  
  [1X3 [33X[0;0YCreating digraphs[133X[101X
  
  [33X[0;0YIn this chapter we describe how to create digraphs.[133X
  
  
  [1X3.1 [33X[0;0YCreating digraphs[133X[101X
  
  [1X3.1-1 IsDigraph[101X
  
  [33X[1;0Y[29X[2XIsDigraph[102X[32X Category[133X
  
  [33X[0;0YEvery   digraph  in  [5XDigraphs[105X  belongs  to  the  category  [10XIsDigraph[110X.  Basic
  attributes   and  operations  for  digraphs  are:  [2XDigraphVertices[102X  ([14X5.1-1[114X),
  [2XDigraphRange[102X  ([14X5.2-5[114X),  [2XDigraphSource[102X  ([14X5.2-5[114X),  [2XOutNeighbours[102X  ([14X5.2-6[114X), and
  [2XDigraphEdges[102X ([14X5.1-3[114X).[133X
  
  [1X3.1-2 IsCayleyDigraph[101X
  
  [33X[1;0Y[29X[2XIsCayleyDigraph[102X[32X Category[133X
  
  [33X[0;0Y[10XIsCayleyDigraph[110X  is  a  subcategory  of  [10XIsDigraph[110X. Digraphs that are Cayley
  digraphs  of a group and that are constructed by the operation [2XCayleyDigraph[102X
  ([14X3.1-10[114X) are constructed in this category.[133X
  
  [1X3.1-3 IsDigraphWithAdjacencyFunction[101X
  
  [33X[1;0Y[29X[2XIsDigraphWithAdjacencyFunction[102X[32X Category[133X
  
  [33X[0;0Y[10XIsDigraphWithAdjacencyFunction[110X  is a subcategory of [10XIsDigraph[110X. Digraphs that
  are [13Xcreated[113X using an adjacency function are constructed in this category.[133X
  
  [1X3.1-4 DigraphType[101X
  
  [33X[1;0Y[29X[2XDigraphType[102X[32X global variable[133X
  [33X[1;0Y[29X[2XDigraphFamily[102X[32X family[133X
  
  [33X[0;0YThe  type  of  all  digraphs  is  [10XDigraphType[110X. The family of all digraphs is
  [10XDigraphFamily[110X.[133X
  
  [1X3.1-5 Digraph[101X
  
  [33X[1;0Y[29X[2XDigraph[102X( [3Xobj[103X[, [3Xsource[103X, [3Xrange[103X] ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraph[102X( [3Xlist[103X, [3Xfunc[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraph[102X( [3XG[103X, [3Xlist[103X, [3Xact[103X, [3Xadj[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [8Xfor a list (i.e. an adjacency list)[108X
        [33X[0;6Yif  [3Xobj[103X is a list of lists of positive integers in the range from [10X1[110X to
        [10XLength([3Xobj[103X[10X)[110X,  then this function returns the digraph with vertices [22XE ^
        0 =[122X[10X[1 .. Length([3Xobj[103X[10X)][110X, and edges corresponding to the entries of [3Xobj[103X.[133X
  
        [33X[0;6YMore  precisely,  there is an edge from vertex [10Xi[110X to [10Xj[110X if and only if [10Xj[110X
        is  in  [10X[3Xobj[103X[10X[i][110X;  the source of this edge is [10Xi[110X and the range is [10Xj[110X. If [10Xj[110X
        occurs in [10X[3Xobj[103X[10X[i][110X with multiplicity [10Xk[110X, then there are [10Xk[110X edges from [10Xi[110X to
        [10Xj[110X.[133X
  
  [8Xfor three lists[108X
        [33X[0;6Yif  [3Xobj[103X  is  a  duplicate-free list, and [3Xsource[103X and [3Xrange[103X are lists of
        equal  length  consisting  of  positive  integers  in  the  list [10X[1 ..
        Length([3Xobj[103X[10X)][110X, then this function returns a digraph with vertices [22XE ^ 0
        =[122X[10X[1  ..  Length([3Xobj[103X[10X)][110X,  and  [10XLength([3Xsource[103X[10X)[110X edges. For each [10Xi[110X in [10X[1 ..
        Length([3Xsource[103X[10X)][110X  there exists an edge with source vertex [10Xsource[i][110X and
        range  vertex  [10Xrange[i][110X.  See  [2XDigraphSource[102X  ([14X5.2-5[114X) and [2XDigraphRange[102X
        ([14X5.2-5[114X).[133X
  
        [33X[0;6YThe vertices of the digraph will be labelled by the elements of [3Xobj[103X.[133X
  
  [8Xfor an integer, and two lists[108X
        [33X[0;6Yif  [3Xobj[103X  is an integer, and [3Xsource[103X and [3Xrange[103X are lists of equal length
        consisting  of  positive  integers  in  the list [10X[1 .. [3Xobj[103X[10X][110X, then this
        function  returns  a  digraph  with  vertices  [22XE  ^ 0 =[122X[10X[1 .. [3Xobj[103X[10X][110X, and
        [10XLength([3Xsource[103X[10X)[110X edges. For each [10Xi[110X in [10X[1 .. Length([3Xsource[103X[10X)][110X there exists
        an  edge  with  source vertex [10Xsource[i][110X and range vertex [10Xrange[i][110X. See
        [2XDigraphSource[102X ([14X5.2-5[114X) and [2XDigraphRange[102X ([14X5.2-5[114X).[133X
  
  [8Xfor a list and a function[108X
        [33X[0;6Yif  [3Xlist[103X  is a list and [3Xfunc[103X is a function taking 2 arguments that are
        elements  of [3Xlist[103X, and [3Xfunc[103X returns [9Xtrue[109X or [9Xfalse[109X, then this operation
        creates  a  digraph with vertices [10X[1 .. Length([3Xlist[103X[10X)][110X and an edge from
        vertex  [10Xi[110X  to  vertex  [10Xj[110X if and only if [10X[3Xfunc[103X[10X([3Xlist[103X[10X[i], [3Xlist[103X[10X[j])[110X returns
        [9Xtrue[109X.[133X
  
  [8Xfor a group, a list, and two functions[108X
        [33X[0;6YThe arguments will be [3XG, list, act, adj[103X.[133X
  
        [33X[0;6YLet [3XG[103X be a group acting on the objects in [3Xlist[103X via the action [3Xact[103X, and
        let  [3Xadj[103X  be  a function taking two objects from [3Xlist[103X as arguments and
        returning  [10Xtrue[110X or [10Xfalse[110X. The function [3Xadj[103X will describe the adjacency
        between  objects  from [3Xlist[103X, which is invariant under the action of [3XG[103X.
        This  variant  of  the constructor returns a digraph with vertices the
        objects of [3Xlist[103X and directed edges [10X[x, y][110X when [10Xf(x, y)[110X is [10Xtrue[110X.[133X
  
        [33X[0;6YThe  action  of  the  group  [3XG[103X on the objects in [3Xlist[103X is stored in the
        attribute  [2XDigraphGroup[102X  ([14X7.2-9[114X),  and  is used to speed up operations
        like [2XDigraphDiameter[102X ([14X5.3-1[114X).[133X
  
  [8Xfor a Grape package graph[108X
        [33X[0;6Yif  [3Xobj[103X is a [5XGrape[105X package graph (i.e. a record for which the function
        [10XIsGraph[110X returns [9Xtrue[109X), then this function returns a digraph isomorphic
        to [3Xobj[103X.[133X
  
  [8Xfor a binary relation[108X
        [33X[0;6Yif  [3Xobj[103X is a binary relation on the points [10X[1 .. n][110X for some posititve
        integer  [22Xn[122X,  then  this  function  returns the digraph defined by [3Xobj[103X.
        Specifically,  this  function  returns a digraph which has [22Xn[122X vertices,
        and  which  has an edge with source [10Xi[110X and range [10Xj[110X if and only if [10X[i,j][110X
        is a pair in the binary relation [3Xobj[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[127X[104X
    [4X[25X>[125X [27X[2, 5, 8, 10], [2, 3, 4, 2, 5, 6, 8, 9, 10], [1],[127X[104X
    [4X[25X>[125X [27X[3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],[127X[104X
    [4X[25X>[125X [27X[1, 5, 9], [1, 2, 7, 8], [3, 5]]);[127X[104X
    [4X[28X<multidigraph with 10 vertices, 38 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph(["a", "b", "c"], ["a"], ["b"]);[127X[104X
    [4X[28X<digraph with 3 vertices, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph(5, [1, 2, 2, 4, 1, 1], [2, 3, 5, 5, 1, 1]);[127X[104X
    [4X[28X<multidigraph with 5 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XPetersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,[127X[104X
    [4X[25X>[125X [27Xfunction(x, y) return Intersection(x, y) = []; end);;[127X[104X
    [4X[25Xgap>[125X [27XDigraph(Petersen);[127X[104X
    [4X[28X<digraph with 10 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27Xb := BinaryRelationOnPoints([127X[104X
    [4X[25X>[125X [27X[[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);[127X[104X
    [4X[28XBinary Relation on 5 points[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph(b);[127X[104X
    [4X[28X<digraph with 5 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([1 .. 10], ReturnTrue);[127X[104X
    [4X[28X<digraph with 10 vertices, 100 edges>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  example  illustrates the uses of the fourth and fifth variants of
  this  constructor. The resulting digraph is a strongly regular graph, and it
  is  actually  the  point  graph  of the van Lint-Schrijver partial geometry,
  [vLS81].  The  algebraic  description  is  taken  from  the seminal paper of
  Calderbank and Kantor [CK86].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := GF(3 ^ 4);[127X[104X
    [4X[28XGF(3^4)[128X[104X
    [4X[25Xgap>[125X [27Xgamma := First(f, x -> Order(x) = 5);[127X[104X
    [4X[28XZ(3^4)^64[128X[104X
    [4X[25Xgap>[125X [27XL := Union([Zero(f)], List(Group(gamma)));[127X[104X
    [4X[28X[ 0*Z(3), Z(3)^0, Z(3^4)^16, Z(3^4)^32, Z(3^4)^48, Z(3^4)^64 ][128X[104X
    [4X[25Xgap>[125X [27Xomega := Union(List(L, x -> List(Difference(L, [x]), y -> x - y)));[127X[104X
    [4X[28X[ 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, [128X[104X
    [4X[28X  Z(3^4)^16, Z(3^4)^21, Z(3^4)^23, Z(3^4)^24, Z(3^4)^29, Z(3^4)^31, [128X[104X
    [4X[28X  Z(3^4)^32, Z(3^4)^37, Z(3^4)^39, Z(3^4)^45, Z(3^4)^47, Z(3^4)^48, [128X[104X
    [4X[28X  Z(3^4)^53, Z(3^4)^55, Z(3^4)^56, Z(3^4)^61, Z(3^4)^63, Z(3^4)^64, [128X[104X
    [4X[28X  Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^77, Z(3^4)^79 ][128X[104X
    [4X[25Xgap>[125X [27Xadj := function(x, y)[127X[104X
    [4X[25X>[125X [27X  return x - y in omega;[127X[104X
    [4X[25X>[125X [27Xend;[127X[104X
    [4X[28Xfunction( x, y ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xdigraph := Digraph(AsList(f), adj);[127X[104X
    [4X[28X<digraph with 81 vertices, 2430 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgroup := Group(Z(3));[127X[104X
    [4X[28X<group with 1 generators>[128X[104X
    [4X[25Xgap>[125X [27Xact := \*;[127X[104X
    [4X[28X<Operation "*">[128X[104X
    [4X[25Xgap>[125X [27Xdigraph := Digraph(group, List(f), act, adj);[127X[104X
    [4X[28X<digraph with 81 vertices, 2430 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.1-6 DigraphByAdjacencyMatrix[101X
  
