n This function returns the list out of out-ne\ ighbours of each\n representative of the orbits of the action of on\n the vertex set of the digraph digraph. <\ P/>\n\n More specifically, if reps is the list of orbit representati\ ves,\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 digra\ ph.\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> 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 := 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> Representati\ veOutNeighbours(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);;\ngap> Gen\ eratorsOfGroup(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 IsDigraph\ Isomorphism 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 returns true if \ the permutation or\n transformation x is an automorphism of the digr\ aph digraph.\n \n\n A permutation or transformation x \ is an isomorphism from a\n digraph src to a digraph ran if the following hold:\n \n \n x is a bije\ ction 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 a\ lso . \n \ \n If col1 and col2, or col, are given, then they mu\ st\n represent vertex colourings; see \n \n for details of the per\ missible values for\n these arguments. The homomorphism must then also have\ the property:\n \n \n \n col1[i] = col2[i ^ x]\ for all vertices i of src,\n for IsDigraphIsomorp\ hism. \n \n col[i] = col[i ^ x] for all ve\ rtices i of digraph,\n for IsDigraphAutomorphism. \ \n \n\n For some digraphs, it can be faster to use IsDi\ graphAutomorphism\n than to test membership in the automorphism group o\ f digraph.\n\n src := Digraph([[1], [1, 2],\ [1, 3]]);\n\ngap> IsDigraphAutomo\ rphism(src, (1, 2, 3));\nfalse\ngap> IsDigraphAutomorphism(src, (2, 3));\ntrue\ \ngap> IsDigraphAutomorphism(src, (2, 3), [2, 1, 1]);\ntrue\ngap> IsDigraphAut\ omorphism(src, (2, 3), [2, 2, 1]);\nfalse\ngap> IsDigraphAutomorphism(src, (2,\ 3)(4, 5));\nfalse\ngap> IsDigraphAutomorphism(src, (1, 4));\nfalse\ngap> IsDi\ graphAutomorphism(src, ());\ntrue\ngap> ran := Digraph([[2, 1], [2], [2, 3]]);\ \n\ngap> IsDigraphIsomorphism(src,\ ran, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ntrue\ngap>\ IsDigraphIsomorphism(ran, src, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(src,\ Digraph([[3], [1, 3], [2]]), (1, 2, 3));\nfalse\ngap> IsDigraphIsomorphism(sr\ c, ran, (1, 2), [1, 2, 3], [2, 1, 3]);\ntrue\ngap> IsDigraphIsomorphism(src, r\ an, (1, 2), [1, 2, 2], [2, 1, 3]);\nfalse\n]]>\n \n\n\n \n \n \n true or fals\ e. \n \n The operation IsDigraphColourin\ g 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 digraph\n digraph\ if list consists of positive integers, the length of\n lis\ t equals the number of vertices in digraph, and for any\n vert\ ices u, v of digraph if u and v are adjacent,\n \ then list[u] >< list[v].\n \n\n A \ transformation t describes a proper colouring of a digraph\n di\ graph, 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]);\nfals\ e\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, -1]);\nfalse\ngap> I\ sDigraphColouring(D, [1, 2, 3]);\nfalse\ngap> IsDigraphColouring(D, IdentityTr\ ansformation);\ntrue\n]]>\n \n \n\n \ \n\n Homomorphisms of digraphs\n\n T\ he following methods exist to find homomorphisms between digraphs.\n If an \ argument to one of these methods is a digraph with multiple edges,\n then t\ he multiplicity of edges will be ignored in order to perform the\n calculat\ ion; the digraph will be treated as if it has no multiple edges.\n\n \n \n The argument user_param.\n \n This function finds homomorphisms from the digraph D1 to\ the digraph\n D2 subject to the conditions imposed by the other arg\ uments as\n described below.\n\n If f and g are homomo\ rphisms found by\n HomomorphismDigraphsFinder, then f cannot \ be obtained from\n g by right multiplying by an automorphism of D\ 2 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 have two arguments\n \ user_param (see below) and a transformation t. The\n f\ unction hook(user_param, t) is called every time\n \ a new homomorphism t is found by\n HomomorphismDigraphsFinde\ r. If the function returns true,\n then HomomorphismDigra\ phsFinder stops and does not find any\n further homomorphisms. This\ feature might be useful if you are\n searching for a homomorphism that\ satisfies some condition that you\n cannot specify via the other argum\ ents to\n HomomorphismDigraphsFinder.\n \n\n I\ f hook is fail, then a default function is used which\n s\ imply adds every new homomorphism found by\n HomomorphismDigraphsFin\ der to user_param, which must be a\n mutable list in this ca\ se.\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_param might be a\n \ transformation semigroup, and hook(user_param, t)\n \ might set user_param to be the closure of user_param and\ \n t. \n\n If the value of hook is fail\ , then the value of\n user_param must be a mutable list.\n \ \n\n max_results\n \n This a\ rgument should be a positive integer or infinity.\n Homomorph\ ismDigraphsFinder 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 hi\ nt is a positive integer, then only homorphisms of rank\n hint are found. \n\n If hint is fail, then no restrictio\ n is put on the rank\n of homomorphisms found.\n \n\n \ injective\n \n This argument should be <\ C>0, 1, or 2. If it is\n 2, then only embeddi\ ngs are found, if it is 1, then only\n injective homomorphisms a\ re found, and if it is 0 there are no\n restrictions imposed by \ this argument.\n \n\n For backwards compatibility, inject\ ive can also be false\n or true which correspond to t\ he 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 vertices of the graph D2<\ /A>.\n HomomorphismDigraphsFinder only finds homomorphisms from\ \n D1 to the subgraph of D2 induced by the vertices\n \ image.\n \n\n partial_map\n \ \n This argument should be a partial map from D1 to \ D2,\n that is, a (not necessarily dense) list of vertices of the di\ graph\n D2 of length no greater than the number vertices in the \ digraph\n D1. HomomorphismDigraphsFinder only finds homom\ orphisms\n extending partial_map (if any).\n \n \ \n colors1\n \n This should be a l\ ist representing possible colours of vertices in the\n digraph D1; see \n \n for details of the permissible value\ s for this argument.\n \n \n 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 argument.\n \ \n order\n \n The 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 lowest degree.\n \ \n \n The optional argument aut_grp should be\ a subgroup of the\n automorphism group of D2. This function ret\ urns unique\n representatives of the homomorphisms found up to right mu\ ltiplication\n by aut_grp. If this argument is not specific, it \ defaults to the\n full automorphism group of D2, which may be co\ stly to calculate.\n \n \n\n \ 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 := Complete\ Digraph(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, [Transformat\ ion([2, 2])],\n> 3, fail, 0, [1 .. 6], [1, 2, 1], fail, fail);\n[ Transformati\ on( [ 2, 2 ] ), \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 2 ] ), \n Tr\ ansformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 3 ] ), \n Transformation( [ 2, 2, \ 2, 3, 4, 5, 6, 2, 2, 4 ] ) ]\ngap> HomomorphismDigraphsFinder(NullDigraph(2), \ NullDigraph(3), fail,\n> [], infinity, fail, 1, [1, 2, 3], fail, fail, fail, f\ ail,\n> Group(()));\n[ IdentityTransformation, Transformation( [ 1, 3, 3 ] ), \ \n Transformation( [ 2, 1 ] ), Transformation( [ 2, 3, 3 ] ), \n Transformat\ ion( [ 3, 1, 3 ] ), Transformation( [ 3, 2, 3 ] ) ]\ngap> HomomorphismDigraphs\ Finder(NullDigraph(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [1, 2, \ 3], fail, fail, fail, fail,\n> Group((1, 2)));\n[ IdentityTransformation, Tran\ sformation( [ 1, 3, 3 ] ), \n Transformation( [ 3, 1, 3 ] ) ]]]>\n \ \n\n\n \n \n A transformation, or f\ ail.\n \n A homomorphism from digraph1 t\ o 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 vert\ ices [i,j] which has\n an edge i->j is mapped to a pair of ve\ rtices [a,b] which\n has an edge a->b. Note that non-adjacen\ t vertices can still be\n mapped to adjacent vertices. \n\n Digra\ phHomomorphism 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> Digr\ aphHomomorphism(gr1, gr1);\nIdentityTransformation\ngap> map := DigraphHomomor\ phism(gr1, gr2);\nTransformation( [ 3, 1, 5, 4, 5 ] )\ngap> IsDigraphHomomorph\ ism(gr1, gr2, map);\ntrue\n]]>\n \n\n\n \ \n \n \n A list of transformations.\n \n HomomorphismsDigraphsRepresentatives finds every\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 automorph\ ism of digraph2.\n \n\n HomomorphismsDigraphs returns \ all homomorphisms between\n digraph1 and digraph2.\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2]\ , [3, 1], [], [4]]);\n\ngap> Homom\ orphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 3, 1 ] ), Transformation( \ [ 1, 3, 3 ] ), \n Transformation( [ 1, 5, 4, 4, 5 ] ), Transformation( [ 2, 2\ , 2 ] ), \n Transformation( [ 3, 1, 3 ] ), Transformation( [ 3, 1, 5, 4, 5 ] \ ), \n Transformation( [ 3, 3, 1 ] ), Transformation( [ 3, 3, 3 ] ) ]\ngap> Ho\ momorphismsDigraphsRepresentatives(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 returns a single inject\ ive 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> Di\ graphMonomorphism(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 same as and\n , except they only return\n injective homomorphisms.