  
  [1X8 [33X[0;0YFinding cliques and independent sets[133X[101X
  
  [33X[0;0YIn [5XDigraphs[105X, a [13Xclique[113X of a digraph is a set of mutually adjacent vertices of
  the  digraph,  and  an  [13Xindependent  set[113X  is  a set of mutually non-adjacent
  vertices  of the digraph. A [13Xmaximal clique[113X is a clique which is not properly
  contained in another clique, and a [13Xmaximal independent set[113X is an independent
  set  which  is not properly contained in another independent set. Using this
  definition in [5XDigraphs[105X, cliques and independent sets are both permitted, but
  not required, to contain vertices at which there is a loop. Another name for
  a clique is a [13Xcomplete subgraph[113X.[133X
  
  [33X[0;0Y[5XDigraphs[105X   provides   extensive  functionality  for  computing  cliques  and
  independent  sets  of  a  digraph,  whether  maximal or not. The fundamental
  algorithm  used  in most of the methods in [5XDigraphs[105X to calculate cliques and
  independent  sets  is  a version of the Bron-Kerbosch algorithm. Calculating
  the  cliques  and  independent  sets  of  a digraph is a well-known and hard
  problem,  and  searching for cliques or independent sets in a digraph can be
  very  lengthy,  even for a digraph with a small number of vertices. [5XDigraphs[105X
  uses several strategies to increase the performance of these calculations.[133X
  
  [33X[0;0YFrom  the  definition  of  cliques and independent sets, it follows that the
  presence  of  loops  and  multiple  edges  in a digraph is irrelevant to the
  existence   of   cliques   and   independent   sets   in  the  digraph.  See
  [2XDigraphHasLoops[102X  ([14X6.2-1[114X)  and  [2XIsMultiDigraph[102X  ([14X6.2-11[114X) for more information
  about  these  properties. Therefore given a digraph [3Xdigraph[103X, the cliques and
  independent sets of [3Xdigraph[103X are equal to the cliques and independent sets of
  the digraph:[133X
  
  [30X    [33X[0;6Y[10XDigraphRemoveLoops(DigraphRemoveAllMultipleEdges([110X[3Xdigraph[103X[10X))[110X.[133X
  
  [33X[0;0YSee  [2XDigraphRemoveLoops[102X  ([14X3.3-25[114X) and [2XDigraphRemoveAllMultipleEdges[102X ([14X3.3-26[114X)
  for  more  information  about  these attributes. Furthermore, the cliques of
  this  digraph are equal to the cliques of the digraph formed by removing any
  edge  [10X[u,v][110X  for  which the corresponding reverse edge [10X[v,u][110X is not present.
  Therefore,  overall,  the cliques of [3Xdigraph[103X are equal to the cliques of the
  symmetric digraph:[133X
  
  [30X    [33X[0;6Y[10XMaximalSymmetricSubdigraphWithoutLoops([110X[3Xdigraph[103X[10X)[110X.[133X
  
  [33X[0;0YSee  [2XMaximalSymmetricSubdigraphWithoutLoops[102X  ([14X3.3-5[114X)  for  more  information
  about this. The [2XAutomorphismGroup[102X ([14X7.2-2[114X) of this symmetric digraph contains
  the  automorphism  group  of [3Xdigraph[103X as a subgroup. By performing the search
  for  maximal  cliques  with  the  help  of this larger automorphism group to
  reduce  the search space, the computation time may be reduced. The functions
  and  attributes  which  return  representatives  of  cliques of [3Xdigraph[103X will
  return orbit representatives of cliques under the action of the automorphism
  group of the [13Xmaximal symmetric subdigraph without loops[113X on sets of vertices.[133X
  
  [33X[0;0YThe  independent  sets of a digraph are equal to the independent sets of the
  [2XDigraphSymmetricClosure[102X  ([14X3.3-12[114X).  Therefore, overall, the independent sets
  of [3Xdigraph[103X are equal to the independent sets of the symmetric digraph:[133X
  
  [30X    [33X[0;6Y[10XDigraphSymmetricClosure(DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(
        [110X[3Xdigraph[103X[10X)))[110X.[133X
  