  [33X[1;0Y[29X[2XDigraphByAdjacencyMatrix[102X( [3Xadj[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xadj[103X is the adjacency matrix of a digraph in the sense of [2XAdjacencyMatrix[102X
  ([14X5.2-1[114X), then this operation returns the digraph which is defined by [3Xadj[103X.[133X
  
  [33X[0;0YAlternatively,  if  [3Xadj[103X  is  a  square  boolean  matrix, then this operation
  returns  the  digraph  with [10XLength([110X[3Xadj[103X[10X)[110X vertices which has the edge [10X[i,j][110X if
  and only if [3Xadj[103X[10X[i][j][110X is [9Xtrue[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDigraphByAdjacencyMatrix([[127X[104X
    [4X[25X>[125X [27X[0, 1, 0, 2, 0],[127X[104X
    [4X[25X>[125X [27X[1, 1, 1, 0, 1],[127X[104X
    [4X[25X>[125X [27X[0, 3, 2, 1, 1],[127X[104X
    [4X[25X>[125X [27X[0, 0, 1, 0, 1],[127X[104X
    [4X[25X>[125X [27X[2, 0, 0, 0, 0]]);[127X[104X
    [4X[28X<multidigraph with 5 vertices, 18 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphByAdjacencyMatrix([[127X[104X
    [4X[25X>[125X [27X[true, false, true],[127X[104X
    [4X[25X>[125X [27X[false, false, true],[127X[104X
    [4X[25X>[125X [27X[false, true, false]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr);[127X[104X
    [4X[28X[ [ 1, 3 ], [ 3 ], [ 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.1-7 DigraphByEdges[101X
  
  [33X[1;0Y[29X[2XDigraphByEdges[102X( [3Xedges[103X[, [3Xn[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xedges[103X  is list of pairs of positive integers, then this function returns
  the  digraph with the minimum number of vertices [10Xm[110X such that its edges equal
  [3Xedges[103X.[133X
  
  [33X[0;0YIf  the optional second argument [3Xn[103X is a positive integer with [10X[3Xn[103X[10X >= m[110X (with [10Xm[110X
  defined  as  above),  then this function returns the digraph with [3Xn[103X vertices
  and edges [3Xedges[103X.[133X
  
  [33X[0;0YSee [2XDigraphEdges[102X ([14X5.1-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDigraphByEdges([127X[104X
    [4X[25X>[125X [27X[[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],[127X[104X
    [4X[25X>[125X [27X [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]]);[127X[104X
    [4X[28X<digraph with 6 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphByEdges([127X[104X
    [4X[25X>[125X [27X[[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],[127X[104X
    [4X[25X>[125X [27X [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);[127X[104X
    [4X[28X<digraph with 12 vertices, 10 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.1-8 EdgeOrbitsDigraph[101X
  
  [33X[1;0Y[29X[2XEdgeOrbitsDigraph[102X( [3XG[103X, [3Xedges[103X[, [3Xn[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA new digraph.[133X
  
  [33X[0;0YIf  [3XG[103X  is a permutation group, [3Xedges[103X is an edge or list of edges, and [3Xn[103X is a
  non-negative integer such that [3XG[103X fixes [10X[1 .. [3Xn[103X[10X][110X setwise, then this operation
  returns  a  new  digraph  with [3Xn[103X vertices and the union of the orbits of the
  edges  in  [3X  edges  [103X under the action of the permutation group [3XG[103X. An edge in
  this context is simply a pair of positive integers.[133X
  
  [33X[0;0YIf  the  optional  third  argument  [3Xn[103X is not present, then the largest moved
  point of the permutation group [3XG[103X is used by default.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph := EdgeOrbitsDigraph(Group((1, 3), (1, 2)(3, 4)),[127X[104X
    [4X[25X>[125X [27X                                [[1, 2], [4, 5]], 5);[127X[104X
    [4X[28X<digraph with 5 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(digraph);[127X[104X
    [4X[28X[ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4, 5 ], [ 1, 3, 5 ], [  ] ][128X[104X
    [4X[25Xgap>[125X [27XRepresentativeOutNeighbours(digraph);[127X[104X
    [4X[28X[ [ 2, 4, 5 ], [  ] ][128X[104X
  [4X[32X[104X
  
  [1X3.1-9 DigraphByInNeighbours[101X
  
  [33X[1;0Y[29X[2XDigraphByInNeighbours[102X( [3Xin[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphByInNeighbors[102X( [3Xin[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xin[103X  is  a  list  of  lists  of  positive  integers  in  the  range [10X[1 ..
  Length([3Xin[103X[10X)][110X,  then this function returns the digraph with vertices [22XE^0=[122X[10X[1 ..
  Length([3Xin[103X[10X)][110X,  and  edges corresponding to the entries of [3Xin[103X. More precisely,
  there is an edge with source vertex [10Xi[110X and range vertex [10Xj[110X if [10Xi[110X is in [10X[3Xin[103X[10X[j][110X.[133X
  
  [33X[0;0YIf  [10Xi[110X  occurs  in [10X[3Xin[103X[10X[j][110X with multiplicity [10Xk[110X, then there are [10Xk[110X multiple edges
  from [10Xi[110X to [10Xj[110X.[133X
  
  [33X[0;0YSee [2XInNeighbours[102X ([14X5.2-7[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphByInNeighbours([[127X[104X
    [4X[25X>[125X [27X[2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10],[127X[104X
    [4X[25X>[125X [27X[1], [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],[127X[104X
    [4X[25X>[125X [27X[1, 5, 9], [1, 2, 7, 8], [3, 5]]);[127X[104X
    [4X[28X<digraph with 10 vertices, 37 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphByInNeighbours([[2, 3, 2], [1], [1, 2, 3]]);[127X[104X
    [4X[28X<multidigraph with 3 vertices, 7 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.1-10 CayleyDigraph[101X
  
  [33X[1;0Y[29X[2XCayleyDigraph[102X( [3XG[103X[, [3Xgens[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YLet  [3XG[103X  be  any group and let [3Xgens[103X be a list of elements of [3XG[103X. This function
  returns  the  Cayley  graph of the group with respect [3Xgens[103X. The vertices are
  the elements of [3XG[103X. There exists an edge from the vertex [10Xu[110X to the vertex [10Xv[110X if
  and only if there exists a generator [10Xg[110X in [3Xgens[103X such that [10Xx * g = y[110X.[133X
  
  [33X[0;0YIf  the optional second argument [3Xgens[103X is not present, then the generators of
  [3XG[103X  are used by default. The digraph created by this operation belongs to the
  category  [2XIsCayleyDigraph[102X  ([14X3.1-2[114X),  the  group  [3XG[103X can be recovered from the
  digraph  using  [2XGroupOfCayleyDigraph[102X ([14X5.4-1[114X), and the generators [3Xgens[103X can be
  obtained using [2XGeneratorsOfCayleyDigraph[102X ([14X5.4-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := DihedralGroup(8);[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XCayleyDigraph(G);[127X[104X
    [4X[28X<digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XG := DihedralGroup(IsPermGroup, 8);[127X[104X
    [4X[28XGroup([ (1,2,3,4), (2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XCayleyDigraph(G);[127X[104X
    [4X[28X<digraph with 8 vertices, 16 edges>[128X[104X
    [4X[25Xgap>[125X [27Xdigraph := CayleyDigraph(G, [()]);[127X[104X
    [4X[28X<digraph with 8 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XGroupOfCayleyDigraph(digraph) = G;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfCayleyDigraph(digraph);[127X[104X
    [4X[28X[ () ][128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YChanging representations[133X[101X
  
  [1X3.2-1 AsBinaryRelation[101X
  
  [33X[1;0Y[29X[2XAsBinaryRelation[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA binary relation.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph  with  a  positive  number of vertices [22Xn[122X, and no
  multiple  edges, then this operation returns a binary relation on the points
  [10X[1..n][110X.  The pair [10X[i,j][110X is in the binary relation if and only if [10X[i,j][110X is an
  edge in [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[3, 2], [1, 2], [2], [3, 4]]);[127X[104X
    [4X[28X<digraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XAsBinaryRelation(gr);[127X[104X
    [4X[28XBinary Relation on 4 points[128X[104X
  [4X[32X[104X
  