\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], \ [2], [3, 1], [], [4]]);\n\ngap> Mo\ nomorphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 5, 4, 4, 5 ] ), \n Tra\ nsformation( [ 3, 1, 5, 4, 5 ] ) ]\ngap> MonomorphismsDigraphsRepresentatives(\ gr1, CompleteDigraph(3));\n[ Transformation( [ 2, 1 ] ) ]\n]]>\n \n\n\n \n \n A transformation, or fail.\n \n DigraphEpimorphism returns a single\ surjective between digra\ ph1 and digraph2\n if one exists, otherwise it returns fail.\n gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphRemoveEdge(Compl\ eteDigraph(3), [1, 2]);\n\ngap> Di\ graphEpimorphism(gr2, gr1);\nfail\ngap> DigraphEpimorphism(gr1, gr2);\nTransfo\ rmation( [ 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 onl\ y return\n surjective homomorphisms.\n g\ r1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphSymmetricClosure(CycleDigraph(3));\n\ngap> EpimorphismsDigraphsRepresentati\ ves(gr1, gr2);\n[ Transformation( [ 3, 1, 2, 1 ] ), Transformation( [ 3, 1, 2,\ 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ) ]\ngap> EpimorphismsDigraphs(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 Tr\ ansformation( [ 1, 3, 2, 1 ] ), Transformation( [ 1, 3, 2, 3 ] ), \n Transfor\ mation( [ 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 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 ] ), Tran\ sformation( [ 3, 2, 3, 1 ] ) ]\n]]>\n \n\ \n\n \n \n A transformation, or fail.\n \n An embedding of a digraph digraph1 into another digraph\n digraph2 is a from digraph1\n \ to digraph2 which has the additional property that a pair of\n ve\ rtices [i, j] which have no edge i -> j in digraph1 are\n\ \n mapped to a pair of vertices [a, b] which have 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 DigraphE\ mbedding returns a single embedding if one exists, otherwise\n it retur\ ns fail.\n gr := ChainDigraph(3);\n\ngap> DigraphEmbedding(gr, CompleteDigraph(4\ ));\nfail\ngap> DigraphEmbedding(gr, Digraph([[3], [1, 4], [1], [3]]));\nTrans\ formation( [ 2, 4, 3, 4 ] )\n]]>\n \n\n\ \n \n \n \n A \ list of transformations.\n \n These operations beha\ ve the same as and\n , except they only return\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[ Transfor\ mation( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 ] ) ]\ngap> EmbeddingsDigr\ aphs(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 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.<\ /Returns>\n \n IsDigraphHomomorphism returns true if the permutation\n or transformation x is a homomorphism from t\ he digraph\n src to the digraph ran.\n \n\n IsDi\ graphEpimorphism returns true if the permutation\n or transforma\ tion x is a surjective homomorphism from the digraph\n src to\ the digraph ran.\n \n\n IsDigraphMonomorphism returns\ true if the permutation\n or transformation x is an injectiv\ e homomorphism from the digraph\n src to the digraph ran.\n \ \n\n IsDigraphEndomorphism returns true if the permutat\ ion\n or transformation x is an endomorphism of the digraph\n \ digraph.\n \n\n A permutation or transformation x is a homomorphism from a\n digraph src to a digraph ran if th\ e 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 eve\ ry i which is not a vertex of src.\n \n \n\ See also . \n\n\n If <\ A>col1 and col2, or col, are given, then they must\n repr\ esent vertex colourings; see \n \n for details of the permissible val\ ues for these argument. The\n homomorphism must then also have the property\ :\n \n \n \n col[i] = col[i ^ x] for all ve\ rtices i of digraph,\n in the case of IsDigraphEndomor\ phism.\n \n col1[i] = col2[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> IsDigraphHomomorphis\ m(src, ran, Transformation([1, 2, 2]));\ntrue\ngap> IsDigraphHomomorphism(src,\ ran, Transformation([2, 1, 2]));\nfalse\ngap> IsDigraphHomomorphism(src, ran,\ Transformation([3, 3, 3]));\nfalse\ngap> IsDigraphHomomorphism(src, src, Tran\ sformation([3, 3, 3]));\ntrue\ngap> IsDigraphHomomorphism(src, ran, Transforma\ tion([1, 2, 2]),\n> [1, 2, 2], [1, 2]);\ntrue\ngap> I\ sDigraphHomomorphism(src, ran, Transformation([1, 2, 2]),\n> \ [2, 1, 1], [1, 2]);\nfalse\ngap> IsDigraphEndomorphism(src, Transforma\ tion([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, Transformation([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 tr\ ansformation x is a embedding of the digraph\n src into the d\ igraph ran, while respecting the colourings\n col1 and col\ 2 if given.\n \n\n A permutation or transformation x is a\ embedding of a digraph\n src into a digraph ran if x is a monomorphism from\n src to ran 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 also . \n\n src := Digraph([[1], [1\ , 2]]);\n\ngap> ran := Digraph([[1\ ], [1, 2], [1, 3]]);\n\ngap> IsDig\ raphMonomorphism(src, ran, ());\ntrue\ngap> IsDigraphEmbedding(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 := Digra\ ph([[1, 2], [1, 2], [1, 3]]);\n\ng\ ap> IsDigraphMonomorphism(src, ran, IdentityTransformation);\ntrue\ngap> IsDig\ raphEmbedding(src, ran, IdentityTransformation);\nfalse]]>\n \n\n\n \n \n true or \ false. \n \n The operation DigraphsRe\ spectsColouring verifies whether or not \n the permutation or transfo\ rmation x respects the vertex colourings\n col1 and col2\ of the digraphs src and range.\n That is, DigraphsR\ espectsColouring returns true if and only if for\n all vertice\ s i of src, col1[i] = col2[i ^ x].\n \n\n <\ Example> 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]]>\n \n \n\n \n \n \n A list of transformations.\ \n \n An endomorphism of digraph is a homomorphism\n \ from digraph back to itself.\n\n \ GeneratorsOfEndomorphismMonoid, called with a single argument,\n \ returns a generating set for the monoid of all endomorphisms of\n digrap\ h. If digraph belongs to \ , then the value of\n GeneratorsOfEndomorphismMonoid will not be rec\ omputed on future\n calls. \n\n If the colors argument is spec\ ified, then\n GeneratorsOfEndomorphismMonoid will return a generatin\ g\n set for the monoid of endomorphisms which respect the given colouring.\ \n The colouring colors can be in one of two forms: \n\n \n \n A list of positive integers of size the number of ve\ rtices 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 wi\ th colour i.\n \n \n\n If the limit arg\ ument is specified, then it will return only the\n first limit homom\ orphisms, where limit must be a positive\n integer or infinity. \n\n gr := Digraph(List([1 .. 3], x -> [1 .\ . 3]));;\ngap> GeneratorsOfEndomorphismMonoid(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(g\ r, 3);\n[ Transformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ), \n Ident\ ityTransformation ]\ngap> gr := CompleteDigraph(3);;\ngap> GeneratorsOfEndomor\ phismMonoid(gr);\n[ Transformation( [ 2, 3, 1 ] ), Transformation( [ 2, 1 ] ),\ \n IdentityTransformation ]\ngap> GeneratorsOfEndomorphismMonoid(gr, [1, 2, \ 2]);\n[ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ]\ngap> Generato\ rsOfEndomorphismMonoid(gr, [[1], [2, 3]]);\n[ Transformation( [ 1, 3, 2 ] ), I\ dentityTransformation ]\n]]>\n \n\n\n \ \n \n A transformation, or\ fail.\n \n A proper colouring of a d\ igraph is a labelling of its vertices in\n such a way that adjacent vertice\ s have different labels. A proper\n n-colouring is a proper\ colouring that uses exactly n\n colours. Equivalently, a proper (<\ C>n-)colouring of a digraph can be\n defined to be a from a digraph onto the\n complete digraph (with n \ vertices); see . Note that a digraph wi\ th loops () does not have a proper n<\ /C>-colouring for\n any value n.\n \n\n If digraph \ is a digraph and n is a non-negative integer, then\n DigraphColou\ ring(digraph, n) returns an epimorphism\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 Note that a digraph with \ at least two vertices has a 2-colouring if and only\n if it is bipartite, s\ ee .\n\n Digraph\ Colouring(CompleteDigraph(5), 4);\nfail\ngap> DigraphColouring(ChainDigraph(10\ ), 1);\nfail\ngap> D := ChainDigraph(10);;\ngap> t := DigraphColouring(D, 2);\ \nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\ngap> IsDigraphColouring(\ D, t); \ntrue\ngap> DigraphGreedyColouring(D);\nTransformation( [ 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 t\ hat adjacent vertices have different labels. Note that a digraph \n with lo\ ops () does not have any proper \n colouring\ .\n \n \n If digraph is a digraph and order is a de\ nse list consisting \n of all of the vertices of digraph (in any ord\ er), then \n DigraphGreedyColouring\n uses a greedy algorithm wi\ th the specified order to obtain some proper\n colouring of digraph,\ which may not use the minimal number of\n colours. \n \n If d\ igraph is a digraph and func is a function whose argument \n is \ a digraph, and that returns a dense list order, then \n DigraphGr\ eedyColouring(digraph, func) returns\n DigraphGreedyCo\ louring(digraph, func(digraph)).\n \n\n If \ the optional second argument (either a list or a function), is not \n speci\ fied, then is used by default. \n <\ P/>\n \n See also\n .\n \n\n 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]]>\n \n\n\n \n \n A list of the vertices.