  [33X[0;0YAgain,  the  automorphism  group  of  this  symmetric  digraph  contains the
  automorphism  group of [3Xdigraph[103X. Since a search for independent sets is equal
  to  a  search  for  cliques in the [2XDigraphDual[102X ([14X3.3-11[114X), the methods used in
  [5XDigraphs[105X  usually  transform a search for independent sets into a search for
  cliques,  as  described  above.  The  functions  and attributes which return
  representatives   of   independent   sets   of  [3Xdigraph[103X  will  return  orbit
  representatives  of  independent  sets  under the action of the automorphism
  group  of  the [13Xsymmetric closure[113X of the digraph formed by removing loops and
  multiple edges.[133X
  
  [33X[0;0YPlease  note that in [5XDigraphs[105X, cliques and independent sets are not required
  to  be maximal. Some authors use the word clique to mean [13Xmaximal[113X clique, and
  some authors use the phrase independent set to mean [13Xmaximal[113X independent set,
  but please note that [5XDigraphs[105X does not use this definition.[133X
  
  
  [1X8.1 [33X[0;0YFinding cliques[133X[101X
  
  [1X8.1-1 IsClique[101X
  
  [33X[1;0Y[29X[2XIsClique[102X( [3Xdigraph[103X, [3Xl[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsMaximalClique[102X( [3Xdigraph[103X, [3Xl[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph  and  [3Xl[103X  is a duplicate-free list of vertices of
  [3Xdigraph[103X,  then  [10XIsClique([110X[3Xdigraph[103X[10X,[110X[3Xl[103X[10X)[110X returns [9Xtrue[109X if [3Xl[103X is a [13Xclique[113X of [3Xdigraph[103X
  and  [9Xfalse[109X  if it is not. Similarly, [10XIsMaximalClique([110X[3Xdigraph[103X[10X,[110X[3Xl[103X[10X)[110X returns [9Xtrue[109X
  if [3Xl[103X is a [13Xmaximal clique[113X of [3Xdigraph[103X and [9Xfalse[109X if it is not.[133X
  
  [33X[0;0YA [13Xclique[113X of a digraph is a set of mutually adjacent vertices of the digraph.
  A  [13Xmaximal  clique[113X  is  a  clique  that is not properly contained in another
  clique.  A  clique  is  permitted,  but not required, to contain vertices at
  which there is a loop.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(4);;[127X[104X
    [4X[25Xgap>[125X [27XIsClique(D, [1, 3, 2]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsMaximalClique(D, [1, 3, 2]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsMaximalClique(D, DigraphVertices(D));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);[127X[104X
    [4X[28X<immutable multidigraph with 4 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsClique(D, [2, 3, 4]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsMaximalClique(D, [1, 2, 4]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(IsMutableDigraph, 4);;[127X[104X
    [4X[25Xgap>[125X [27XIsClique(D, [1, 3, 2]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X8.1-2 CliquesFinder[101X
  
  [33X[1;0Y[29X[2XCliquesFinder[102X( [3Xdigraph[103X, [3Xhook[103X, [3Xuser_param[103X, [3Xlimit[103X, [3Xinclude[103X, [3Xexclude[103X, [3Xmax[103X, [3Xsize[103X, [3Xreps[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe argument [3Xuser_param[103X.[133X
  
  [33X[0;0YThis function finds cliques of the digraph [3Xdigraph[103X subject to the conditions
  imposed  by  the  other  arguments as described below. Note that a clique is
  represented by the immutable list of the vertices that it contains.[133X
  
  [33X[0;0YLet  [10XG[110X  denote the automorphism group of the maximal symmetric subdigraph of
  [3Xdigraph[103X     without     loops    (see    [2XAutomorphismGroup[102X    ([14X7.2-2[114X)    and
  [2XMaximalSymmetricSubdigraphWithoutLoops[102X ([14X3.3-5[114X)).[133X
  
  [8X[3Xhook[103X[8X[108X
        [33X[0;6YThis argument should be a function or [9Xfail[109X.[133X
  
        [33X[0;6YIf  [3Xhook[103X  is  a function, then it should have two arguments [3Xuser_param[103X
        (see below) and a clique [10Xc[110X. The function [10X[3Xhook[103X[10X([3Xuser_param[103X[10X, c)[110X is called
        every time a new clique [10Xc[110X is found by [10XCliquesFinder[110X.[133X
  
        [33X[0;6YIf  [3Xhook[103X  is  [9Xfail[109X,  then  a default function is used that simply adds
        every new clique found by [10XCliquesFinder[110X to [3Xuser_param[103X, which must be a
        list in this case.[133X
  