  [1X3.2-2 AsDigraph[101X
  
  [33X[1;0Y[29X[2XAsDigraph[102X( [3Xtrans[103X[, [3Xn[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph, or [9Xfail[109X.[133X
  
  [33X[0;0YIf  [3Xtrans[103X is a transformation, and [3Xn[103X is a non-negative integer such that the
  restriction  of [3Xtrans[103X to [10X[1 .. [3Xn[103X[10X][110X defines a transformation of [10X[1 .. [3Xn[103X[10X][110X, then
  [10XAsDigraph[110X  returns  the functional digraph with [3Xn[103X vertices defined by [3Xtrans[103X.
  See [2XIsFunctionalDigraph[102X ([14X6.1-7[114X).[133X
  
  [33X[0;0YSpecifically, the digraph returned by [10XAsDigraph[110X has [3Xn[103X edges: for each vertex
  [10Xx[110X  in  [10X[1  ..  [3Xn[103X[10X][110X, there is a unique edge with source [10Xx[110X; this edge has range
  [10Xx^[3Xtrans[103X[10X[110X.[133X
  
  [33X[0;0YIf  the  optional  second argument [3Xn[103X is not supplied, then the degree of the
  transformation  [3Xtrans[103X  is used by default. If the restriction of [3Xtrans[103X to [10X[1
  .. [3Xn[103X[10X][110X does not define a transformation of [10X[1 .. [3Xn[103X[10X][110X, then [10XAsDigraph([3Xtrans[103X[10X, [3Xn[103X[10X)[110X
  returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := Transformation([4, 3, 3, 1, 7, 9, 10, 4, 2, 3]);[127X[104X
    [4X[28XTransformation( [ 4, 3, 3, 1, 7, 9, 10, 4, 2, 3 ] )[128X[104X
    [4X[25Xgap>[125X [27XAsDigraph(f);[127X[104X
    [4X[28X<digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XAsDigraph(f, 4);[127X[104X
    [4X[28X<digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XAsDigraph(f, 5);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X3.2-3 Graph[101X
  
  [33X[1;0Y[29X[2XGraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA [5XGrape[105X package graph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is a digraph without multiple edges, then this operation returns
  a [5XGrape[105X package graph that is isomorphic to [3Xdigraph[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  multidigraph, then since [5XGrape[105X does not support multiple
  edges, the multiple edges will be reduced to a single edge in the result. In
  order  words,  for  a  multidigraph  this  operation will return the same as
  [10XGraph(DigraphRemoveAllMultipleEdges([110X[3Xdigraph[103X[10X))[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPetersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,[127X[104X
    [4X[25X>[125X [27Xfunction(x, y) return Intersection(x, y) = []; end);[127X[104X
    [4X[28Xrec( adjacencies := [ [ 3, 5, 8 ] ], group := Group([ (1,2,3,5,7)[128X[104X
    [4X[28X  (4,6,8,9,10), (2,4)(6,9)(7,10) ]), isGraph := true, [128X[104X
    [4X[28X  names := [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], [128X[104X
    [4X[28X      [ 2, 4 ], [ 1, 5 ], [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ], [128X[104X
    [4X[28X  order := 10, representatives := [ 1 ], [128X[104X
    [4X[28X  schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27XDigraph(Petersen);[127X[104X
    [4X[28X<digraph with 10 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27XGraph(last);[127X[104X
    [4X[28Xrec( adjacencies := [ [ 3, 5, 8 ] ], group := Group([ (1,2,3,5,7)[128X[104X
    [4X[28X  (4,6,8,9,10), (2,4)(6,9)(7,10) ]), isGraph := true, [128X[104X
    [4X[28X  names := [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], [128X[104X
    [4X[28X      [ 2, 4 ], [ 1, 5 ], [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ], [128X[104X
    [4X[28X  order := 10, representatives := [ 1 ], [128X[104X
    [4X[28X  schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )[128X[104X
  [4X[32X[104X
  
  [1X3.2-4 AsGraph[101X
  
  [33X[1;0Y[29X[2XAsGraph[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA [5XGrape[105X package graph.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a digraph, then this method returns the same as [2XGraph[102X ([14X3.2-3[114X),
  except that the result will be stored as a mutable attribute of [3Xdigraph[103X.[133X
  
  [33X[0;0YIf [10XAsGraph([110X[3Xdigraph[103X[10X)[110X is called subsequently, then the same [5XGAP[105X object will be
  returned as before.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xd := Digraph([[1, 2], [3], []]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27Xg := AsGraph(d);[127X[104X
    [4X[28Xrec( adjacencies := [ [ 1, 2 ], [ 3 ], [  ] ], group := Group(()), [128X[104X
    [4X[28X  isGraph := true, names := [ 1 .. 3 ], order := 3, [128X[104X
    [4X[28X  representatives := [ 1, 2, 3 ], schreierVector := [ -1, -2, -3 ] )[128X[104X
  [4X[32X[104X
  
  [1X3.2-5 AsTransformation[101X
  
  [33X[1;0Y[29X[2XAsTransformation[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA transformation, or [9Xfail[109X[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  functional  digraph,  then  [10XAsTransformation[110X returns the
  transformation which is defined by [3Xdigraph[103X. See [2XIsFunctionalDigraph[102X ([14X6.1-7[114X).
  Otherwise, [10XAsTransformation([110X[3Xdigraph[103X[10X)[110X returns [9Xfail[109X.[133X
  
  [33X[0;0YIf    [3Xdigraph[103X    is   a   functional   digraph   with   [22Xn[122X   vertices,   then
  [10XAsTransformation([110X[3Xdigraph[103X[10X)[110X will return the transformation [10Xf[110X of degree at most
  [22Xn[122X  where  for  each [22X1 ≤ i ≤ n[122X, [10Xi ^ f[110X is equal to the unique out-neighbour of
  vertex [10Xi[110X in [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1], [3], [2]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(3);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XAsTransformation(gr);[127X[104X
    [4X[28XTransformation( [ 2, 3, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XAsPermutation(last);[127X[104X
    [4X[28X(1,2,3)[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 3], [], []]);[127X[104X
    [4X[28X<digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XAsTransformation(gr);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YNew digraphs from old[133X[101X
  
  [1X3.3-1 DigraphCopy[101X
  
  [33X[1;0Y[29X[2XDigraphCopy[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThis  function  returns  a  new  copy  of  [3Xdigraph[103X,  retaining  none  of the
  attributes or properties of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(10);[127X[104X
    [4X[28X<digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphCopy(gr) = gr;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-2 InducedSubdigraph[101X
  
  [33X[1;0Y[29X[2XInducedSubdigraph[102X( [3Xdigraph[103X, [3Xverts[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a digraph, and [3Xverts[103X is a subset of the vertices of [3Xdigraph[103X,
  then  this operation returns a digraph constructed from [3Xdigraph[103X by retaining
  precisely  those  vertices  in [3Xverts[103X, and those edges whose source and range
  vertices are both contained in [3Xverts[103X.[133X
  
  [33X[0;0YThe  vertices  of  the  induced  subdigraph  are  [10X[1..Length(verts)][110X but the
  original vertex labels can be accessed via [2XDigraphVertexLabels[102X ([14X5.1-9[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 1, 2, 3, 4, 4], [1, 3, 4], [3, 1], [1, 1]]);[127X[104X
    [4X[28X<multidigraph with 4 vertices, 13 edges>[128X[104X
    [4X[25Xgap>[125X [27XInducedSubdigraph(gr, [1, 3, 4]);[127X[104X
    [4X[28X<multidigraph with 3 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(last);[127X[104X
    [4X[28X[ 1 .. 3 ][128X[104X
  [4X[32X[104X
  
  [1X3.3-3 ReducedDigraph[101X
  
  [33X[1;0Y[29X[2XReducedDigraph[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThis  function  returns  a  digraph  isomorphic to the subdigraph of [3Xdigraph[103X
  induced  by the set of non-isolated vertices, i.e. the set of those vertices
  of  [3Xdigraph[103X  which  are  the  source  or  range of some edge in [3Xdigraph[103X. See
  [2XInducedSubdigraph[102X ([14X3.3-2[114X).[133X
  
  [33X[0;0YThe  vertex  and edge labels of the graph are preserved. A vertex in the new
  digraph  can  be matched to the corresponding vertex in [3Xdigraph[103X by using the
  label.[133X
  
  [33X[0;0YThe ordering of the vertices is preserved.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xd := Digraph([[1, 2], [], [], [1, 4], []]);[127X[104X
    [4X[28X<digraph with 5 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27Xr := ReducedDigraph(d);[127X[104X
    [4X[28X<digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(r);[127X[104X
    [4X[28X[ [ 1, 2 ], [  ], [ 1, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(d);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 1, 2 ], [ 4, 1 ], [ 4, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(r);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabel(r, 3);[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabel(r, 2);[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X3.3-4 MaximalSymmetricSubdigraph[101X
  
  [33X[1;0Y[29X[2XMaximalSymmetricSubdigraph[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XMaximalSymmetricSubdigraphWithoutLoops[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a digraph, then [10XMaximalSymmetricSubdigraph[110X returns a symmetric
  digraph without multiple edges which has the same vertex set as [3Xdigraph[103X, and
  whose  edge  list  is  formed  from  [3Xdigraph[103X by ignoring the multiplicity of
  edges,  and  by  ignoring edges [10X[u,v][110X for which there does not exist an edge
  [10X[v,u][110X.[133X
  
  [33X[0;0YThe  digraph returned by [10XMaximalSymmetricSubdigraphWithoutLoops[110X is the same,
  except that loops are removed.[133X
  