\n\ \n DigraphWelshPowellOrder returns a list of all of t\ he vertices of \n the digraph digraph ordered according to the sum o\ f the number of \n out- and in-neighbours, from highest to lowest. \n \n\n 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]]>\n \n\n\n \ \n \n A non-negati\ ve integer.\n \n A proper colouring of a dig\ raph 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 de\ fined to be a from a digraph onto a \ complete digraph. \n\n If digraph is a digraph without loops (se\ e , then ChromaticNumber returns \ the least\n non-negative integer n such that there is a proper colou\ ring of\n digraph with n colours. In other words, for a digr\ aph with at\n least one vertex, ChromaticNumber returns the least nu\ mber n\n such that DigraphColouring(digraph, n) does n\ ot return\n fail. See .\n\n Chr\ omaticNumber(NullDigraph(10));\n1\ngap> ChromaticNumber(CompleteDigraph(10));\ \n10\ngap> ChromaticNumber(CompleteBipartiteDigraph(5, 5));\n2\ngap> Chromatic\ Number(Digraph([[], [3], [5], [2, 3], [4]]));\n3\ngap> ChromaticNumber(NullDig\ raph(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 DigraphCore returns a \ list of vertices\n corresponding to the core of D. In parti\ cular, the subdigraph\n of D induced by this list is isomorphic to\ the core of D. \n\n The core of a digraph D is the\ minimal subdigraph C of\n D which is a homomorphic image of \ D. The core of a digraph \n is unique up to isomorphism.\n D := DigraphSymmetricClosure(CycleDigraph(8));\n\ngap> DigraphCore(D);\n[ 1, 2 ]\ \ngap> D := PetersenGraph();\n\n\ gap> DigraphCore(D);\n[ 1 .. 10 ]\ngap> D := Digraph(IsMutableDigraph, [[3], [\ 3], [4], [5], [2]]);\n\ngap> Digraph\ Core(D);\n[ 2, 3, 4, 5 ]\n]]>\n \n \n\ \n \n\n\n \n \n Finding cliques and independen\ t sets\n\n In &Digraphs;, a clique of a digraph is a set o\ f mutually adjacent\n vertices of the digraph, and an independent set is a set of mutually\n non-adjacent vertices of the digraph. A maximal\ clique is a clique\n which is not properly contained in another clique\ , and a maximal\n independent set is an independent set which is n\ ot properly contained\n in another independent set. Using this definition \ in &Digraphs;, cliques\n and independent sets are both permitted, but not r\ equired, to contain vertices\n at which there is a loop. Another name for \ a clique is a complete\n subgraph.\n \n\n &Digraphs; pro\ vides 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 se\ ts is a version of the Bron-Kerbosch algorithm.\n Calculating the cliques a\ nd independent sets of a digraph is a well-known\n and hard problem, and se\ arching 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 sev\ eral strategies to increase the performance\n of these calculations.\n <\ P/>\n\n From the definition of cliques and independent sets, it follows tha\ t the\n presence of loops and multiple edges in a digraph 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 dig\ raph, the cliques and independent sets of digraph are\n equal to\ the cliques and independent sets of the digraph:\n\n \n \ \n DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(digraph)).\n \n \n\n See and for more inf\ ormation about these\n attributes. Furthermore, the cliques of this digrap\ h are equal to the\n cliques of the digraph formed by removing any edge \ [u,v] for which\n the corresponding reverse edge [v,u] is not pr\ esent. Therefore,\n overall, the cliques of digraph are equal to th\ e cliques of the\n symmetric digraph:\n\n \n \n \ MaximalSymmetricSubdigraphWithoutLoops(digraph).\n <\ /Item>\n \n\n See for more\n information about this. The of this symmetric digraph contains the\n\ automorphism group of digraph as a subgroup. By performing the\n \ search for maximal cliques with the help of this larger automorphism group\n\ to reduce the search space, the computation time may be reduced. The\n \ functions and attributes which return representatives of cliques of\n d\ igraph will return orbit representatives of cliques under the\n action \ of the automorphism group of the maximal symmetric subdigraph\n withou\ t 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(DigraphRemoveLoops(DigraphRemov\ eAllMultipleEdges(\n digraph))).\n \n \ \n\n Again, the automorphism group of this symmetric digraph contain\ s the\n automorphism group of digraph. Since a search for independen\ t sets\n is equal to a search for cliques in the , the\n methods used in &Digraphs; usually transform a search for independ\ ent sets\n into a search for cliques, as described above. The functions an\ d\n attributes which return representatives of independent sets of\n \ digraph will return orbit representatives of independent sets under\n t\ he action of the automorphism group of the symmetric closure 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 m\ aximal. Some authors use the word clique to mean\n maximal clique, a\ nd some authors use the phrase independent set to\n mean maximal ind\ ependent set, but please note that &Digraphs; does\n not use this definitio\ n.\n \n\n Finding cliques\n \n\n\ntrue or false.\n\n If digraph is a digraph and l is a duplicate-free l\ ist of\n vertices of digraph, then\n IsClique(digraph,l) returns true\n if l is a clique\ of digraph and false if it is\n not. Similarly,\n IsM\ aximalClique(digraph,l) returns\n true if l is a maximal clique of digraph and\n false<\ /K> if it is not. \n\n A clique of a digraph is a set of mutual\ ly adjacent vertices of the\n digraph. A maximal clique is a clique \ that is not properly\n contained in another clique. A clique is permitted, \ but not required, to\n contain vertices at which there is a loop.\n D := CompleteDigraph(4);;\ngap> IsClique(D, [1, 3, 2]);\n\ true\ngap> IsMaximalClique(D, [1, 3, 2]);\nfalse\ngap> IsMaximalClique(D, Digr\ aphVertices(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]);\ntrue\ngap> D := Compl\ eteDigraph(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 digraph subje\ ct to the\n conditions imposed by the other arguments as described below. N\ ote\n that a clique is represented by the immutable list of the vertices th\ at\n it contains.\n \n\n Let G denote the automorphism gro\ up of the maximal symmetric\n subdigraph of digraph without loops (s\ ee and ).\n\n \n \ hook\n \n This argument should be a fun\ ction or fail. \n\n If hook is a function, then it sho\ uld have two arguments\n user_param (see below) and a clique \ c. The function\n hook(user_param, c) is calle\ d every time a new\n clique c is found by CliquesFinder.<\ P/>\n\n If hook is fail, then a default function is used \ that\n simply adds every new clique found by CliquesFinder to\n \ user_param, which must be a list in this case.\n \n\ \n user_param\n \n If hook i\ s a function, then user_param can be any &GAP;\n object. The obj\ ect user_param is used as the first argument for\n the function \ hook. For example, user_param might be a\n list, and <\ A>hook(user_param, c) might add the size\n of the clique\ c to the list user_param. \n\n If the value of ho\ ok is fail, then the value of\n user_param must be a \ list.\n \n\n limit\n \n \ This argument should be a positive integer or infinity.\n Cl\ iquesFinder will return after it has found\n limit cliques o\ r the search is complete.\n \n\n include and \ exclude\n \n These arguments should each be a (po\ ssibly 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 cliques containing \ all of the\n vertices in include and containing none of the vert\ ices in\n exclude. \n\n Note that the search may be m\ uch more efficient if each of these lists\n is invariant under the acti\ on 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 false then non-m\ aximal cliques may be\n found.\n \n\n size\n \n This argument should be fail or a posit\ ive integer.\n If size is a positive integer then CliquesFind\ er will\n only search for cliques that contain precisely size vertices.\n If size is fail then cliques of any size ma\ y be found.\n \n\n reps\n \n \ This argument should be true or false. \n\n If <\ A>reps is true then the arguments include and\n ex\ clude are each required to be invariant under the action of\n G<\ /C> on sets of vertices. In this case, CliquesFinder will\n fin\ d representatives of the orbits of the desired cliques under the\n acti\ on of G, although representatives may be returned that\n ar\ e in the same orbit.\n If reps is false then CliquesFinde\ r will not take this into\n consideration. \n\n For a dig\ raph such that G is non-trivial, the search for\n clique represe\ ntatives can be much more efficient than the search for\n all cliques.\ \n \n \n\n This function uses a version of the Bron-K\ erbosch algorithm.