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Digraphs

Version 0.15.4

Jan De Beule
Email: jdebeule@cage.ugent.be
Homepage: http://homepages.vub.ac.be/~jdbeule

Julius Jonušas
Email: jj252@st-andrews.ac.uk
Homepage: http://www-circa.mcs.st-andrews.ac.uk/~julius

James D. Mitchell
Email: jdm3@st-andrews.ac.uk
Homepage: http://goo.gl/ZtViV6

Michael Torpey
Email: mct25@st-andrews.ac.uk
Homepage: http://www-circa.mcs.st-andrews.ac.uk/~mct25

Wilf A. Wilson
Email: gap@wilf-wilson.net
Homepage: http://wilf.me

Stuart Burrell

Luke Elliott

Christopher Jefferson

Markus Pfeiffer

Chris Russell

Finn Smith

Abstract

The Digraphs package is a GAP package containing methods for graphs, digraphs, and multidigraphs.

Copyright

© 2014-19 by Jan De Beule, Julius Jonušas, James D. Mitchell, Michael Torpey, Wilf A. Wilson et al.

Digraphs is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.

Acknowledgements

We would like to thank Christopher Jefferson for his help in including bliss in Digraphs. This package's methods for computing digraph homomorphisms are based on work by Max Neunhöffer, and independently Artur Schäfer.

Contents

1 The Digraphs package
 1.1 Introduction

  1.1-1 Definitions
2 Installing Digraphs
 2.1 For those in a hurry
 2.2 Optional package dependencies

  2.2-1 The Grape package
 2.3 Compiling the kernel module
 2.4 Rebuilding the documentation

  2.4-1 DigraphsMakeDoc
 2.5 Testing your installation

  2.5-1 DigraphsTestInstall
  2.5-2 DigraphsTestStandard
3 Creating digraphs
 3.1 Creating digraphs

  3.1-1 IsDigraph
  3.1-2 IsCayleyDigraph
  3.1-3 IsDigraphWithAdjacencyFunction
  3.1-4 DigraphType
  3.1-5 Digraph
  3.1-6 DigraphByAdjacencyMatrix
  3.1-7 DigraphByEdges
  3.1-8 EdgeOrbitsDigraph
  3.1-9 DigraphByInNeighbours
  3.1-10 CayleyDigraph
 3.2 Changing representations

  3.2-1 AsBinaryRelation
  3.2-2 AsDigraph
  3.2-3 Graph
  3.2-4 AsGraph
  3.2-5 AsTransformation
 3.3 New digraphs from old

  3.3-1 DigraphCopy
  3.3-2 InducedSubdigraph
  3.3-3 ReducedDigraph
  3.3-4 MaximalSymmetricSubdigraph
  3.3-5 MaximalAntiSymmetricSubdigraph
  3.3-6 UndirectedSpanningTree
  3.3-7 QuotientDigraph
  3.3-8 DigraphReverse
  3.3-9 DigraphDual
  3.3-10 DigraphSymmetricClosure
  3.3-11 DigraphReflexiveTransitiveClosure
  3.3-12 DigraphReflexiveTransitiveReduction
  3.3-13 DigraphAddVertex
  3.3-14 DigraphAddVertices
  3.3-15 DigraphAddEdge
  3.3-16 DigraphAddEdgeOrbit
  3.3-17 DigraphAddEdges
  3.3-18 DigraphRemoveVertex
  3.3-19 DigraphRemoveVertices
  3.3-20 DigraphRemoveEdge
  3.3-21 DigraphRemoveEdgeOrbit
  3.3-22 DigraphRemoveEdges
  3.3-23 DigraphRemoveLoops
  3.3-24 DigraphRemoveAllMultipleEdges
  3.3-25 DigraphReverseEdges
  3.3-26 DigraphDisjointUnion
  3.3-27 DigraphEdgeUnion
  3.3-28 DigraphJoin
  3.3-29 LineDigraph
  3.3-30 LineUndirectedDigraph
  3.3-31 DoubleDigraph
  3.3-32 BipartiteDoubleDigraph
  3.3-33 DigraphAddAllLoops
  3.3-34 DistanceDigraph
  3.3-35 DigraphClosure
 3.4 Random digraphs

  3.4-1 RandomDigraph
  3.4-2 RandomMultiDigraph
  3.4-3 RandomTournament
 3.5 Standard examples