  [33X[0;0YSee [2XIsSymmetricDigraph[102X ([14X6.1-10[114X), [2XIsMultiDigraph[102X ([14X6.1-8[114X), and [2XDigraphHasLoops[102X
  ([14X6.1-1[114X) for more information.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 2], [1, 3], [4], [3, 1]]);[127X[104X
    [4X[28X<multidigraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnot IsSymmetricDigraph(gr) and IsMultiDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr);[127X[104X
    [4X[28X[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xsym := MaximalSymmetricSubdigraph(gr);[127X[104X
    [4X[28X<digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(sym) and not IsMultiDigraph(sym);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(sym);[127X[104X
    [4X[28X[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-5 MaximalAntiSymmetricSubdigraph[101X
  
  [33X[1;0Y[29X[2XMaximalAntiSymmetricSubdigraph[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph,  then  [10XMaximalAntiSymmetricSubdigraph[110X returns a
  anti-symmetric subdigraph of [3Xdigraph[103X which does not have multiple edges, has
  the  same  vertex set as [3Xdigraph[103X, and whose edge list is formed from [3Xdigraph[103X
  by ignoring the multiplicity of edges, and by having either an edge from the
  vertex  [10Xu[110X  to  the vertex [10Xv[110X, or the edge from [10Xv[110X to [10Xu[110X (but not both) whenever
  both edges belong to [3Xdigraph[103X.[133X
  
  [33X[0;0YSee [2XIsAntisymmetricDigraph[102X ([14X6.1-2[114X) for more information.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 2], [1, 3], [4], [3, 1]]);[127X[104X
    [4X[28X<multidigraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnot IsAntiSymmetricDigraph(D) and IsMultiDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := MaximalAntiSymmetricSubdigraph(D);[127X[104X
    [4X[28X<digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsAntiSymmetricDigraph(D) and not IsMultiDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(D);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-6 UndirectedSpanningTree[101X
  
  [33X[1;0Y[29X[2XUndirectedSpanningTree[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XUndirectedSpanningForest[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph, or [9Xfail[109X.[133X
  
  [33X[0;0YIf    [3Xdigraph[103X    is   a   digraph   with   at   least   one   vertex,   then
  [10XUndirectedSpanningForest[110X  returns  an undirected spanning forest of [3Xdigraph[103X,
  otherwise   this  attribute  returns  [9Xfail[109X.  See  [2XIsUndirectedSpanningForest[102X
  ([14X4.1-2[114X) for the definition of an undirected spanning forest.[133X
  
  [33X[0;0YIf   [3Xdigraph[103X   is   a   digraph   with   at   least  one  vertex  and  whose
  [2XMaximalSymmetricSubdigraph[102X  ([14X3.3-4[114X)  is  connected  (see  [2XIsConnectedDigraph[102X
  ([14X6.3-3[114X)), then [10XUndirectedSpanningTree[110X returns an undirected spanning tree of
  [3Xdigraph[103X, otherwise this attribute returns [9Xfail[109X. See [2XIsUndirectedSpanningTree[102X
  ([14X4.1-2[114X) for the definition of an undirected spanning tree.[133X
  
  [33X[0;0YNote  that for a digraph that has an undirected spanning tree, the attribute
  [10XUndirectedSpanningTree[110X   returns   the   same   digraph   as  the  attribute
  [10XUndirectedSpanningForest[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]]);[127X[104X
    [4X[28X<multidigraph with 4 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XUndirectedSpanningTree(gr);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xforest := UndirectedSpanningForest(gr);[127X[104X
    [4X[28X<digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(forest);[127X[104X
    [4X[28X[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsUndirectedSpanningForest(gr, forest);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDigraphConnectedComponents(forest).comps;[127X[104X
    [4X[28X[ [ 1, 2 ], [ 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphConnectedComponents(MaximalSymmetricSubdigraph(gr)).comps;[127X[104X
    [4X[28X[ [ 1, 2 ], [ 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XUndirectedSpanningForest(MaximalSymmetricSubdigraph(gr))[127X[104X
    [4X[25X>[125X [27X= forest;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := CompleteDigraph(4);[127X[104X
    [4X[28X<digraph with 4 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27Xtree := UndirectedSpanningTree(gr);[127X[104X
    [4X[28X<digraph with 4 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsUndirectedSpanningTree(gr, tree);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xtree = UndirectedSpanningForest(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUndirectedSpanningForest(EmptyDigraph(0));[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X3.3-7 QuotientDigraph[101X
  
  [33X[1;0Y[29X[2XQuotientDigraph[102X( [3Xdigraph[103X, [3Xp[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a digraph, and [3Xp[103X is a partition of the vertices of [3Xdigraph[103X,
  then  this  operation  returns a new digraph constructed by amalgamating all
  vertices of [3Xdigraph[103X which lie in the same part of [3Xp[103X.[133X
  
  [33X[0;0YA  partition  of  the  vertices  of  [3Xdigraph[103X is a list of non-empty disjoint
  lists,  such  that the union of all the sub-lists is equal to the vertex set
  of  [3Xdigraph[103X.  In  particular,  each  vertex  must  appear  in  precisely one
  sub-list.[133X
  
  [33X[0;0YThe vertices of [3Xdigraph[103X in part [10Xi[110X of [3Xp[103X will become vertex [10Xi[110X in the quotient,
  and  every edge of [3Xdigraph[103X with source in part [10Xi[110X and range in part [10Xj[110X becomes
  an  edge  from  [10Xi[110X  to  [10Xj[110X in the quotient. In particular, this means that the
  quotient of a digraph without multiple edges can have multiple edges.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 1], [4], [1], [1, 3, 4]]);[127X[104X
    [4X[28X<digraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(gr);[127X[104X
    [4X[28X[ 1 .. 4 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 1 ], [ 2, 4 ], [ 3, 1 ], [ 4, 1 ], [ 4, 3 ], [128X[104X
    [4X[28X  [ 4, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xp := [[1], [2, 4], [3]];[127X[104X
    [4X[28X[ [ 1 ], [ 2, 4 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xqr := QuotientDigraph(gr, p);[127X[104X
    [4X[28X<multidigraph with 3 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(qr);[127X[104X
    [4X[28X[ 1 .. 3 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(qr);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 1 ], [ 2, 2 ], [ 2, 1 ], [ 2, 3 ], [ 2, 2 ], [128X[104X
    [4X[28X  [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XQuotientDigraph(EmptyDigraph(0), []);[127X[104X
    [4X[28X<digraph with 0 vertices, 0 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-8 DigraphReverse[101X
  
  [33X[1;0Y[29X[2XDigraphReverse[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a digraph, then this operation returns a digraph constructed
  from [3Xdigraph[103X by reversing the orientation of every edge.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);[127X[104X
    [4X[28X<digraph with 5 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphReverse(gr);[127X[104X
    [4X[28X<digraph with 5 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(last);[127X[104X
    [4X[28X[ [ 2, 3, 4 ], [ 4, 5 ], [ 1, 2, 5 ], [ 4 ], [ 2, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 4], [1], [4], [3, 4]]);[127X[104X
    [4X[28X<digraph with 4 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 4 ], [ 2, 1 ], [ 3, 4 ], [ 4, 3 ], [ 4, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(DigraphReverse(gr));[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 1 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ], [ 4, 4 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-9 DigraphDual[101X
  
  [33X[1;0Y[29X[2XDigraphDual[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a digraph without multiple edges, then this returns the [13Xdual[113X
  of [3Xdigraph[103X. The [13Xdual[113X is sometimes called the [13Xcomplement[113X.[133X
  
  [33X[0;0YThe  [13Xdual[113X  of [3Xdigraph[103X has the same vertices as [3Xdigraph[103X, and there is an edge
  in the dual from [10Xi[110X to [10Xj[110X whenever there is no edge from [10Xi[110X to [10Xj[110X in [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 3], [], [4, 6], [5], [],[127X[104X
    [4X[25X>[125X [27X[7, 8, 9], [], [], []]);[127X[104X
    [4X[28X<digraph with 9 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphDual(gr);[127X[104X
    [4X[28X<digraph with 9 vertices, 73 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-10 DigraphSymmetricClosure[101X
  
  [33X[1;0Y[29X[2XDigraphSymmetricClosure[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph, then this attribute gives the minimal symmetric
  digraph which has the same vertices and contains all the edges of [3Xdigraph[103X.[133X
  
  [33X[0;0YA  digraph  is  [13Xsymmetric[113X if its adjacency matrix [2XAdjacencyMatrix[102X ([14X5.2-1[114X) is
  symmetric.  For  a digraph with multiple edges this means that there are the
  same number of edges from a vertex [10Xu[110X to a vertex [10Xv[110X as there are from [10Xv[110X to [10Xu[110X;
  see [2XIsSymmetricDigraph[102X ([14X6.1-10[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2, 3], [2, 4], [1], [3, 4]]);[127X[104X
    [4X[28X<digraph with 4 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr1 := DigraphSymmetricClosure(gr);[127X[104X
    [4X[28X<digraph with 4 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(gr1);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XList(OutNeighbours(gr1), AsSet);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 2], [1]]);[127X[104X
    [4X[28X<multidigraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr1 := DigraphSymmetricClosure(gr);[127X[104X
    [4X[28X<multidigraph with 2 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr1);[127X[104X
    [4X[28X[ [ 2, 2 ], [ 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-11 DigraphReflexiveTransitiveClosure[101X
  
  [33X[1;0Y[29X[2XDigraphReflexiveTransitiveClosure[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDigraphTransitiveClosure[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a digraph with no multiple edges, then these attributes return
  the (reflexive) transitive closure of [3Xdigraph[103X.[133X
  
  [33X[0;0YA  digraph  is  [13Xreflexive[113X  if  it  has  a  loop  at  every vertex, and it is
  [13Xtransitive[113X  if  whenever [10X[i,j][110X and [10X[j,k][110X are edges of [3Xdigraph[103X, [10X[i,k][110X is also
  an  edge.  The  [13X(reflexive)  transitive  closure[113X of a digraph [3Xdigraph[103X is the
  least (reflexive and) transitive digraph containing [3Xdigraph[103X.[133X
  