\n D := CompleteDigraph(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> 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, f\ alse);\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[ ]\ngap> D := CompleteDigraph(IsMutableDigraph, 5);\n\ngap> user_param := [];;\ngap> f := function(a, b) # C\ alculate size of clique\n> AddSet(user_param, Size(b));\n> end;;\ngap> Cliqu\ esFinder(D, f, user_param, infinity, [], [], false, fail,\n> \ true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFinder(D, fail, [], 5, [2, 4], [3], fal\ se, 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, fail, [], infinity, [], [], true, 5, fal\ se);\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> Clique\ sFinder(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 functions returns a c\ lique\n of digraph if one exists that satisfies the arguments, else \ it\n returns fail. A clique is defined by the set 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 vertices of digraph\ , and\n the optional argument size must be a positive integer. \ By default,\n include and exclude are empty. These functions \ will search for\n a clique of digraph that includes the vertices of \ include\n but does not include any vertices in exclude; if th\ e argument\n size is supplied, then additionally the clique will be \ required to\n contain precisely size vertices. \n\n If incl\ ude is not a clique, then these functions return fail.\n Otherwi\ se, 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 DigraphClique an\ d\n DigraphMaximalClique greedily enlarge the clique include<\ /A>\n until it can not continue. The result is guaranteed to be a maxi\ mal\n clique. This may or may not return an answer more quickly than us\ ing\n . with a limit of 1.\n \ \n Three arguments\n \n If three a\ rguments are supplied, then DigraphClique greedily\n enlarges th\ e clique include until it can not continue, although\n this cliq\ ue may not be maximal. \n\n Given three arguments, DigraphMaximal\ Clique returns the maximal\n clique returned by DigraphMaximalCl\ iques(digraph,\n include, exclude, 1) if one exists,\n else fail.\n \n <\ Mark>Four arguments\n \n If four arguments are suppli\ ed, then DigraphClique returns the\n clique returned by Digra\ phCliques(digraph,\n include, excl\ ude, 1,\n size) if one exists, else fail. This clique may\n not be maximal. Given four arguments, Digra\ phMaximalClique\n returns the maximal clique returned by\n <\ C>DigraphMaximalCliques(digraph, include,\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\ngap> DigraphClique(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 ]\ngap> Digrap\ hClique(D, [1], [2]);\n[ 1, 4 ]\ngap> DigraphMaximalClique(D, [1], [3, 4]);\nf\ ail\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 ]]]>\n \ \n\n\n \n \n \n \n \n \n \n An immutable list of lists of positi\ ve integers.\n \n If digraph is digraph, the\ n these functions and attributes use to r\ eturn cliques of digraph. A\n clique is defined by the set of vertic\ es that it contains; see and . \n\n The optional arguments include and exclud\ e must each be a\n (possibly empty) list of vertices of digraph,\ the optional argument\n limit must be either a positive integer or \ infinity, and the\n optional argument size must be a positive\ integer. If not\n specified, then include and exclude are c\ hosen to be empty\n lists, and limit is set to infinity. \ \n\n The functions will return as many suitable cliques 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 contain any of the\n \ vertices of exclude. The argument size restricts the search\n \ to those cliques that contain precisely size vertices.\n If the fu\ nction or attribute has Maximal in its name, then only\n maximal cli\ ques will be returned; otherwise non-maximal cliques may be\n returned. \n\n Let G denote the automorphism group of maximal symmetric subdi\ graph\n of digraph without loops (see and ).\n\n \n Distinct cliques\n\ \n DigraphMaximalCliques and DigraphCliques e\ ach return a\n duplicate-free list of at most limit cliques of <\ A>digraph\n that satisfy the arguments. \n\n The computat\ ion may be significantly faster if include and\n exclude \ are invariant under the action of G\n on sets of vertices.\n \ \n\n Orbit representatives of cliques\n \ \n To use DigraphMaximalCliquesReps or DigraphCliquesReps\ ,\n the arguments include and exclude must each be invari\ ant\n under the action of G on sets of vertices. \n\n \ If this is the case, then DigraphMaximalCliquesReps and\n Dig\ raphCliquesReps each return a duplicate-free list of at most\n l\ imit 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 orbit 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 := Autom\ orphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ngap> DigraphMaxima\ lCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]\ngap> DigraphMaximalCliqu\ esReps(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> D\ igraphMaximalCliquesReps(D, [3, 4], [], 1);\n[ [ 3, 4 ] ]\ngap> DigraphMaximal\ Cliques(D, [1, 2], [5, 6], 1, 2);\n[ ]\ngap> DigraphCliques(D, [1], [5, 6], i\ nfinity, 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);\ntrue\ngap> G := Automorp\ hismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ngap> DigraphMaximalCl\ iques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]]]>\n \n\n\n \n \n A non-negative integer.\n \n I\ f digraph is a digraph, then CliqueNumber(digraph)\n r\ eturns the largest integer n such that digraph contains a\n c\ lique with n vertices as an induced subdigraph.\n \n\n A c\ lique of a digraph is a set of mutually adjacent vertices of the\n digr\ aph. Loops and multiple edges are ignored for the purpose of\n determining \ the clique number of a digraph. \n D := CompleteDi\ graph(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(IsMutableDigraph, 4);;\ngap> \ CliqueNumber(D);\n4]]>\n\n\n\n \n\n Finding independent sets\n \ \n \n \n true or false<\ /K>.\n \n If digraph is a digraph and l is a duplicate-free list of\n vertices of digraph, then\n I\ sIndependentSet(digraph,l) returns\n tru\ e if l is an independent set of digraph and\n f\ alse if it is not. Similarly,\n IsMaximalIndependentSet(digr\ aph,l)\n returns true if l is a m\ aximal independent set of\n digraph and false if it is no\ t. \n\n An independent set of a digraph is a set of mutually no\ n-adjacent\n vertices of the digraph. A maximal independent set is a\ n independent\n set that is not properly contained in another independent s\ et. An\n independent set is permitted, but not required, to contain vertice\ s at\n which there is a loop.\n D := CycleDigra\ ph(4);;\ngap> IsIndependentSet(D, [1]);\ntrue\ngap> IsMaximalIndependentSet(D,\ [1]);\nfalse\ngap> IsIndependentSet(D, [1, 4, 3]);\nfalse\ngap> IsIndependent\ Set(D, [2, 4]);\ntrue\ngap> IsMaximalIndependentSet(D, [2, 4]);\ntrue\ngap> D \ := CycleDigraph(IsMutableDigraph, 4);;\ngap> IsIndependentSet(D, [1]);\ntrue]]\ >\n \n\n\n \n \n \n An immutable list of positive integer\ s, or fail.\n \n If digraph is a digr\ aph, then these functions returns an independent\n set of digraph if\ one exists that satisfies the arguments, else it\n returns fail. A\ n independent set is defined by the set of vertices\n that it contains; see\ and . \n\n The optional arguments include and exclude mu\ st each be a\n (possibly empty) duplicate-free list of vertices of digra\ ph, and the\n optional argument size must be a positive integer.\ By default,\n include and exclude are empty. These function\ s will search\n for an independent set of digraph that includes the \ vertices of\n include but does not include any vertices in exclud\ e;\n if the argument size is supplied, then additionally the ind\ ependent\n set will be required to contain precisely size vertices.<\ P/>\n\n If include is not an independent set, then these functions r\ eturn\n fail. Otherwise, the functions behave in the following way,\ \n depending on the number of arguments:\n\n \n One or \ two arguments\n \n If one or two arguments are suppli\ ed, then DigraphIndependentSet\n and DigraphMaximalIndependen\ tSet greedily enlarge the\n independent set include until it\ can not continue. The result\n is guaranteed to be a maximal independ\ ent 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 DigraphIndependentSet\n greed\ ily enlarges the independent set include until it can not\n cont\ inue, although this independent set may not be maximal. \n\n Given t\ hree arguments, DigraphMaximalIndependentSet returns the\n maxim\ al independent set returned by\n DigraphMaximalIndependentSets(<\ A>digraph,\n include, exclude, 1) if one exists,\n else fail.\n \n Four a\ rguments\n \n If four arguments are supplied, then DigraphIndependentSet\n returns the independent set returned by\n \ DigraphIndependentSets(digraph, include,\ \n exclude, 1, size) if one exists, els\ e\n fail. This independent set may not be maximal. Given fou\ r\n arguments, DigraphMaximalIndependentSet returns the maximal\ \n independent set returned by\n DigraphMaximalIndependentSet\ s(digraph,\n include, exclude<\ C>, 1,\n size) if one exists, else fail.\n \ \n \n D := ChainDigraph(6);\n\ngap> DigraphIndependentSet(D);\n[ 6, \ 4, 2 ]\ngap> DigraphMaximalIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphIndepe\ ndentSet(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> D\ igraphMaximalIndependentSet(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 ]\ngap> D\ := ChainDigraph(IsMutableDigraph, 6);\n\ngap> DigraphIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphMaximalIndepend\ entSet(D);\n[ 6, 4, 2 ]\ngap> DigraphIndependentSet(D, [2, 4]);\n[ 2, 4, 6 ]\n\ gap> DigraphMaximalIndependentSet(D, [1, 3]);\n[ 1, 3, 6 ]\ngap> DigraphIndepe\ ndentSet(D, [2, 4], [6]);\n[ 2, 4 ]\ngap> DigraphMaximalIndependentSet(D, [2, \ 4], [6]);\nfail\ngap> DigraphIndependentSet(D, [1], [], 2);\n[ 1, 3 ]\ngap> Di\ graphMaximalIndependentSet(D, [1], [], 2);\nfail\ngap> DigraphMaximalIndepende\ ntSet(D, [1], [], 3);\n[ 1, 3, 5 ]]]>\n \n\n\n \n \n \n \n \n \n \n An immutabl\ e list of lists of positive integers.