  3.5-1 ChainDigraph
  3.5-2 CompleteDigraph
  3.5-3 CompleteBipartiteDigraph
  3.5-4 CompleteMultipartiteDigraph
  3.5-5 CycleDigraph
  3.5-6 EmptyDigraph
  3.5-7 JohnsonDigraph
4 Operators
 4.1 Operators for digraphs

  4.1-1 IsSubdigraph
  4.1-2 IsUndirectedSpanningTree
5 Attributes and operations
 5.1 Vertices and edges

  5.1-1 DigraphVertices
  5.1-2 DigraphNrVertices
  5.1-3 DigraphEdges
  5.1-4 DigraphNrEdges
  5.1-5 DigraphSinks
  5.1-6 DigraphSources
  5.1-7 DigraphTopologicalSort
  5.1-8 DigraphVertexLabel
  5.1-9 DigraphVertexLabels
  5.1-10 DigraphEdgeLabel
  5.1-11 DigraphEdgeLabels
  5.1-12 DigraphInEdges
  5.1-13 DigraphOutEdges
  5.1-14 IsDigraphEdge
  5.1-15 IsMatching
 5.2 Neighbours and degree

  5.2-1 AdjacencyMatrix
  5.2-2 CharacteristicPolynomial
  5.2-3 BooleanAdjacencyMatrix
  5.2-4 DigraphAdjacencyFunction
  5.2-5 DigraphRange
  5.2-6 OutNeighbours
  5.2-7 InNeighbours
  5.2-8 OutDegrees
  5.2-9 InDegrees
  5.2-10 OutDegreeOfVertex
  5.2-11 OutNeighboursOfVertex
  5.2-12 InDegreeOfVertex
  5.2-13 InNeighboursOfVertex
  5.2-14 DigraphLoops
  5.2-15 PartialOrderDigraphMeetOfVertices
 5.3 Reachability and connectivity

  5.3-1 DigraphDiameter
  5.3-2 DigraphShortestDistance
  5.3-3 DigraphShortestDistances
  5.3-4 DigraphLongestDistanceFromVertex
  5.3-5 DigraphDistanceSet
  5.3-6 DigraphGirth
  5.3-7 DigraphUndirectedGirth
  5.3-8 DigraphConnectedComponents
  5.3-9 DigraphConnectedComponent
  5.3-10 DigraphStronglyConnectedComponents
  5.3-11 DigraphStronglyConnectedComponent
  5.3-12 DigraphBicomponents
  5.3-13 ArticulationPoints
  5.3-14 DigraphPeriod
  5.3-15 DigraphFloydWarshall
  5.3-16 IsReachable
  5.3-17 DigraphPath
  5.3-18 DigraphShortestPath
  5.3-19 IteratorOfPaths
  5.3-20 DigraphAllSimpleCircuits
  5.3-21 DigraphLongestSimpleCircuit
  5.3-22 DigraphLayers
  5.3-23 DigraphDegeneracy
  5.3-24 DigraphDegeneracyOrdering
  5.3-25 HamiltonianPath
 5.4 Cayley graphs of groups

  5.4-1 GroupOfCayleyDigraph
  5.4-2 GeneratorsOfCayleyDigraph
 5.5 Associated semigroups

  5.5-1 AsSemigroup
 5.6 Planarity

  5.6-1 KuratowskiPlanarSubdigraph
  5.6-2 KuratowskiOuterPlanarSubdigraph
  5.6-3 PlanarEmbedding
  5.6-4 OuterPlanarEmbedding
  5.6-5 SubdigraphHomeomorphicToK23
6 Properties of digraphs
 6.1 Edge properties

  6.1-1 DigraphHasLoops
  6.1-2 IsAntisymmetricDigraph
  6.1-3 IsBipartiteDigraph
  6.1-4 IsCompleteBipartiteDigraph
  6.1-5 IsCompleteDigraph
  6.1-6 IsEmptyDigraph
  6.1-7 IsFunctionalDigraph
  6.1-8 IsMultiDigraph
  6.1-9 IsReflexiveDigraph
  6.1-10 IsSymmetricDigraph
  6.1-11 IsTournament
  6.1-12 IsTransitiveDigraph
  6.1-13 IsPreorderDigraph
  6.1-14 IsPartialOrderDigraph
  6.1-15 IsMeetSemilatticeDigraph
 6.2 Regularity

  6.2-1 IsInRegularDigraph
  6.2-2 IsOutRegularDigraph
  6.2-3 IsRegularDigraph
  6.2-4 IsDistanceRegularDigraph
 6.3 Connectivity and cycles