  [33X[0;0YLet  [22Xn[122X  be  the  number  of  vertices of [3Xdigraph[103X, and let [22Xm[122X be the number of
  edges.  For an arbitrary digraph, these attributes will use a version of the
  Floyd-Warshall   algorithm,   with   complexity   [22XO(n^3)[122X.   However,  for  a
  topologically  sortable  digraph [see [2XDigraphTopologicalSort[102X ([14X5.1-7[114X)], these
  attributes  will  use  methods with complexity [22XO(m + n + m ⋅ n)[122X when this is
  faster.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphFromDiSparse6String(".H`eOWR`Ul^");[127X[104X
    [4X[28X<digraph with 9 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsReflexiveDigraph(gr) or IsTransitiveDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr);[127X[104X
    [4X[28X[ [ 4, 6 ], [ 1, 3 ], [  ], [ 5 ], [  ], [ 7, 8, 9 ], [  ], [  ], [128X[104X
    [4X[28X  [  ] ][128X[104X
    [4X[25Xgap>[125X [27Xtrans := DigraphTransitiveClosure(gr);[127X[104X
    [4X[28X<digraph with 9 vertices, 18 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(trans);[127X[104X
    [4X[28X[ [ 4, 5, 6, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ], [  ], [ 5 ], [128X[104X
    [4X[28X  [  ], [ 7, 8, 9 ], [  ], [  ], [  ] ][128X[104X
    [4X[25Xgap>[125X [27Xreflextrans := DigraphReflexiveTransitiveClosure(gr);[127X[104X
    [4X[28X<digraph with 9 vertices, 27 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(reflextrans);[127X[104X
    [4X[28X[ [ 1, 4, 5, 6, 7, 8, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 3 ], [128X[104X
    [4X[28X  [ 4, 5 ], [ 5 ], [ 6, 7, 8, 9 ], [ 7 ], [ 8 ], [ 9 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-12 DigraphReflexiveTransitiveReduction[101X
  
  [33X[1;0Y[29X[2XDigraphReflexiveTransitiveReduction[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphTransitiveReduction[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is a topologically sortable digraph [see [2XDigraphTopologicalSort[102X
  ([14X5.1-7[114X)]   with   no  multiple  edges,  then  these  operations  return  the
  (reflexive) transitive reduction of [3Xdigraph[103X.[133X
  
  [33X[0;0YThe  (reflexive)  transitive reduction of such a digraph is the unique least
  subgraph  such  that  the  (reflexive) transitive closure of the subgraph is
  equal    to   the   (reflexive)   transitive   closure   of   [3Xdigraph[103X   [see
  [2XDigraphReflexiveTransitiveClosure[102X ([14X3.3-11[114X)]. In order words, it is the least
  subgraph of [3Xdigraph[103X which retains the same reachability as [3Xdigraph[103X.[133X
  
  [33X[0;0YLet  [22Xn[122X  be  the number of vertices of an arbitrary digraph, and let [22Xm[122X be the
  number of edges. Then these operations use methods with complexity [22XO(m + n +
  m ⋅ n)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2, 3], [3], [3]]);;[127X[104X
    [4X[25Xgap>[125X [27XDigraphHasLoops(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr1 := DigraphReflexiveTransitiveReduction(gr);[127X[104X
    [4X[28X<digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphHasLoops(gr1);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr1);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ], [  ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphTransitiveReduction(gr);[127X[104X
    [4X[28X<digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphHasLoops(gr2);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr2);[127X[104X
    [4X[28X[ [ 2, 1 ], [ 3 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphReflexiveTransitiveClosure(gr)[127X[104X
    [4X[25X>[125X [27X = DigraphReflexiveTransitiveClosure(gr1);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDigraphTransitiveClosure(gr)[127X[104X
    [4X[25X>[125X [27X = DigraphTransitiveClosure(gr2);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-13 DigraphAddVertex[101X
  
  [33X[1;0Y[29X[2XDigraphAddVertex[102X( [3Xdigraph[103X[, [3Xlabel[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThe  operation  returns  a  new digraph constructed from [3Xdigraph[103X by adding a
  single new vertex.[133X
  
  [33X[0;0YIf  the  optional second argument [3Xlabel[103X is a [5XGAP[105X object, then the new vertex
  will be labelled [3Xlabel[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CompleteDigraph(3);[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertex(gr);[127X[104X
    [4X[28X<digraph with 4 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(new);[127X[104X
    [4X[28X[ 1 .. 4 ][128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertex(gr, Group([(1, 2)]));[127X[104X
    [4X[28X<digraph with 4 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(new);[127X[104X
    [4X[28X[ 1, 2, 3, Group([ (1,2) ]) ][128X[104X
  [4X[32X[104X
  
  [1X3.3-14 DigraphAddVertices[101X
  
  [33X[1;0Y[29X[2XDigraphAddVertices[102X( [3Xdigraph[103X, [3Xm[103X[, [3Xlabels[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YFor  a  non-negative  integer  [3Xm[103X,  this  operation  returns  a  new  digraph
  constructed from [3Xdigraph[103X by adding [3Xm[103X new vertices.[133X
  
  [33X[0;0YIf  the  optional  third argument [3Xlabels[103X is a list of length [3Xm[103X consisting of
  [5XGAP[105X objects, then the new vertices will be labelled according to this list.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CompleteDigraph(3);[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertices(gr, 3);[127X[104X
    [4X[28X<digraph with 6 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertices(new);[127X[104X
    [4X[28X[ 1 .. 6 ][128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphAddVertices(gr, 2, [Group([(1, 2)]), "d"]);[127X[104X
    [4X[28X<digraph with 5 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(new);[127X[104X
    [4X[28X[ 1, 2, 3, Group([ (1,2) ]), "d" ][128X[104X
    [4X[25Xgap>[125X [27XDigraphAddVertices(gr, 0) = gr;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-15 DigraphAddEdge[101X
  
  [33X[1;0Y[29X[2XDigraphAddEdge[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xedge[103X is a pairs of vertices of [3Xdigraph[103X, then this operation returns a new
  digraph  constructed  from  [3Xdigraph[103X by adding a new edge with source [3Xedge[103X[10X[1][110X
  and range [3Xedge[103X[10X[2][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := Digraph([[2], [3], []]);[127X[104X
    [4X[28X<digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr1);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphAddEdge(gr1, [3, 1]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr2);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr3 := DigraphAddEdge(gr2, [2, 3]);[127X[104X
    [4X[28X<multidigraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr3);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-16 DigraphAddEdgeOrbit[101X
  
  [33X[1;0Y[29X[2XDigraphAddEdgeOrbit[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA new digraph.[133X
  
  [33X[0;0YThis  operation  returns  a  new digraph with the same vertices and edges as
  [3Xdigraph[103X  and  with additional edges consisting of the orbit of the edge [3Xedge[103X
  under  the action of the [2XDigraphGroup[102X ([14X7.2-9[114X) of [3Xdigraph[103X. If [3Xedge[103X is already
  an edge in [3Xdigraph[103X, then [3Xdigraph[103X is returns unchanged.[133X
  
  [33X[0;0YAn edge is simply a pair of vertices of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := CayleyDigraph(DihedralGroup(8));[127X[104X
    [4X[28X<digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);[127X[104X
    [4X[28X<digraph with 8 vertices, 32 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr1);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 8 ], [ 2, 6 ], [128X[104X
    [4X[28X  [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [128X[104X
    [4X[28X  [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [128X[104X
    [4X[28X  [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr2);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 8 ], [128X[104X
    [4X[28X  [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], [128X[104X
    [4X[28X  [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], [128X[104X
    [4X[28X  [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], [128X[104X
    [4X[28X  [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], [128X[104X
    [4X[28X  [ 8, 5 ], [ 8, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr3 := DigraphRemoveEdgeOrbit(gr2, [1, 8]);[127X[104X
    [4X[28X<digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr3 = gr1;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-17 DigraphAddEdges[101X
  
  [33X[1;0Y[29X[2XDigraphAddEdges[102X( [3Xdigraph[103X, [3Xedges[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xedges[103X  is  a (possibly empty) list of pairs of vertices of [3Xdigraph[103X, then
  this  operation returns a new digraph constructed from [3Xdigraph[103X by adding the
  edges  specified  by  [3Xedges[103X.  More precisely, for every [10Xedge[110X in [3Xedges[103X, a new
  edge will be added with source [10Xedge[1][110X and range [10Xedges[2][110X.[133X
  
  [33X[0;0YIf an edge is included in [3Xedges[103X with multiplicity [10Xk[110X, then it will be added [10Xk[110X
  times.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfunc := function(n)[127X[104X
    [4X[25X>[125X [27X local source, range, i;[127X[104X
    [4X[25X>[125X [27X source := [];[127X[104X
    [4X[25X>[125X [27X range  := [];[127X[104X
    [4X[25X>[125X [27X for i in [1 .. n - 2] do[127X[104X
    [4X[25X>[125X [27X   Add(source, i);[127X[104X
    [4X[25X>[125X [27X   Add(range, i + 1);[127X[104X
    [4X[25X>[125X [27X od;[127X[104X
    [4X[25X>[125X [27X return Digraph(n, source, range);[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27Xgr := func(1024);[127X[104X
    [4X[28X<digraph with 1024 vertices, 1022 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphAddEdges(gr,[127X[104X
    [4X[25X>[125X [27X[[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);[127X[104X
    [4X[28X<multidigraph with 1024 vertices, 1026 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-18 DigraphRemoveVertex[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveVertex[102X( [3Xdigraph[103X, [3Xv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xv[103X  is  a  vertex  of  [3Xdigraph[103X, then this operation returns a new digraph
  constructed  from  [3Xdigraph[103X  by  removing vertex [3Xv[103X, along with any edge whose
  source or range vertex is [3Xv[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  has  [10Xn[110X  vertices,  then  the  vertices  of  the new digraph are
  [10X[1..n-1][110X,  but  the  original labels can be accessed via [2XDigraphVertexLabels[102X
  ([14X5.1-9[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph(["a", "b", "c"],[127X[104X
    [4X[25X>[125X [27X                 ["a", "a", "b", "c", "c"],[127X[104X
    [4X[25X>[125X [27X                 ["b", "c", "a", "a", "c"]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(gr);[127X[104X
    [4X[28X[ "a", "b", "c" ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 3, 1 ], [ 3, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphRemoveVertex(gr, 2);[127X[104X
    [4X[28X<digraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(new);[127X[104X
    [4X[28X[ "a", "c" ][128X[104X
  [4X[32X[104X
  