\n \n If di\ graph is digraph, then these functions and attributes use to return independent sets of digraph. An\n ind\ ependent set is defined by the set of vertices that it contains; see\n and . \n\n The optional arguments include and exclude must ea\ ch be a\n (possibly empty) list of vertices of digraph, the optional\ argument\n limit must be either a positive integer 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 e\ mpty\n lists, and limit is set to infinity. \n\n The f\ unctions will return as many suitable independent sets as possible, up\n to\ the number limit. These functions will find independent sets\n tha\ t contain all of the vertices of include but do not\n contain any of\ the vertices of exclude The argument size\n restricts the se\ arch to those cliques that contain precisely size\n vertices. If th\ e function or attribute has Maximal in its name,\n then only maximal\ independent sets will be returned; otherwise non-maximal\n independent set\ s may be returned.\n \n\n Let G denote the \n of the of the digraph formed from\n digraph by removing loops\ and ignoring the multiplicity of edges.\n\n \n Distinct i\ ndependent sets\n \n DigraphMaximalIndependentSets\ and DigraphIndependentSets\n each return a duplicate-free l\ ist of at most limit independent\n sets of digraph that s\ atisfy the arguments. \n\n The computation may be significantly fast\ er if include and\n exclude are invariant under the actio\ n of G on sets of\n vertices.\n \n\n Repr\ esentatives of distinct orbits of independent sets\n \n \ To use DigraphMaximalIndependentSetsReps or\n DigraphIndep\ endentSetsReps, the arguments include and\n exclude m\ ust each be invariant under the action of G on\n sets of vertice\ s. \n\n If this is the case, then DigraphMaximalIndependentSetsRe\ ps 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 representatives are not guar\ anteed to be in distinct orbits.\n However, if lim is not specif\ ied, or fewer than lim\n results are returned, then there will b\ e at least one representative\n from each orbit of maximal independent \ sets.\n \n \n\n D := CycleDi\ graph(5);\n\ngap> DigraphMaximalIndep\ endentSetsReps(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 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> DigraphMaximalIndependentSetsReps(D);\n[ [ 1,\ 3 ] ]\ngap> DigraphIndependentSetsReps(D);\n[ [ 1 ], [ 1, 3 ] ]\ngap> Set(Dig\ raphMaximalIndependentSets(D));\n[ [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 2, 5 ], [ 3\ , 5 ] ]\ngap> DigraphMaximalIndependentSets(D, [1]);\n[ [ 1, 3 ], [ 1, 4 ] ]\n\ gap> 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 \ \n\n pickled (.p or .pickle)\n \ \n Digraphs are pickled using the &IO; package. This is particul\ arly good\n when the is non-trivial.\n \ \n \n\n ReadDigraphs(\n> Concat\ enation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 10);\n\ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/\ data/graph5.g6.gz\"), 17);\n\ngap\ > ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/data/tree9.4.txt\"));\n[ <\ immutable digraph with 9 vertices, 8 edges>, \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \ \n , \n , \n ]]]>\n \n\n\n \n \n \n If digraphs is a list of digraphs or a digraph and filena\ me\n is a string or an &IO; file object created using ,\n then WriteDigraphs writes the digraphs to the file rep\ resented by\n filename. If the supplied filename ends in one of the\ extensions\n .gz, .bz2, or .xz, then the file will be\ compressed\n appropriately. Excluding these extensions, if the file ends \ with an\n extension in the list below, the corresponding graph format will \ be used to\n encode it. If such an extension is not included, an appropria\ te format\n will be chosen intelligently, and an extension appended, to min\ imise 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 SetInfoLevel. \ \n\n The currently supported file formats are:\n \n gra\ ph6 (.g6)\n \n\tA standard and widely-used format for undire\ cted graphs, with no support\n\tfor loops or multiple edges. Only symmetric g\ raphs are allowed -- each\n\tedge is combined with its converse edge to produc\ e 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 m\ ultiple edges.\n\tHowever, its use is still limited to symmetric graphs. This\ format is\n\tbetter-suited to \"sparse\" graphs -- those with few edges per v\ ertex.\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\tmultiple edges ar\ e not. Best compression with \"dense\" graphs.\n \n di\ sparse6 (.ds6)\n \n\tAny type of digraph can be encoded in d\ isparse6: directions, loops, and\n\tmultiple edges are all allowed. Similar t\ o sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\n \ \n\n plain text (.txt)\n \n\tThis is a h\ uman-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 separat\ ed by two spaces. \n\n See for\ a more flexible way to store\n digraphs in a plain text file. \nReadPlainTextDigraph(\"data-local/\ soc-Epinions1.txt.gz\", \"\\t\", 1, '#');\n-->\n \n pickled (.p or .pickle)\n \n Digraphs \ are pickled using the &IO; package. This is particularly good\n when th\ e is non-trivial.\n \n \n\n\ 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 := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ \ngap> WriteDigraphs(filename, grs, \"w\");\nIO_OK\ngap> ReadDigraphs(filename\ );\n[ , \n , \n ]]]>\n \n\n\n \n \n An iter\ ator.\n \n If filename is a string represent\ ing the name of a file containing\n encoded digraphs, then IteratorFromD\ igraphFile returns an iterator\n for which the value of is the\n next digraph encoded in the file.\n\ \n\n If the optional argument decoder is specified and is a \ function\n which decodes a string into a digraph, then IteratorFromDigra\ phFile\n will use decoder to decode the digraphs contained in filename.\n \n\n The purpose of this function is to easily allo\ w looping over digraphs\n encoded in a file when loading all of the encoded\ digraphs would require\n too much memory. \n\n To see what file typ\ es are available, see .\n\n filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\nga\ p> file := DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> Wri\ teDigraphs(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 decodes a digraph in\n \ a plain text format. \n\n DigraphPlainTextLineEncoder returns \ a function which takes a single\n digraph as an argument. The function ret\ urns a string describing the edges\n of that digraph; each edge is written \ as a pair of integers separated by the\n string delimiter2, and the \ edges themselves are separated by the\n string delimiter1. Digra\ phPlainTextLineDecoder returns the\n corresponding decoder function, wh\ ich 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 DigraphPlainText\ LineDecoder, it will return a function which decodes\n a single edge, r\ eturning its contents as a list of integers. \n\n The argument offset\ should be an integer, which will describe a\n number to be added to ea\ ch 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 of 1. \n\n N\ ote 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 gr := Digraph([[2, 3], [1], [1]]);\n\ngap> enc := DigraphPlainTextLineEncoder(\" \", \" \", -1);;\ngap> \ dec := DigraphPlainTextLineDecoder(\" \", \" \", 1);;\ngap> enc(gr);\n\"0 1 \ 0 2 1 0 2 0\"\ngap> dec(last);\n\ \n]]>\n \n\n\n \n \n A digraph.\ \n \n This function takes a string str, decodes it, an\ d then returns the\n tournament [see ] which it\ defines, according to\n the following rules. \n\n The characters of\ the string str represent the entries in the upper\n triangle of a t\ ournament's adjacency matrix. The number of vertices\n n will be det\ ected from the length of the string and will be as\n large as possible. \n\n The first character represents the possible edge 1 -> 2, the \ second\n represents 1 -> 3 and so on until 1 -> n; then the f\ ollowing\n character represents 2 -> 3, and so on up to the characte\ r which\n represents the edge n-1 -> n. \n\n If a character o\ f 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 edg\ es are encoded one-by-one.\n\n gr := TournamentLin\ eDecoder(\"100001\");\n\ngap> OutN\ eighbours(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, decodes 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 entries in the upper\ \n triangle of a digraph's adjacency matrix. The number of vertices 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 on until 1 -\ > n; then the following\n character represents 2 -> 3, and so on\ up to the character which\n represents the edge n-1 -> n. If a cha\ racter 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 pr\ esent. 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 topologically sorted.\n\ \n gr := AdjacencyMatrixUpperTriangleLineDecoder(\ \"100001\");\n\ngap> OutNeighbours\ (gr);\n[ [ 2 ], [ ], [ 4 ], [ ] ]\ngap> gr := AdjacencyMatrixUpperTriangleLi\ neDecoder(\"111111x111\");\n\ngap>\ OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 3, 4 ], [ 4, 5 ], [ 5 ], [ ] ]\n]]><\ /Example>\n \n\n\n \n \n A digraph.