  
  
                                    [1X[5XDigraphs[105X[101X
  
  
                              [1XMethods for digraphs[101X
  
  
                                 Version 0.13.0
  
  
                                  Jan De Beule
  
                                 Julius Jonušas
  
                               James D. Mitchell
  
                                 Michael Torpey
  
                                 Wilf A. Wilson
  
                                 Stuart Burrell
  
                                  Luke Elliott
  
                             Christopher Jefferson
  
                                Markus Pfeiffer
  
                                 Chris Russell
  
                                   Finn Smith
  
  
  
  Jan De Beule
      Email:    [7Xmailto:jdebeule@cage.ugent.be[107X
      Homepage: [7Xhttp://homepages.vub.ac.be/~jdbeule[107X
  Julius Jonušas
      Email:    [7Xmailto:jj252@st-andrews.ac.uk[107X
      Homepage: [7Xhttp://www-circa.mcs.st-andrews.ac.uk/~julius[107X
  James D. Mitchell
      Email:    [7Xmailto:jdm3@st-andrews.ac.uk[107X
      Homepage: [7Xhttp://goo.gl/ZtViV6[107X
  Michael Torpey
      Email:    [7Xmailto:mct25@st-andrews.ac.uk[107X
      Homepage: [7Xhttp://www-circa.mcs.st-andrews.ac.uk/~mct25[107X
  Wilf A. Wilson
      Email:    [7Xmailto:gap@wilf-wilson.net[107X
      Homepage: [7Xhttp://wilf.me[107X
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  [5XDigraphs[105X  package  is  a  [5XGAP[105X  package  containing  methods for graphs,
  digraphs, and multidigraphs.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y©  2014-18  by  Jan  De  Beule,  Julius  Jonušas, James D. Mitchell, Michael
  Torpey, Wilf A. Wilson et al.[133X
  
  [33X[0;0Y[5XDigraphs[105X  is  free  software; you can redistribute it and/or modify it under
  the      terms      of      the      GNU      General     Public     License
  ([7Xhttp://www.fsf.org/licenses/gpl.html[107X)  as  published  by  the Free Software
  Foundation;  either  version 3 of the License, or (at your option) any later
  version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe  would  like to thank Christopher Jefferson for his help in including the
  [5Xbliss[105X  tool  in  the  package.  This package's methods for computing digraph
  homomorphisms  are  based on work by Max Neunhöffer, and independently Artur
  Schäfer.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (digraphs)[101X
  