  [1X3.3-19 DigraphRemoveVertices [101X
  
  [33X[1;0Y[29X[2XDigraphRemoveVertices [102X( [3Xdigraph[103X, [3Xverts[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xverts[103X  is a (possibly empty) duplicate-free list of vertices of [3Xdigraph[103X,
  then  this  operation  returns  a  new  digraph  constructed from [3Xdigraph[103X by
  removing  every  vertex  in [3Xverts[103X, along with any edge whose source or range
  vertex is in [3Xverts[103X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  has  [10Xn[110X vertices, then the vertices of the new digraph are [10X[1 ..
  n-Length([3Xverts[103X[10X)][110X,   but   the   original   labels   can   be   accessed  via
  [2XDigraphVertexLabels[102X ([14X5.1-9[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);[127X[104X
    [4X[28X<digraph with 5 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XSetDigraphVertexLabels(gr, ["a", "b", "c", "d", "e"]);[127X[104X
    [4X[25Xgap>[125X [27Xnew := DigraphRemoveVertices(gr, [2, 4]);[127X[104X
    [4X[28X<digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphVertexLabels(new);[127X[104X
    [4X[28X[ "a", "c", "e" ][128X[104X
  [4X[32X[104X
  
  [1X3.3-20 DigraphRemoveEdge[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveEdge[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf one of the following holds:[133X
  
  [30X    [33X[0;6Y[3Xdigraph[103X  is  a  digraph  with no multiple edges, and [3Xedge[103X is a pair of
        vertices of [3Xdigraph[103X, or[133X
  
  [30X    [33X[0;6Y[3Xdigraph[103X is any digraph and [3Xedge[103X is the index of an edge of [3Xdigraph[103X,[133X
  
  [33X[0;0Ythen  this  operation  returns  a  new  digraph  constructed from [3Xdigraph[103X by
  removing  the  edges  specified by [3Xedges[103X. If, in the first case, the pair of
  vertices  [3Xedge[103X  does  not  specify  an  edge  of [3Xdigraph[103X, then a new copy of
  [3Xdigraph[103X will be returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(250000);[127X[104X
    [4X[28X<digraph with 250000 vertices, 250000 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphRemoveEdge(gr, [250000, 1]);[127X[104X
    [4X[28X<digraph with 250000 vertices, 249999 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphRemoveEdge(gr, 10);[127X[104X
    [4X[28X<digraph with 250000 vertices, 249998 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-21 DigraphRemoveEdgeOrbit[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveEdgeOrbit[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA new digraph.[133X
  
  [33X[0;0YThis  operation  returns a new digraph with the same vertices as [3Xdigraph[103X and
  with  the  orbit  of  the  edge  [3Xedge[103X  (under the action of the [2XDigraphGroup[102X
  ([14X7.2-9[114X) of [3Xdigraph[103X) removed. If [3Xedge[103X is not an edge in [3Xdigraph[103X, then [3Xdigraph[103X
  is returned unchanged.[133X
  
  [33X[0;0YAn edge is simply a pair of vertices of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := CayleyDigraph(DihedralGroup(8));[127X[104X
    [4X[28X<digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);[127X[104X
    [4X[28X<digraph with 8 vertices, 32 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr1);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 8 ], [ 2, 6 ], [128X[104X
    [4X[28X  [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [128X[104X
    [4X[28X  [ 5, 3 ], [ 5, 2 ], [ 5, 8 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [128X[104X
    [4X[28X  [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 8, 7 ], [ 8, 6 ], [ 8, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr2);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 8 ], [128X[104X
    [4X[28X  [ 2, 6 ], [ 2, 3 ], [ 3, 5 ], [ 3, 4 ], [ 3, 7 ], [ 3, 2 ], [128X[104X
    [4X[28X  [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 3 ], [ 5, 2 ], [128X[104X
    [4X[28X  [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 5 ], [ 6, 2 ], [ 6, 7 ], [128X[104X
    [4X[28X  [ 7, 8 ], [ 7, 1 ], [ 7, 3 ], [ 7, 6 ], [ 8, 7 ], [ 8, 6 ], [128X[104X
    [4X[28X  [ 8, 5 ], [ 8, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr3 := DigraphRemoveEdgeOrbit(gr2, [1, 8]);[127X[104X
    [4X[28X<digraph with 8 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr3 = gr1;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-22 DigraphRemoveEdges[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveEdges[102X( [3Xdigraph[103X, [3Xedges[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf one of the following holds:[133X
  
  [30X    [33X[0;6Y[3Xdigraph[103X  is  a  digraph with no multiple edges, and [3Xedges[103X is a list of
        pairs of vertices of [3Xdigraph[103X, or[133X
  
  [30X    [33X[0;6Y[3Xdigraph[103X  is  any  digraph  and  [3Xedges[103X is a list of indices of edges of
        [3Xdigraph[103X,[133X
  
  [33X[0;0Ythen  this  operation  returns  a  new  digraph  constructed from [3Xdigraph[103X by
  removing  all  of  the  edges  specified  by  [3Xedges[103X  [see  [2XDigraphRemoveEdge[102X
  ([14X3.3-20[114X)].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(250000);[127X[104X
    [4X[28X<digraph with 250000 vertices, 250000 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphRemoveEdges(gr, [[250000, 1]]);[127X[104X
    [4X[28X<digraph with 250000 vertices, 249999 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphRemoveEdges(gr, [10]);[127X[104X
    [4X[28X<digraph with 250000 vertices, 249998 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-23 DigraphRemoveLoops[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveLoops[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph,  then  this  operation  returns  a  new digraph
  constructed  from  [3Xdigraph[103X  by  removing  every loop. A loop is an edge with
  equal source and range.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2, 4], [1, 4], [3, 4], [1, 4, 5], [1, 5]]);[127X[104X
    [4X[28X<digraph with 5 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphRemoveLoops(gr);[127X[104X
    [4X[28X<digraph with 5 vertices, 8 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-24 DigraphRemoveAllMultipleEdges[101X
  
  [33X[1;0Y[29X[2XDigraphRemoveAllMultipleEdges[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph,  then  this  operation  returns  a  new digraph
  constructed  from  [3Xdigraph[103X by removing all multiple edges. The result is the
  largest subdigraph of [3Xdigraph[103X which does not contain multiple edges.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := Digraph([[1, 2, 3, 2], [1, 1, 3], [2, 2, 2]]);[127X[104X
    [4X[28X<multidigraph with 3 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphRemoveAllMultipleEdges(gr1);[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr2);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 3 ], [ 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-25 DigraphReverseEdges[101X
  
  [33X[1;0Y[29X[2XDigraphReverseEdges[102X( [3Xdigraph[103X, [3Xedges[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDigraphReverseEdge[102X( [3Xdigraph[103X, [3Xedge[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a digraph without multiple edges, and [3Xedges[103X is either:[133X
  
  [30X    [33X[0;6Ya  list  of  pairs  of  vertices  of [3Xdigraph[103X (the entries of each pair
        corresponding to the source and the range of an edge, respectively),[133X
  
  [30X    [33X[0;6Ya list of positions of elements in the list [2XDigraphEdges[102X ([14X5.1-3[114X),[133X
  
  [33X[0;0Ythen  [10XDigraphReverseEdges[110X  returns a new digraph constructed from [3Xdigraph[103X by
  reversing the orientation of every edge specified by [3Xedges[103X. If only one edge
  is  to  be  reversed,  then  [10XDigraphReverseEdge[110X can be used instead. In this
  case,  the  second  argument should just be a single vertex-pair or a single
  position.[133X
  
  [33X[0;0YNote  that  even  though  [3Xdigraph[103X cannot have multiple edges, the output may
  have multiple edges.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphFromDiSparse6String(".Tg?i@s?t_e?_qEsC");[127X[104X
    [4X[28X<digraph with 21 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 7, 19 ], [ 9, 1 ], [128X[104X
    [4X[28X  [ 11, 2 ], [ 21, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphReverseEdges(gr, [1, 2, 4]);[127X[104X
    [4X[28X<digraph with 21 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr = DigraphReverseEdges(gr2, [[7, 1], [2, 1], [21, 5]]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := DigraphReverseEdge(gr, 5);[127X[104X
    [4X[28X<digraph with 21 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 = DigraphReverseEdge(gr, [7, 19]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-26 DigraphDisjointUnion[101X
  
  [33X[1;0Y[29X[2XDigraphDisjointUnion[102X( [3Xgr1[103X, [3Xgr2[103X, [3X...[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphDisjointUnion[102X( [3Xlist[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIn the first form, if [3Xgr1[103X, [3Xgr2[103X, etc. are digraphs, then [10XDigraphDisjointUnion[110X
  returns  their  disjoint  union.  In the second form, if [3Xlist[103X is a non-empty
  list  of  digraphs,  then [10XDigraphDisjointUnion[110X returns the disjoint union of
  the digraphs contained in the list.[133X
  
  [33X[0;0YFor  a  disjoint  union of digraphs, the vertex set is the disjoint union of
  the vertex sets, and the edge list is the disjoint union of the edge lists.[133X
  