  [8X[3Xuser_param[103X[8X[108X
        [33X[0;6YIf  [3Xhook[103X  is  a  function,  then [3Xuser_param[103X can be any [5XGAP[105X object. The
        object [3Xuser_param[103X is used as the first argument for the function [3Xhook[103X.
        For example, [3Xuser_param[103X might be a list, and [10X[3Xhook[103X[10X([3Xuser_param[103X[10X, c)[110X might
        add the size of the clique [10Xc[110X to the list [3Xuser_param[103X.[133X
  
        [33X[0;6YIf  the  value of [3Xhook[103X is [9Xfail[109X, then the value of [3Xuser_param[103X must be a
        list.[133X
  
  [8X[3Xlimit[103X[8X[108X
        [33X[0;6YThis  argument should be a positive integer or [9Xinfinity[109X. [10XCliquesFinder[110X
        will  return  after  it  has  found  [3Xlimit[103X  cliques  or  the search is
        complete.[133X
  
  [8X[3Xinclude[103X[8X and [3Xexclude[103X[8X[108X
        [33X[0;6YThese  arguments should each be a (possibly empty) duplicate-free list
        of vertices of [3Xdigraph[103X (i.e. positive integers less than the number of
        vertices of [3Xdigraph[103X).[133X
  
        [33X[0;6Y[10XCliquesFinder[110X  will  only  look  for  cliques  containing  all  of the
        vertices in [3Xinclude[103X and containing none of the vertices in [3Xexclude[103X.[133X
  
        [33X[0;6YNote that the search may be much more efficient if each of these lists
        is invariant under the action of [10XG[110X on sets of vertices.[133X
  
  [8X[3Xmax[103X[8X[108X
        [33X[0;6YThis   argument  should  be  [9Xtrue[109X  or  [9Xfalse[109X.  If  [3Xmax[103X  is  true  then
        [10XCliquesFinder[110X  will  only  search for [13Xmaximal[113X cliques. If [9Xmax[109X is [9Xfalse[109X
        then non-maximal cliques may be found.[133X
  
  [8X[3Xsize[103X[8X[108X
        [33X[0;6YThis  argument  should  be  [9Xfail[109X  or  a positive integer. If [3Xsize[103X is a
        positive  integer then [10XCliquesFinder[110X will only search for cliques that
        contain  precisely  [3Xsize[103X vertices. If [3Xsize[103X is [9Xfail[109X then cliques of any
        size may be found.[133X
  
  [8X[3Xreps[103X[8X[108X
        [33X[0;6YThis argument should be [9Xtrue[109X or [9Xfalse[109X.[133X
  
        [33X[0;6YIf  [3Xreps[103X  is  [9Xtrue[109X  then  the  arguments  [3Xinclude[103X and [3Xexclude[103X are each
        required to be invariant under the action of [10XG[110X on sets of vertices. In
        this  case,  [10XCliquesFinder[110X  will find representatives of the orbits of
        the  desired  cliques  under the action of [10XG[110X, [13Xalthough representatives
        may  be  returned  that  are  in the same orbit[113X. If [3Xreps[103X is false then
        [10XCliquesFinder[110X will not take this into consideration.[133X
  
        [33X[0;6YFor  a  digraph  such  that  [10XG[110X  is  non-trivial, the search for clique
        representatives  can  be  much  more efficient than the search for all
        cliques.[133X
  
  [33X[0;0YThis function uses a version of the Bron-Kerbosch algorithm.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(5);[127X[104X
    [4X[28X<immutable complete digraph with 5 vertices>[128X[104X
    [4X[25Xgap>[125X [27Xuser_param := [];;[127X[104X
    [4X[25Xgap>[125X [27Xf := function(a, b)  # Calculate size of clique[127X[104X
    [4X[25X>[125X [27X  AddSet(user_param, Size(b));[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, f, user_param, infinity, [], [], false, fail,[127X[104X
    [4X[25X>[125X [27X                 true);[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5 ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], 5, [2, 4], [3], false, fail, false);[127X[104X
    [4X[28X[ [ 2, 4 ], [ 1, 2, 4 ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], 2, [2, 4], [3], false, fail, false);[127X[104X
    [4X[28X[ [ 2, 4 ], [ 1, 2, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], infinity, [], [], true, 5, false);[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], infinity, [1, 3], [], false, 3, false);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1, 3, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], infinity, [1, 3], [], true, 3, false);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(IsMutableDigraph, 5);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 20 edges>[128X[104X
    [4X[25Xgap>[125X [27Xuser_param := [];;[127X[104X
    [4X[25Xgap>[125X [27Xf := function(a, b)  # Calculate size of clique[127X[104X
    [4X[25X>[125X [27X  AddSet(user_param, Size(b));[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, f, user_param, infinity, [], [], false, fail,[127X[104X
    [4X[25X>[125X [27X                 true);[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5 ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], 5, [2, 4], [3], false, fail, false);[127X[104X
    [4X[28X[ [ 2, 4 ], [ 1, 2, 4 ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], 2, [2, 4], [3], false, fail, false);[127X[104X
    [4X[28X[ [ 2, 4 ], [ 1, 2, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], infinity, [], [], true, 5, false);[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], infinity, [1, 3], [], false, 3, false);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1, 3, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XCliquesFinder(D, fail, [], infinity, [1, 3], [], true, 3, false);[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [1X8.1-3 DigraphClique[101X
  