  1 [33X[0;0YThe [5XDigraphs[105X package[133X
    1.1 [33X[0;0YIntroduction[133X
      1.1-1 [33X[0;0YDefinitions[133X
  2 [33X[0;0YInstalling [5XDigraphs[105X[133X
    2.1 [33X[0;0YFor those in a hurry[133X
    2.2 [33X[0;0YOptional package dependencies[133X
      2.2-1 [33X[0;0YThe Grape package[133X
    2.3 [33X[0;0YCompiling the kernel module[133X
    2.4 [33X[0;0YRebuilding the documentation[133X
      2.4-1 DigraphsMakeDoc
    2.5 [33X[0;0YTesting your installation[133X
      2.5-1 DigraphsTestInstall
      2.5-2 DigraphsTestStandard
  3 [33X[0;0YCreating digraphs[133X
    3.1 [33X[0;0YCreating digraphs[133X
      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 [33X[0;0YChanging representations[133X
      3.2-1 AsBinaryRelation
      3.2-2 AsDigraph
      3.2-3 Graph
      3.2-4 AsGraph
      3.2-5 AsTransformation
    3.3 [33X[0;0YNew digraphs from old[133X
      3.3-1 DigraphCopy
      3.3-2 InducedSubdigraph
      3.3-3 ReducedDigraph
      3.3-4 MaximalSymmetricSubdigraph
      3.3-5 UndirectedSpanningTree
      3.3-6 QuotientDigraph
      3.3-7 DigraphReverse
      3.3-8 DigraphDual
      3.3-9 DigraphSymmetricClosure
      3.3-10 DigraphReflexiveTransitiveClosure
      3.3-11 DigraphReflexiveTransitiveReduction
      3.3-12 DigraphAddVertex
      3.3-13 DigraphAddVertices
      3.3-14 DigraphAddEdge
      3.3-15 DigraphAddEdgeOrbit
      3.3-16 DigraphAddEdges
      3.3-17 DigraphRemoveVertex
      3.3-18 DigraphRemoveVertices 
      3.3-19 DigraphRemoveEdge
      3.3-20 DigraphRemoveEdgeOrbit
      3.3-21 DigraphRemoveEdges
      3.3-22 DigraphRemoveLoops
      3.3-23 DigraphRemoveAllMultipleEdges
      3.3-24 DigraphReverseEdges
      3.3-25 DigraphDisjointUnion
      3.3-26 DigraphEdgeUnion
      3.3-27 DigraphJoin
      3.3-28 LineDigraph
      3.3-29 LineUndirectedDigraph
      3.3-30 DoubleDigraph
      3.3-31 BipartiteDoubleDigraph
      3.3-32 DigraphAddAllLoops
      3.3-33 DistanceDigraph
      3.3-34 DigraphClosure
    3.4 [33X[0;0YRandom digraphs[133X
      3.4-1 RandomDigraph
      3.4-2 RandomMultiDigraph
      3.4-3 RandomTournament
    3.5 [33X[0;0YStandard examples[133X
      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 [33X[0;0YOperators[133X
    4.1 [33X[0;0YOperators for digraphs[133X
      4.1-1 IsSubdigraph
      4.1-2 IsUndirectedSpanningTree
  5 [33X[0;0YAttributes and operations[133X
    5.1 [33X[0;0YVertices and edges[133X
      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 [33X[0;0YNeighbours and degree[133X
      5.2-1 AdjacencyMatrix
      5.2-2 BooleanAdjacencyMatrix
      5.2-3 DigraphAdjacencyFunction
      5.2-4 DigraphRange
      5.2-5 OutNeighbours
      5.2-6 InNeighbours
      5.2-7 OutDegrees
      5.2-8 InDegrees
      5.2-9 OutDegreeOfVertex
      5.2-10 OutNeighboursOfVertex
      5.2-11 InDegreeOfVertex
      5.2-12 InNeighboursOfVertex
      5.2-13 DigraphLoops
      5.2-14 PartialOrderDigraphMeetOfVertices
    5.3 [33X[0;0YReachability and connectivity[133X
      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 IteratorOfPaths
      5.3-19 DigraphAllSimpleCircuits
      5.3-20 DigraphLongestSimpleCircuit
      5.3-21 DigraphLayers
      5.3-22 DigraphDegeneracy
      5.3-23 DigraphDegeneracyOrdering
      5.3-24 HamiltonianPath
    5.4 [33X[0;0YCayley graphs of groups[133X
      5.4-1 GroupOfCayleyDigraph
      5.4-2 GeneratorsOfCayleyDigraph
  6 [33X[0;0YProperties of digraphs[133X
    6.1 [33X[0;0YEdge properties[133X
      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 [33X[0;0YRegularity[133X
      6.2-1 IsInRegularDigraph
      6.2-2 IsOutRegularDigraph
      6.2-3 IsRegularDigraph
      6.2-4 IsDistanceRegularDigraph
    6.3 [33X[0;0YConnectivity and cycles[133X
      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
  7 [33X[0;0YHomomorphisms[133X
    7.1 [33X[0;0YActing on digraphs[133X
      7.1-1 OnDigraphs
      7.1-2 OnMultiDigraphs
    7.2 [33X[0;0YIsomorphisms and canonical labellings[133X
      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
    7.3 [33X[0;0YHomomorphisms of digraphs[133X
      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 GeneratorsOfEndomorphismMonoid
      7.3-9 DigraphColouring
      7.3-10 DigraphEmbedding
      7.3-11 ChromaticNumber
      7.3-12 IsDigraphHomomorphism
      7.3-13 IsDigraphEmbedding
  8 [33X[0;0YFinding cliques and independent sets[133X
    8.1 [33X[0;0YFinding cliques[133X
      8.1-1 IsClique
      8.1-2 CliquesFinder
      8.1-3 DigraphClique
      8.1-4 DigraphMaximalCliques
      8.1-5 CliqueNumber
    8.2 [33X[0;0YFinding independent sets[133X
      8.2-1 IsIndependentSet
      8.2-2 DigraphIndependentSet
      8.2-3 DigraphMaximalIndependentSets
  9 [33X[0;0YVisualising and IO[133X
    9.1 [33X[0;0YVisualising a digraph[133X
      9.1-1 Splash
      9.1-2 DotDigraph
      9.1-3 DotSymmetricDigraph
      9.1-4 DotPartialOrderDigraph
      9.1-5 DotPreorderDigraph
    9.2 [33X[0;0YReading and writing graphs to a file[133X
      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 [33X[0;0YGrape to Digraphs Command Map[133X
    A.1 [33X[0;0YFunctions to construct and modify graphs[133X
    A.2 [33X[0;0YFunctions to inspect graphs, vertices and edges[133X
    A.3 [33X[0;0YFunctions to determine regularity properties of graphs[133X
    A.4 [33X[0;0YSome special vertex subsets of a graph[133X
    A.5 [33X[0;0YFunctions to construct new graphs from old[133X
    A.6 [33X[0;0YVertex-Colouring and Complete Subgraphs[133X
    A.7 [33X[0;0YAutomorphism groups and isomorphism testing for graphs[133X
  
  
  [32X