  [33X[0;0YMore  specifically, for a collection of digraphs [3Xgr1[103X, [3Xgr2[103X, [10X...[110X, the disjoint
  union  with  have  [10XDigraphNrVertices([110X[3Xgr1[103X[10X)[110X  [10X+[110X  [10XDigraphNrVertices([110X[3Xgr2[103X[10X)[110X  [10X+[110X  [10X...[110X
  vertices.  The  edges  of [3Xgr1[103X will remain unchanged, whilst the edges of the
  [10Xi[110Xth  digraph,  [3Xgr[103X[10X[i][110X, will be changed so that they belong to the vertices of
  the disjoint union corresponding to [3Xgr[103X[10X[i][110X. In particular, the edges of [3Xgr[103X[10X[i][110X
  will have their source and range increased by [10XDigraphNrVertices([110X[3Xgr1[103X[10X)[110X [10X+[110X [10X...[110X [10X+[110X
  [10XDigraphNrVertices([110X[3Xgr[103X[10X[i-1])[110X.[133X
  
  [33X[0;0YNote that previously set [2XDigraphVertexLabels[102X ([14X5.1-9[114X) will be lost.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := CycleDigraph(3);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr1);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ], [ 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr2 := CompleteDigraph(3);[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr2);[127X[104X
    [4X[28X[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xunion := DigraphDisjointUnion(gr1, gr2);[127X[104X
    [4X[28X<digraph with 6 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(union);[127X[104X
    [4X[28X[ [ 2 ], [ 3 ], [ 1 ], [ 5, 6 ], [ 4, 6 ], [ 4, 5 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-27 DigraphEdgeUnion[101X
  
  [33X[1;0Y[29X[2XDigraphEdgeUnion[102X( [3Xgr1[103X, [3Xgr2[103X, [3X...[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphEdgeUnion[102X( [3Xlist[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIn  the  first  form,  if [3Xgr1[103X, [3Xgr2[103X, etc. are digraphs, then [10XDigraphEdgeUnion[110X
  returns their edge union. In the second form, if [3Xlist[103X is a non-empty list of
  digraphs,  then  [10XDigraphEdgeUnion[110X  returns  the  edge  union of the digraphs
  contained in the list.[133X
  
  [33X[0;0YThe vertex set of the edge union of a collection of digraphs is the [13Xunion[113X of
  the vertex sets, whilst the edge list of the edge union is the [13Xconcatenation[113X
  of  the edge lists. The number of vertices of the edge union is equal to the
  [13Xmaximum[113X  number  of  vertices  of  one of the digraphs, whilst the number of
  edges  of  the  edge union will equal the [13Xsum[113X of the number of edges of each
  digraph.[133X
  
  [33X[0;0YNote that previously set [2XDigraphVertexLabels[102X ([14X5.1-9[114X) will be lost.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(10);[127X[104X
    [4X[28X<digraph with 10 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdgeUnion(gr, gr);[127X[104X
    [4X[28X<multidigraph with 10 vertices, 20 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr1 := Digraph([[2], [1]]);[127X[104X
    [4X[28X<digraph with 2 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := Digraph([[2, 3], [2], [1]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27Xunion := DigraphEdgeUnion(gr1, gr2);[127X[104X
    [4X[28X<multidigraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(union);[127X[104X
    [4X[28X[ [ 2, 2, 3 ], [ 1, 2 ], [ 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xunion = DigraphByEdges([127X[104X
    [4X[25X>[125X [27XConcatenation(DigraphEdges(gr1), DigraphEdges(gr2)));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-28 DigraphJoin[101X
  
  [33X[1;0Y[29X[2XDigraphJoin[102X( [3Xgr1[103X, [3Xgr2[103X, [3X...[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphJoin[102X( [3Xlist[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIn  the first form, if [3Xgr1[103X, [3Xgr2[103X, etc. are digraphs, then [10XDigraphJoin[110X returns
  their  join.  In  the  second form, if [3Xlist[103X is a non-empty list of digraphs,
  then [10XDigraphJoin[110X returns the join of the digraphs contained in the list.[133X
  
  [33X[0;0YThe join of a collection of digraphs [3Xgr1[103X, [3Xgr2[103X, [10X...[110X is formed by first taking
  the  [2XDigraphDisjointUnion[102X ([14X3.3-26[114X) of the collection. In the disjoint union,
  if  [22Xi ≠ j[122X then there are no edges between vertices corresponding to digraphs
  [3Xgr[103X[10X[i][110X and [3Xgr[103X[10X[j][110X in the collection; the join is created by including all such
  edges.[133X
  
  [33X[0;0YFor example, the join of two empty digraphs is a complete bipartite digraph.[133X
  
  [33X[0;0YNote that previously set [2XDigraphVertexLabels[102X ([14X5.1-9[114X) will be lost.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CompleteDigraph(3);[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteDigraph(DigraphJoin(gr, gr));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := CycleDigraph(3);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphJoin(gr, gr2);[127X[104X
    [4X[28X<digraph with 6 vertices, 27 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-29 LineDigraph[101X
  
  [33X[1;0Y[29X[2XLineDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XEdgeDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YGiven  a  digraph  [3Xdigraph[103X,  the  operation  returns the digraph obtained by
  associating  a vertex with each edge of [3Xdigraph[103X, and creating an edge from a
  vertex  [9Xv[109X  to  a  vertex  [9Xu[109X  if  and only if the terminal vertex of the edge
  associated with [9Xv[109X is the start vertex of the edge associated with [9Xu[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLineDigraph(CompleteDigraph(3));[127X[104X
    [4X[28X<digraph with 6 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XLineDigraph(ChainDigraph(3));[127X[104X
    [4X[28X<digraph with 2 vertices, 1 edge>[128X[104X
  [4X[32X[104X
  
  [1X3.3-30 LineUndirectedDigraph[101X
  
  [33X[1;0Y[29X[2XLineUndirectedDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XEdgeUndirectedDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YGiven  a  symmetric  digraph  [3Xdigraph[103X,  the  operation returns the symmetric
  digraph obtained by associating a vertex with each edge of [3Xdigraph[103X, ignoring
  directions and multiplicites, and adding an edge between two vertices if and
  only if the corresponding edges have a vertex in common.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLineUndirectedDigraph(CompleteDigraph(3));[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XLineUndirectedDigraph(DigraphSymmetricClosure(ChainDigraph(3)));[127X[104X
    [4X[28X<digraph with 2 vertices, 2 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-31 DoubleDigraph[101X
  
  [33X[1;0Y[29X[2XDoubleDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YLet [3Xdigraph[103X be a digraph with vertex set [10XV[110X. This function returns the double
  digraph  of  [3Xdigraph[103X.  The  vertex  set of the double digraph is the orginal
  vertex  set  together  with  a duplicate. The edges are [10X[u_1, v_2][110X and [10X[u_2,
  v_1][110X if and only if [10X[u, v][110X is an edge in [3Xdigraph[103X, together with the original
  edges and their duplicates.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgamma := Digraph([[2], [3], [1]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDoubleDigraph(gamma);[127X[104X
    [4X[28X<digraph with 6 vertices, 12 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-32 BipartiteDoubleDigraph[101X
  
  [33X[1;0Y[29X[2XBipartiteDoubleDigraph[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YLet  [3Xdigraph[103X  be  a  digraph  with  vertex  set [10XV[110X. This function returns the
  bipartite double digraph of [3Xdigraph[103X. The vertex set of the double digraph is
  the  orginal  vertex set together with a duplicate. The edges are [10X[u_1, v_2][110X
  and  [10X[u_2,  v_1][110X  if and only if [10X[u, v][110X is an edge in [3Xdigraph[103X. The resulting
  graph  is  bipartite,  since  the  orignal  edges  are  not  included in the
  resulting digraph.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgamma := Digraph([[2], [3], [1]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XBipartiteDoubleDigraph(gamma);[127X[104X
    [4X[28X<digraph with 6 vertices, 6 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-33 DigraphAddAllLoops[101X
  
  [33X[1;0Y[29X[2XDigraphAddAllLoops[102X( [3Xdigraph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YFor  a  digraph  [3Xdigraph[103X this operation return a copy of [3Xdigraph[103X such that a
  loop is added for every vertex which did not have a loop in [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := EmptyDigraph(13);[127X[104X
    [4X[28X<digraph with 13 vertices, 0 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphAddAllLoops(gr);[127X[104X
    [4X[28X<digraph with 13 vertices, 13 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr);[127X[104X
    [4X[28X[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ], [ 8 ], [ 9 ], [128X[104X
    [4X[28X  [ 10 ], [ 11 ], [ 12 ], [ 13 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2, 3], [1, 3], [1]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphAddAllLoops(gr);[127X[104X
    [4X[28X<digraph with 3 vertices, 8 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 1, 3 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.3-34 DistanceDigraph[101X
  
  [33X[1;0Y[29X[2XDistanceDigraph[102X( [3Xdigraph[103X, [3Xi[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDistanceDigraph[102X( [3Xdigraph[103X, [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YThe  first  argument  is  a  digraph,  the second argument is a non-negative
  integer  or a list of positive integers. This operation returns a digraph on
  the same set of vertices as [3Xdigraph[103X, with two vertices being adjacent if and
  only  if  the  distance  between  them in [3Xdigraph[103X equals [3Xi[103X or is a number in
  [3Xlist[103X. See [2XDigraphShortestDistance[102X ([14X5.3-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdigraph := DigraphFromSparse6String([127X[104X
    [4X[25X>[125X [27X":]n?AL`BC_DEbEF`GIaGHdIJeGKcKL_@McDHfILaBJfHMjKM");[127X[104X
    [4X[28X<digraph with 30 vertices, 90 edges>[128X[104X
    [4X[25Xgap>[125X [27XDistanceDigraph(digraph, 1);[127X[104X
    [4X[28X<digraph with 30 vertices, 90 edges>[128X[104X
    [4X[25Xgap>[125X [27XDistanceDigraph(digraph, [1, 2]);[127X[104X
    [4X[28X<digraph with 30 vertices, 270 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.3-35 DigraphClosure[101X
  