\n \n If str is a string consisting of at least two non-negative inte\ gers\n separated by spaces, then this function will attempt to return the d\ igraph\n which it defines as a TCode string. \n\n The first integer \ of the string defines the number of vertices v in\n the digraph, and\ the second defines the number of edges e. The\n following 2e integers should be vertex numbers in the range [0 ..\n v-1]. The\ se integers are read in pairs and define the digraph's\n edges. This funct\ ion will return an error if str has fewer than\n 2e+2 entries\ . \n\n Note that the vertex numbers will be incremented by 1 in the dig\ raph\n returned. Hence the string fragment 0 6 will describe the ed\ ge\n [1,7].\n gr := TCodeDecoder(\"3 2 0\ 2 2 1\");\n\ngap> OutNeighbours(g\ r);\n[ [ 3 ], [ ], [ 2 ] ]\ngap> gr := TCodeDecoder(\"12 3 0 10 5 2 8 8\");\n\ \ngap> OutNeighbours(gr);\n[ [ 11\ ], [ ], [ ], [ ], [ ], [ 3 ], [ ], [ ], [ 9 ], [ ], \n [ ], [ ] ]\n\ ]]>\n \n\n\n \n \n \n A string.\n \n P\ lainTextString takes a single digraph, and returns a string\n describin\ g the edges of that digraph. DigraphFromPlainTextString\n takes suc\ h a string and returns the digraph which it describes. Each edge\n is writ\ ten as a pair of integers separated by a single space. The edges\n themsel\ ves are separated by a double space. Vertex numbers are reduced by\n 1 whe\ n 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 lost. \n\n \ \n The operation DigraphFromPlainTextString\n takes an optiona\ l first argument \n or , which specifies whether the output digraph shall\n be mutab\ le or immutable. If no first argument is provided, then an immutable\n digr\ aph is returned by default.\n\n gr := Digraph([[2,\ 3], [1], [1]]);\n\ngap> PlainText\ String(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> DigraphFromPlainTextString(last);\n<\ immutable digraph with 3 vertices, 4 edges>\n]]>\n \n\ \n\n \n \n \n \n These \ functions write and read a single digraph in a human-readable plain\n text \ format as follows: each line contains a single edge, and each edge is\n wri\ tten as a pair of integers separated by the string delimiter. \n\n \ filename should be the name of a file which will be written to or\n \ read from, and offset should be an integer which is added to each\n \ vertex number as it is written or read. For example, if\n WritePlainT\ extDigraph is called with offset -1, then\n the vertices \ will be numbered in the file starting from 0 instead of 1 -\n ReadPlainT\ extDigraph would then need to be called with offset\n 1 t\ o convert back to the original graph. \n\n ignore should be a lis\ t of characters which will be ignored when\n reading the graph.\n gr := Digraph([[1, 2, 3], [1, 1], [2]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(\ ), \"/tst/out/plain.txt\");;\ngap> WritePlainTextDigraph(filename, gr, \",\", \ -1);\ngap> ReadPlainTextDigraph(filename, \",\", 1, ['/', '%']);\n\n]]>\n \n\n\n \n \n \n \n These operations write or read the single symmetric digraph d\ igraph\n to or from a file in DIMACS format, as appropriate. The operat\ ion\n WriteDIMACSDigraph records the vertices and edges of digrap\ h.\n The vertex labels of digraph will be recorded only if they \ are\n integers. See and . \n\n The first argument filename shou\ ld 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 e\ nds in one of .gz, .bz2,\n or .xz, then the file is co\ mpressed, or decompressed, appropriately.\n \n\n The DIMACS format i\ s described as follows. Each line in the DIMACS file has\n one of four typ\ es:\n \n \n A line beginning with c and foll\ owed by any number of characters\n is a comment line, and is ignored.\n\ \n \n A line beginning with p defines th\ e numbers of vertices and edges\n the digraph. This line has the forma\ t p edge <nr_vertices>\n <nr_edges>, where <\ nr_vertices> and\n <nr_edges> are replaced by the r\ elevant integers. There\n must be exactly one such line in the file, an\ d it must occur before any\n of the following kinds of line. \n\n \ Although it is required to be present, the value of\n <nr_ed\ ges> will be ignored. The correct number of edges\n will be dedu\ ced from the rest of the information in the file.\n \n \ \n A line of the form e <v> <w>, where <v><\ /C>\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&g\ t;. 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 per\ mitted.\n \n \n A line of the form n <v>\ ; <label>, where\n <v> is an integer in the rang\ e [1 ..\n <nr_vertices>] and <label> is an \ integer,\n signifies that the vertex <v> has the label\n \ <label> in the digraph. If a label is not specified for a\n\ vertex, then ReadDIMACSDigraph will assign the label\n <\ C>1, according to the DIMACS specification.\n \n \n\ \n A detailed definition of the DIMACS format can be found at\n htt\ p://mat.gsia.cmu.edu/COLOR/general/ccformat.ps, in Section\n 2.1. Not\ e that optional descriptor lines, as described in Section 2.1, will\n be ig\ nored.\n \n\n\n gr := Digraph([[2], [1, 3, \ 4], [2, 4], [2, 3]]);\n\ngap> file\ name := Concatenation(DIGRAPHS_Dir(),\n> \"/tst/o\ ut/dimacs.dimacs\");;\ngap> WriteDIMACSDigraph(filename, gr);;\ngap> ReadDIMAC\ SDigraph(filename);\n]]>\ \n \n\n\n \n\n \n\n\n\n \n Grape to Digraphs Command \ Map\n \n Below is a table of &Grape; commands with the &Digraphs; \ counterparts.\n The sections in this chapter correspond to the chapters in th\ e &Grape;\n manual.\n\n \n \n Functions to constru\ ct and modify 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 &Grape; command\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 <\ /Alt>\n \n \n \n\n \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-negative integ\ ers. Since digraphs in &Digraphs; are immutable,\n there is no reas\ on (although it is possible) to create a null digraph\n and then ad\ d edges, as you might be used to doing in &Grape;.\n \n \ \n \n \n\n \n \n \ CompleteGraph\n \n \n \n \n \n \n Note that, unlike in &Grape;, the argument to is simply a non-negative integer\n \ indicating the number of vertices. By default a complete digraph in\n \ &Digraphs; does not have any loops. If you want a complete digraph\n \ with loops at every vertex, then use .\n \n \n \n <\ HorLine/>\n\n \n \n JohnsonGraph\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 operation behaves the same as\n CayleyGraph\ in &Grape;.\n \n \n \n \n\n \n \n AddEdgeOrbit\n \n \n \n \n \n \n Note that, un\ like in &Grape;, digraphs in &Digraphs; are immutable,\n and this o\ peration returns a new digraph, i.e. it does not modify the\n digra\ ph given as an argument.\n \n \n \n \ \n\n \n \n RemoveEdgeOrbit\ \n \n \n \n \n \n \n \ Note 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 \ AssignVertexNames\n \n \n \n \n and\n \ \n \n \n \n \n Use to set an individual\n vertex label a\ nd to set all of\n the verte\ x labels.\n \n \n \n \n\ \n \n\n \n \n Functions to i\ nspect graphs, vertices and edges\n \n \n \ \n The table in this section contains more information when viewe\ d in\n html format.\n \n \n\n \n \n\n \n \n &Grape; comm\ and\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 \n \n \n \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 the list of v\ ertices of a digraph, or if v is known to be\n a positive in\ teger, then you can simply do\n v <= DigraphNrVertices\ (digraph)\n \n \n \n <\ HorLine/>\n\n \n \n VertexName\n \ \n \n \n \ \n \n \n Unlike <\ C>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 behaves the\n \ same as Vertices in &Grape;.\n \n \n \n \n\n \n \n <\ C>VertexDegree\n \n \n \n \n (see also: )\n \n \n \n \n The operation behaves the same as\n VertexDegree in &Grap\ e;.\n \n \n \n \n\n \ \n \n VertexDegrees\n \n \ \n \n \n (see also:\n ,\n \ ,\n ,\ \n ,\n )\n \n \n \n \n The operation behave\ s 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 behave\ s 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 behaves the same\n \ as Adjacency in &Grape;.\n \n \ \n \n \n\n \n \n \ IsEdge\n \n \n \n \n \n \ \n The arguments of\n \n can be a digra\ ph 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 \n \ \n \n The operation 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; except that i\ n the case that the\n argument is a forest, infinity is retu\ rned 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 attribute behaves the\n same as Is\ ConnectedGraph in &Grape;.\n \n \n \n \n\n \n \n IsBipartite\n \n \n \n \n and \n \n \n \n \n The &Grape; function IsBipartite is equivale\ nt to\n and .\n \n \n \n \n\n \n \n IsNullGraph\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 IsCompleteGraph\ \n \n \n \ \n \n \n \n \ The &Grape; function IsCompleteGraph is roughly\n equivalent\ to\n except that in &Digraphs;\n\ a complete digraph has no loops.\n \n \ \n \n \n\n\n \n \n\n \n \n Functions to determine regularity properties of \ graphs\n \n \n \n The table i\ n 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 <\ /Item>\n \n \n\n \n \n \ IsRegularGraph\n \n \n \n \n (see al\ so\n \n and\n \ )\n \n \n \ \n \n The &Grape; function \ 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 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 GlobalParamete\ rs has no equivalent\n function in &Digraphs;, although this mi\ ght change in the future.