  6.3-1 IsAcyclicDigraph
  6.3-2 IsChainDigraph
  6.3-3 IsConnectedDigraph
  6.3-4 IsBiconnectedDigraph
  6.3-5 IsStronglyConnectedDigraph
  6.3-6 IsAperiodicDigraph
  6.3-7 IsDirectedTree
  6.3-8 IsUndirectedTree
  6.3-9 IsEulerianDigraph
  6.3-10 IsHamiltonianDigraph
  6.3-11 IsCycleDigraph
 6.4 Planarity

  6.4-1 IsPlanarDigraph
  6.4-2 IsOuterPlanarDigraph
7 Homomorphisms
 7.1 Acting on digraphs

  7.1-1 OnDigraphs
  7.1-2 OnMultiDigraphs
 7.2 Isomorphisms and canonical labellings

  7.2-1 DigraphsUseNauty
  7.2-2 AutomorphismGroup
  7.2-3 BlissAutomorphismGroup
  7.2-4 NautyAutomorphismGroup
  7.2-5 AutomorphismGroup
  7.2-6 BlissCanonicalLabelling
  7.2-7 BlissCanonicalLabelling
  7.2-8 BlissCanonicalDigraph
  7.2-9 DigraphGroup
  7.2-10 DigraphOrbits
  7.2-11 DigraphOrbitReps
  7.2-12 DigraphSchreierVector
  7.2-13 DigraphStabilizer
  7.2-14 IsIsomorphicDigraph
  7.2-15 IsIsomorphicDigraph
  7.2-16 IsomorphismDigraphs
  7.2-17 IsomorphismDigraphs
  7.2-18 RepresentativeOutNeighbours
  7.2-19 IsDigraphIsomorphism
  7.2-20 IsDigraphColouring
 7.3 Homomorphisms of digraphs

  7.3-1 HomomorphismDigraphsFinder
  7.3-2 DigraphHomomorphism
  7.3-3 HomomorphismsDigraphs
  7.3-4 DigraphMonomorphism
  7.3-5 MonomorphismsDigraphs
  7.3-6 DigraphEpimorphism
  7.3-7 EpimorphismsDigraphs
  7.3-8 DigraphEmbedding
  7.3-9 EmbeddingsDigraphs
  7.3-10 IsDigraphHomomorphism
  7.3-11 IsDigraphEmbedding
  7.3-12 GeneratorsOfEndomorphismMonoid
  7.3-13 DigraphColouring
  7.3-14 DigraphGreedyColouring
  7.3-15 DigraphWelshPowellOrder
  7.3-16 ChromaticNumber
8 Finding cliques and independent sets
 8.1 Finding cliques

  8.1-1 IsClique
  8.1-2 CliquesFinder
  8.1-3 DigraphClique
  8.1-4 DigraphMaximalCliques
  8.1-5 CliqueNumber
 8.2 Finding independent sets

  8.2-1 IsIndependentSet
  8.2-2 DigraphIndependentSet
  8.2-3 DigraphMaximalIndependentSets
9 Visualising and IO
 9.1 Visualising a digraph

  9.1-1 Splash
  9.1-2 DotDigraph
  9.1-3 DotSymmetricDigraph
  9.1-4 DotPartialOrderDigraph
  9.1-5 DotPreorderDigraph
 9.2 Reading and writing graphs to a file

  9.2-1 DigraphFromGraph6String
  9.2-2 Graph6String
  9.2-3 DigraphFile
  9.2-4 ReadDigraphs
  9.2-5 WriteDigraphs
  9.2-6 IteratorFromDigraphFile
  9.2-7 DigraphPlainTextLineEncoder
  9.2-8 TournamentLineDecoder
  9.2-9 AdjacencyMatrixUpperTriangleLineDecoder
  9.2-10 TCodeDecoder
  9.2-11 PlainTextString
  9.2-12 WritePlainTextDigraph
  9.2-13 WriteDIMACSDigraph
A Grape to Digraphs Command Map
 A.1 Functions to construct and modify graphs
 A.2 Functions to inspect graphs, vertices and edges
 A.3 Functions to determine regularity properties of graphs
 A.4 Some special vertex subsets of a graph
 A.5 Functions to construct new graphs from old
 A.6 Vertex-Colouring and Complete Subgraphs
 A.7 Automorphism groups and isomorphism testing for graphs
References
Index

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