  [33X[1;0Y[29X[2XDigraphClosure[102X( [3Xdigraph[103X, [3Xk[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph[133X
  
  [33X[0;0YGiven  a  symmetric  loopless  digraph  with  no multiple edges [3Xdigraph[103X, the
  [13X[3Xk[103X-closure of [3Xdigraph[103X[113X is defined to be the unique smallest symmetric loopless
  digraph  [10XC[110X  with  no multiple edges on the vertices of [3Xdigraph[103X that contains
  all  the  edges  of  [3Xdigraph[103X  and satsifies the property that the sum of the
  degrees  of  every  two  non-adjacenct  vertices  in  [10XC[110X  is less than [3Xk[103X. See
  [2XIsSymmetricDigraph[102X   ([14X6.1-10[114X),   [2XDigraphHasLoops[102X   ([14X6.1-1[114X),   [2XIsMultiDigraph[102X
  ([14X6.1-8[114X), and [2XOutDegreeOfVertex[102X ([14X5.2-10[114X).[133X
  
  [33X[0;0YThe operation [10XDigraphClosure[110X returns the [3Xk[103X-closure of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CompleteDigraph(6);;[127X[104X
    [4X[25Xgap>[125X [27XDigraphRemoveEdges(gr, [[1, 2], [2, 1]]);;[127X[104X
    [4X[25Xgap>[125X [27Xclosure := DigraphClosure(gr, 6);[127X[104X
    [4X[28X<digraph with 6 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteDigraph(closure);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X3.4 [33X[0;0YRandom digraphs[133X[101X
  
  [1X3.4-1 RandomDigraph[101X
  
  [33X[1;0Y[29X[2XRandomDigraph[102X( [3Xn[103X[, [3Xp[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xn[103X is a positive integer, then this function returns a random digraph with
  [3Xn[103X vertices and without multiple edges. The result may or may not have loops.[133X
  
  [33X[0;0YIf the optional second argument [3Xp[103X is a float with value [22X0 ≤[122X [3X p [103X [22X≤ 1[122X, then an
  edge will exist between each pair of vertices with probability approximately
  [3Xp[103X.  If [3Xp[103X is not specified, then a random probability will be assumed (chosen
  with uniform probability).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRandomDigraph(1000);[127X[104X
    [4X[28X<digraph with 1000 vertices, 364444 edges>[128X[104X
    [4X[25Xgap>[125X [27XRandomDigraph(10000, 0.023);[127X[104X
    [4X[28X<digraph with 10000 vertices, 2300438 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.4-2 RandomMultiDigraph[101X
  
  [33X[1;0Y[29X[2XRandomMultiDigraph[102X( [3Xn[103X[, [3Xm[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf [3Xn[103X is a positive integer, then this function returns a random digraph with
  [3Xn[103X  vertices.  If  the optional second argument [3Xm[103X is a positive integer, then
  the  digraph  will  have  [3Xm[103X edges. If [3Xm[103X is not specified, then the number of
  edges  will  be chosen randomly (with uniform probability) from the range [10X[1
  .. [110X[22Xn choose 2[122X[10X][110X.[133X
  
  [33X[0;0YThe  method  used  by  this  function  chooses each edge from the set of all
  possible edges with uniform probability. No effort is made to avoid creating
  multiple  edges, so it is possible (but not guaranteed) that the result will
  have multiple edges. The result may or may not have loops.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRandomMultiDigraph(1000);[127X[104X
    [4X[28X<multidigraph with 1000 vertices, 216659 edges>[128X[104X
    [4X[25Xgap>[125X [27XRandomMultiDigraph(1000, 950);[127X[104X
    [4X[28X<multidigraph with 1000 vertices, 950 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.4-3 RandomTournament[101X
  
  [33X[1;0Y[29X[2XRandomTournament[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a non-negative integer, this function returns a random tournament
  with [3Xn[103X vertices. See [2XIsTournament[102X ([14X6.1-11[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRandomTournament(10);[127X[104X
    [4X[28X<digraph with 10 vertices, 45 edges>[128X[104X
  [4X[32X[104X
  
  
  [1X3.5 [33X[0;0YStandard examples[133X[101X
  
  [1X3.5-1 ChainDigraph[101X
  
  [33X[1;0Y[29X[2XChainDigraph[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a positive integer, this function returns a chain with [3Xn[103X vertices
  and  [10X[3Xn[103X[10X  -  1[110X edges. Specifically, for each vertex [10Xi[110X (with [10Xi[110X < [10Xn[110X), there is a
  directed edge with source [10Xi[110X and range [10Xi + 1[110X.[133X
  
  [33X[0;0YThe [2XDigraphReflexiveTransitiveClosure[102X ([14X3.3-11[114X) of a chain represents a total
  order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XChainDigraph(42);[127X[104X
    [4X[28X<digraph with 42 vertices, 41 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-2 CompleteDigraph[101X
  
  [33X[1;0Y[29X[2XCompleteDigraph[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  is a non-negative integer, this function returns the complete digraph
  with [3Xn[103X vertices. See [2XIsCompleteDigraph[102X ([14X6.1-5[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCompleteDigraph(20);[127X[104X
    [4X[28X<digraph with 20 vertices, 380 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-3 CompleteBipartiteDigraph[101X
  
  [33X[1;0Y[29X[2XCompleteBipartiteDigraph[102X( [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YA  complete bipartite digraph is a digraph whose vertices can be partitioned
  into  two non-empty vertex sets, such there exists a unique edge with source
  [10Xi[110X and range [10Xj[110X if and only if [10Xi[110X and [10Xj[110X lie in different vertex sets.[133X
  
  [33X[0;0YIf  [3Xm[103X  and  [3Xn[103X  are  positive  integers,  this  function returns the complete
  bipartite digraph with vertex sets of sizes [3Xm[103X (containing the vertices [10X[1 ..
  m][110X) and [3Xn[103X (containing the vertices [10X[m + 1 .. m + n][110X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCompleteBipartiteDigraph(2, 3);[127X[104X
    [4X[28X<digraph with 5 vertices, 12 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-4 CompleteMultipartiteDigraph[101X
  
  [33X[1;0Y[29X[2XCompleteMultipartiteDigraph[102X( [3Xorders[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YFor  a list [3Xorders[103X of [10Xn[110X positive integers, this function returns the digraph
  containing  [10Xn[110X  independent  sets  of  vertices  of  orders  [10X[[3Xl[103X[10X[1]  .. [3Xl[103X[10X[n]][110X.
  Moreover,  each  vertex is adjacent to every other not contained in the same
  independent set.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCompleteMultipartiteDigraph([5, 4, 2]);[127X[104X
    [4X[28X<digraph with 11 vertices, 76 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-5 CycleDigraph[101X
  
  [33X[1;0Y[29X[2XCycleDigraph[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a  positive integer, this function returns a [13Xcycle[113X digraph with [3Xn[103X
  vertices and [3Xn[103X edges. Specifically, for each vertex [10Xi[110X (with [10Xi[110X < [10Xn[110X), there is
  a directed edge with source [10Xi[110X and range [10Xi + 1[110X. In addition, there is an edge
  with source [10Xn[110X and range [10X1[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCycleDigraph(1);[127X[104X
    [4X[28X<digraph with 1 vertex, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27XCycleDigraph(123);[127X[104X
    [4X[28X<digraph with 123 vertices, 123 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-6 EmptyDigraph[101X
  
  [33X[1;0Y[29X[2XEmptyDigraph[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNullDigraph[102X( [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  is  a  non-negative  integer, this function returns the [13Xempty[113X or [13Xnull[113X
  digraph with [3Xn[103X vertices. An empty digraph is one with no edges.[133X
  
  [33X[0;0Y[10XNullDigraph[110X is a synonym for [10XEmptyDigraph[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XEmptyDigraph(20);[127X[104X
    [4X[28X<digraph with 20 vertices, 0 edges>[128X[104X
    [4X[25Xgap>[125X [27XNullDigraph(10);[127X[104X
    [4X[28X<digraph with 10 vertices, 0 edges>[128X[104X
  [4X[32X[104X
  
  [1X3.5-7 JohnsonDigraph[101X
  
  [33X[1;0Y[29X[2XJohnsonDigraph[102X( [3Xn[103X, [3Xk[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA digraph.[133X
  
  [33X[0;0YIf  [3Xn[103X  and  [3Xk[103X  are  non-negative  integers,  then  this  operation returns a
  symmetric digraph which corresponds to the undirected [13XJohnson graph[113X [22XJ(n, k)[122X.[133X
  
  [33X[0;0YThe  [13XJohnson  graph[113X  [22XJ(n,  k)[122X has vertices given by all the [3Xk[103X-subsets of the
  range  [10X[1  ..  [3Xk[103X[10X][110X,  and  two  vertices  are  connected  by an edge iff their
  intersection has size [22X[3Xk[103X - 1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := JohnsonDigraph(3, 1);[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr);[127X[104X
    [4X[28X[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr := JohnsonDigraph(4, 2);[127X[104X
    [4X[28X<digraph with 6 vertices, 24 edges>[128X[104X
    [4X[25Xgap>[125X [27XOutNeighbours(gr);[127X[104X
    [4X[28X[ [ 2, 3, 4, 5 ], [ 1, 3, 4, 6 ], [ 1, 2, 5, 6 ], [ 1, 2, 5, 6 ], [128X[104X
    [4X[28X  [ 1, 3, 4, 6 ], [ 2, 3, 4, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XJohnsonDigraph(1, 0);[127X[104X
    [4X[28X<digraph with 1 vertex, 0 edges>[128X[104X
  [4X[32X[104X
  