\n \n \n \n \ \n\n \n \n IsDistanceRegular<\ /C>\n \n \n \n \n \n \n \ The &Grape; function IsDistanceRegular is\n equiva\ lent to .\n \n <\ /Item>\n \n \n\n \n \n \ CollapsedAdjacencyMat\n \n \n No\ ne\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 \n The &Grape; function OrbitalGraphColadjMats has \ no equivalent\n function in &Digraphs;, although this might change \ in the future.\n \n \n \n \n\n \n \n VertexTransitiveDRGs\n \ \n \n None\n \n \ \n \n The &Grape; function VertexTran\ sitiveDRGs 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 contains more inf\ ormation when viewed in\n html format.\n \n \n\n <\ Table Align=\"|l|l|l|\">\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 ConnectedComponent\n \ \n \n \n \n \n \n \ When applied to a simple di/graph and a vertex,\n ConnectedCom\ ponent in &Grape; is equivalent to\n .\n can al\ so be applied to\n non-simple graphs (in the language of &Grape;).\ \n \n \n \n \n\n <\ Row>\n \n ConnectedComponents\n \n \ \n \n <\ /Item>\n \n \n When applie\ d to a simple di/graph,\n ConnectedComponents 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 \ 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 simple (in t\ he language of &Grape;), i.e. the\n bipartite graphs in &Grape; are\ a proper subset of the bipartite\n digraphs in &Digraphs;.\n \ \n \n \n \n\n \n \ \n DistanceSet\n \n \n \ \n \n \n \n DistanceSet in &Grape; is equivalent 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 IndependentSet\ \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 old\n \n \n \n The table in this section c\ ontains 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 InducedSubgr\ aph\n \n \n \n \n \n \n \ in &Digraphs; does not allow 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 t\ o\n include one.\n \n \n \n\ \n\n \n \n DistanceGraph\n \n \n \n \n \n \n DistanceGraph in &Grape; is equivalent to\ \n i\ n\n &Digraphs;.\n \n \n \n \ \n\n \n \n ComplementGraph\n \n \n \n \ \n \n \n in &Digraphs; is equivalent to\n Complem\ entGraph 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 plans to\n \ include one.\n \n \n \n <\ HorLine/>\n\n \n \n EdgeGraph\n \ \n \n \n \ \n \n \n Edg\ eGraph in &Grape; is equivalent to\n in &Digraphs;.\n \n \n \n \ \n\n \n \n SwitchedGraph\ \n \n \n None\n \n \n \n There is no analogous functio\ n 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 e\ quivalent 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 &Dig\ raphs; multiple edges may be\n created, whereas in &Grape; 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 &Grape; is equiva\ lent to\n in &Digraphs;.\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 CollapsedIndep\ endentOrbitsGraph\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 CollapsedComp\ leteOrbitsGraph\n \n \n None\n \ \n \n \n There is n\ o analogous function in &Digraphs;, although this feature\n might b\ e implemented in the future.\n \n \n \ \n \n\n \n \n NewGroupGraph<\ /C>\n \n \n None\n \n \ \n \n There is no analogous func\ tion in &Digraphs;. However, the same\n results would be possible \ by calling on\n the digraph to produce \ a copy gr, and then applying\n SetDigraphGroup(gr, G)\ .\n \n \n \n \n\n \n\n \n\n \n \n Vertex-Colouring a\ nd Complete Subgraphs\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 VertexColouring\n \n \n \ \n <\ /Item>\n \n \n VertexCo\ louring in &Grape; is equivalent to in &Digraphs;,\n except it re\ turns a transformation rather than a list of vertex\n colors.\n \ \n \n \n \n\n \n\ \n CompleteSubgraphs\n \n <\ Item>\n \n \n \n \n The functionality of Complet\ eSubgraphs in &Grape; is provided\n by and \n in &Digraphs;.\n\ \n \n \n \n\n \n \n CompleteSubgraphsOfGivenSize\n \n \n \n \n \n \n Much of the func\ tionality of CompleteSubgraphsOfGivenSize in\n &Grape; is pr\ ovided by and in &Digraphs;, except that the\n functionality \ relating to vertex weights and partial colourings is\n not implemen\ ted.\n \n \n \n \n\n \ \n \n\n \n \n Automorphism grou\ ps and isomorphism testing for 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 Comment\ s\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 GraphIsomorphism<\ /C> in &Grape; is equivalent to in &Digraphs;.\n Note however that Gra\ phIsomorphism's optional argument\n firstunbindcanon is \ not applicable, since the canonical\n labellings of a digraph are n\ ot portable between sessions.\n \n \n \ \n \n\n \n \n IsIsomorphicGr\ aph\n \n \n \n \n \n <\ Alt Only=\"HTML\">\n IsIsomorphicGraph in &Grape; is equival\ ent to \ in &Digraphs;.\n Note however that IsIsomorphicGraph's opti\ onal argument\n firstunbindcanon is not applicable, since th\ e canonical\n labellings of a digraph are not portable between sess\ ions.\n \n \n \n \n\n \ \n \n GraphIsomorphismClassRepresentatives\n \n \n None\n \n \ \n \n There is no analogous funct\ ion in &Digraphs;, although this feature\n might be implemented in \ the future.\n \n \n \n \ \n\n \n \n\n \n\n\n\n\n\n", inputorigins := [ [ 1, "/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main.\ xml", 1 ], [ 36, "/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main\ .xml", 2 ], [ 90, "/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main\ .xml", 3 ], [ 132, "/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main\ .xml", 4 ], [ 208, "/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main\ .xml", 5 ], [ 256, "/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main\ .xml", 6 ], [ 298, "/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main\ .xml", 7 ], [ 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"/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main\ .xml", 50 ], [ 474643, "/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main\ .xml", 51 ], [ 474644, "/builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/doc/main\ .xml", 52 ] ], labels := rec( AdjacencyMatrix := [ "5.2-1", "chap5_mj.html#X7DC2CD70830BEE60" ], AdjacencyMatrixMutableCopy := [ "5.2-1", "chap5_mj.html#X7DC2CD70830BEE60" ], AdjacencyMatrixUpperTriangleLineDecoder := [ "9.2-10", "chap9_mj.html#X7DF22D1687D096D6" ], ArticulationPoints := [ "5.3-14", "chap5_mj.html#X7DDE06E47E605DD7" ], AsBinaryRelation := [ "3.2-1", "chap3_mj.html#X7FBC4BDB82FBEDD2" ], AsDigraph := [ "3.2-2", "chap3_mj.html#X83CA27E67F685EB3" ], AsGraph := [ "3.2-4", "chap3_mj.html#X7C4F13E080EC16B0" ], AsMonoid := [ "5.5-1", "chap5_mj.html#X87D5C60D7B0C1309" ], ("AsSemigroup (for a filter and a digraph)") := [ "5.5-1", "chap5_mj.html#X87D5C60D7B0C1309" ], ("AsSemigroup (for a filter, semilattice digraph, and two lists)") := [ "5.5-2", "chap5_mj.html#X7C6D5EC27C51066B" ], AsTransformation := [ "3.2-5", "chap3_mj.html#X7C5360B2799943F3" ], ("Attributes and operations") := [ "5", "chap5_mj.html#X8739F6CD78C90B14" ], ("AutomorphismGroup (for a digraph and a homogeneous list)") := [ "7.2-5", "chap7_mj.html#X877732B1783C391B" ], ("AutomorphismGroup (for a digraph)") := [ "7.2-2", "chap7_mj.html#X858C32127A190175" ], ("AutomorphismGroup (for a digraph, homogeneous list, and list)") := [ "7.2-6", "chap7_mj.html#X7DA2C4FE837FFE01" ], BipartiteDoubleDigraph := [ "3.3-37", "chap3_mj.html#X7C6E6CB284982C7A" ], ("BlissAutomorphismGroup (for a digraph and homogenous list)") := [ "7.2-3", "chap7_mj.html#X7E7B0D88865A89F6" ], ("BlissAutomorphismGroup (for a digraph)") := [ "7.2-3", "chap7_mj.html#X7E7B0D88865A89F6" ], ("BlissAutomorphismGroup (for a digraph, homogenous list, and list)") :=\ [ "7.2-3", "chap7_mj.html#X7E7B0D88865A89F6" ], BlissCanonicalDigraph := [ "7.2-9", "chap7_mj.html#X877B1D377EC197D7" ], ("BlissCanonicalLabelling (for a digraph and a list)") := [ "7.2-8", "chap7_mj.html#X87DA265D803DB337" ], ("BlissCanonicalLabelling (for a digraph)") := [ "7.2-7", "chap7_mj.html#X7D676FE67A6684FF" ], BooleanAdjacencyMatrix := [ "5.2-3", "chap5_mj.html#X8507DC4F794780C1" ] , BooleanAdjacencyMatrixMutableCopy := [ "5.2-3", "chap5_mj.html#X8507DC4F794780C1" ], Bridges := [ "5.3-15", "chap5_mj.html#X84D5A125848BD800" ], CayleyDigraph := [ "3.1-12", "chap3_mj.html#X7FCADADC7EC28478" ], ChainDigraph := [ "3.5-1", "chap3_mj.html#X870594FC866AC88E" ], CharacteristicPolynomial := [ "5.2-2", "chap5_mj.html#X87FA0A727CDB060B" ], ChromaticNumber := [ "7.3-17", "chap7_mj.html#X807D49358529A37C" ], CliqueNumber := [ "8.1-5", "chap8_mj.html#X78427A8B81FEB457" ], CliquesFinder := [ "8.1-2", "chap8_mj.html#X7A3BC9FF8169ABCD" ], ("Compiling the kernel module") := [ "2.3", "chap2_mj.html#X849F6196875A6DF5" ], CompleteBipartiteDigraph := [ "3.5-3", "chap3_mj.html#X8795B0AD856014FA" ], CompleteDigraph := 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and\ operations", ("Compiling the kernel module") := "Compiling th\ e kernel module", ("Creating digraphs") := "Creating digraphs", Definitions := "Definitions", ("Finding cliques and independent sets") := "Fin\ ding cliques and independent sets", ("For those in a hurry") := "For those in a 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 := 474668, 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 ], 74, 2, "acknowledgements", "X82A988D47DFAFCFA" ], [ "Table of Contents", ".-4", [ 0, 0, 4 ], 81, 3, "table of contents", "X8537FEB07AF2BEC8" ], [ "\033[1X\033[33X\033[0;-2YThe \033[5XDigraphs\033[105X\033[101X\027\\ 033[1X\027 package\033[133X\033[101X", "1", [ 1, 0, 0 ], 1, 5, "the digraphs package", "X7F202ABA780E595D" ], [ "\033[1X\033[33X\033[0;-2YIntroduction\033[133X\033[101X", "1.1", [ 1, 1, 0 ], 4, 5, "introduction", "X7DFB63A97E67C0A1" ], [ "\033[1X\033[33X\033[0;-2YDefinitions\033[133X\033[101X", "1.1-1", [ 1, 1, 1 ], 75, 6, "definitions", "X84541F61810C741D" ], [ "\033[1X\033[33X\033[0;-2YInstalling \033[5XDigraphs\033[105X\033[101X\\ 027\033[1X\027\033[133X\033[101X", "2", [ 2, 0, 0 ], 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336, 37, 38, 338, 337, 340, 339, 39, 342, 341, 343, 346, 345, 344, 347, 348, 349, 351, 350, 353, 352, 355, 354, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 369, 368, 367, 366, 371, 370, 40, 372, 373, 374, 375, 376, 378, 377, 379, 381, 380, 382, 384, 383, 385, 387, 388, 392, 391, 390, 389, 386, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 41, 42, 406, 405, 407, 409, 408, 411, 410, 413, 414, 412, 415, 416, 43, 418, 417, 419, 420, 421, 423, 424, 425, 426, 422, 44, 45, 427, 429, 428, 430, 431, 433, 432, 434, 46, 436, 435, 440, 437, 438, 439, 444, 443, 442, 441, 445, 446, 447, 448, 450, 449, 451, 452, 453, 455, 454, 457, 456, 458, 459, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57 ], start := 0, stop := 474667, toctext := "1 \n The Digraphs package\n \n 1.1\ Introduction\n\n\n   1.1-1 De\ finitions\n\n \n \n<\ a href=\"chap2_mj.html#X817088B27F90D596\">2 Installin\ g Digraphs\n\ 2.1 For those in a hurry\n\n \n 2.2 \ Optional package dependencies\n\n\n &\ nbsp; 2.2-1 The Grape package\n\n \n 2.3 Compiling th\ e 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 <\ span class=\"nocss\">  2.5-1 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 NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io), Alice Niemeyer (http://www.maths.uwa.edu.au/~alice/), and Pascal Schweitzer (https://lii.rwth-aachen.de/~schweitzer). Homepage: http://TODO/ ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading orb 4.8.3 (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 1.3.1 (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 A. Wilson (https://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 A. Wilson (https://wilf.me). Homepage: https://digraphs.github.io/Digraphs Report issues at https://github.com/digraphs/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── #I File: /builddir/build/BUILD/digraphs-1.3.1/../pkg/digraphs-1.3.1/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.GY8quv + umask 022 + cd /builddir/build/BUILD + '[' /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64 '!=' / ']' + rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64 ++ dirname /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64 + mkdir -p /builddir/build/BUILDROOT + mkdir /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64 + cd digraphs-1.3.1 + mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.1/doc + cp -a bin data gap tst VERSIONS PackageInfo.g init.g makedoc.g read.g /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.1 + 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-1.3.1-1.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.1/doc + /usr/lib/rpm/find-debuginfo.sh -j4 --strict-build-id -m -i --build-id-seed 1.3.1-1.fc33 --unique-debug-suffix -1.3.1-1.fc33.riscv64 --unique-debug-src-base gap-pkg-digraphs-1.3.1-1.fc33.riscv64 --run-dwz --dwz-low-mem-die-limit 10000000 --dwz-max-die-limit 50000000 -S debugsourcefiles.list /builddir/build/BUILD/digraphs-1.3.1 explicitly decompress any DWARF compressed ELF sections in /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.1/bin/riscv64-redhat-linux-gnu-default64-kv7/digraphs.so extracting debug info from /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.1/bin/riscv64-redhat-linux-gnu-default64-kv7/digraphs.so original debug info size: 5604kB, size after compression: 5456kB /usr/lib/rpm/sepdebugcrcfix: Updated 1 CRC32s, 0 CRC32s did match. 982 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-1.3.1-1.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.kAwBYz + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.3.1 + export LC_ALL=C.UTF-8 + LC_ALL=C.UTF-8 + cd tst + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.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 NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io), Alice Niemeyer (http://www.maths.uwa.edu.au/~alice/), and Pascal Schweitzer (https://lii.rwth-aachen.de/~schweitzer). Homepage: http://TODO/ ───────────────────────────────────────────────────────────────────────────── ─────────────��─────────────��──────────────────────────��─────────────��──────── Loading orb 4.8.3 (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 1.3.1 (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 A. Wilson (https://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 A. Wilson (https://wilf.me). Homepage: https://digraphs.github.io/Digraphs Report issues at https://github.com/digraphs/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: testinstall.tst msecs: 6785 true gap> Digraphs package: standard/attr.tst msecs: 46687 true gap> Digraphs package: standard/cliques.tst msecs: 9508 true gap> Digraphs package: standard/constructors.tst msecs: 1215 true gap> Digraphs package: standard/display.tst msecs: 174 true gap> Digraphs package: standard/examples.tst msecs: 412 true gap> Digraphs package: standard/grahom.tst msecs: 8742 true gap> Digraphs package: standard/grape.tst msecs: 407 true gap> Digraphs package: standard/io.tst msecs: 44129 true gap> Digraphs package: standard/isomorph.tst msecs: 13533 true gap> Digraphs package: standard/labels.tst msecs: 202 true gap> Digraphs package: standard/oper.tst msecs: 12841 true gap> Digraphs package: standard/orbits.tst msecs: 83 true gap> Digraphs package: standard/planar.tst msecs: 1203 true gap> Digraphs package: standard/prop.tst msecs: 58615 true gap> gap> + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.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 NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io), Alice Niemeyer (http://www.maths.uwa.edu.au/~alice/), and Pascal Schweitzer (https://lii.rwth-aachen.de/~schweitzer). Homepage: http://TODO/ ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading orb 4.8.3 (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 1.3.1 (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 A. Wilson (https://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 A. Wilson (https://wilf.me). Homepage: https://digraphs.github.io/Digraphs Report issues at https://github.com/digraphs/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: standard/digraph.tst msecs: 7923 gap> + find /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64/usr/lib/gap/pkg/digraphs-1.3.1 -size 0 -delete + RPM_EC=0 ++ jobs -p + exit 0 Processing files: gap-pkg-digraphs-1.3.1-1.fc33.riscv64 Executing(%doc): /bin/sh -e /var/tmp/rpm-tmp.Qd7Jzv + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.3.1 + DOCDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.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-1.3.1-1.fc33.riscv64/usr/share/doc/gap-pkg-digraphs + cp -pr CHANGELOG.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64/usr/share/doc/gap-pkg-digraphs + cp -pr README.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64/usr/share/doc/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Executing(%license): /bin/sh -e /var/tmp/rpm-tmp.GwGG4v + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.3.1 + LICENSEDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.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-1.3.1-1.fc33.riscv64/usr/share/licenses/gap-pkg-digraphs + cp -pr GPL /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64/usr/share/licenses/gap-pkg-digraphs + cp -pr LICENSE /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64/usr/share/licenses/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Provides: bundled(bliss) = 0.73 gap-pkg-digraphs = 1.3.1-1.fc33 gap-pkg-digraphs(riscv-64) = 1.3.1-1.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-1.3.1-1.fc33.noarch Provides: gap-pkg-digraphs-doc = 1.3.1-1.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-1.3.1-1.fc33.riscv64 Provides: gap-pkg-digraphs-debugsource = 1.3.1-1.fc33 gap-pkg-digraphs-debugsource(riscv-64) = 1.3.1-1.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-1.3.1-1.fc33.riscv64 Provides: debuginfo(build-id) = b10678f9aa1f32666c9334c597bf2d480f4e1152 gap-pkg-digraphs-debuginfo = 1.3.1-1.fc33 gap-pkg-digraphs-debuginfo(riscv-64) = 1.3.1-1.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) = 1.3.1-1.fc33 Checking for unpackaged file(s): /usr/lib/rpm/check-files /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64 Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debugsource-1.3.1-1.fc33.riscv64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-1.3.1-1.fc33.riscv64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debuginfo-1.3.1-1.fc33.riscv64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-doc-1.3.1-1.fc33.noarch.rpm Executing(%clean): /bin/sh -e /var/tmp/rpm-tmp.lZuWBy + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.3.1 + /usr/bin/rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.3.1-1.fc33.riscv64 + RPM_EC=0 ++ jobs -p + exit 0 Child return code was: 0