  [33X[1;0Y[29X[2XDigraphClique[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xsize[103X]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphMaximalClique[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xsize[103X]]] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn immutable list of positive integers, or [9Xfail[109X.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a digraph, then these functions returns a clique of [3Xdigraph[103X if
  one  exists  that satisfies the arguments, else it returns [9Xfail[109X. A clique is
  defined  by  the  set of vertices that it contains; see [2XIsClique[102X ([14X8.1-1[114X) and
  [2XIsMaximalClique[102X ([14X8.1-1[114X).[133X
  
  [33X[0;0YThe  optional  arguments [3Xinclude[103X and [3Xexclude[103X must each be a (possibly empty)
  duplicate-free  list  of vertices of [3Xdigraph[103X, and the optional argument [3Xsize[103X
  must be a positive integer. By default, [3Xinclude[103X and [3Xexclude[103X are empty. These
  functions  will search for a clique of [3Xdigraph[103X that includes the vertices of
  [3Xinclude[103X  but  does not include any vertices in [3Xexclude[103X; if the argument [3Xsize[103X
  is  supplied,  then  additionally  the  clique  will  be required to contain
  precisely [3Xsize[103X vertices.[133X
  
  [33X[0;0YIf [3Xinclude[103X is not a clique, then these functions return [9Xfail[109X. Otherwise, the
  functions behave in the following way, depending on the number of arguments:[133X
  
  [8XOne or two arguments[108X
        [33X[0;6YIf   one  or  two  arguments  are  supplied,  then  [10XDigraphClique[110X  and
        [10XDigraphMaximalClique[110X  greedily  enlarge  the  clique  [3Xinclude[103X until it
        cannot continue. The result is guaranteed to be a maximal clique. This
        may   or   may   not   return   an  answer  more  quickly  than  using
        [2XDigraphMaximalCliques[102X ([14X8.1-4[114X) with a limit of 1.[133X
  
  [8XThree arguments[108X
        [33X[0;6YIf  three arguments are supplied, then [10XDigraphClique[110X greedily enlarges
        the  clique [3Xinclude[103X until it cannot continue, although this clique may
        not be maximal.[133X
  
        [33X[0;6YGiven three arguments, [10XDigraphMaximalClique[110X returns the maximal clique
        returned by [10XDigraphMaximalCliques([110X[3Xdigraph[103X[10X, [110X[3Xinclude[103X[10X, [110X[3Xexclude[103X[10X, 1)[110X if one
        exists, else [9Xfail[109X.[133X
  
  [8XFour arguments[108X
        [33X[0;6YIf  four arguments are supplied, then [10XDigraphClique[110X returns the clique
        returned  by [10XDigraphCliques([110X[3Xdigraph[103X[10X, [110X[3Xinclude[103X[10X, [110X[3Xexclude[103X[10X, 1, [110X[3Xsize[103X[10X)[110X if one
        exists, else [9Xfail[109X. This clique may not be maximal.[133X
  
        [33X[0;6YGiven  four arguments, [10XDigraphMaximalClique[110X returns the maximal clique
        returned  by [10XDigraphMaximalCliques([110X[3Xdigraph[103X[10X, [110X[3Xinclude[103X[10X, [110X[3Xexclude[103X[10X, 1, [110X[3Xsize[103X[10X)[110X
        if one exists, else [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);[127X[104X
    [4X[28X<immutable digraph with 5 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XDigraphClique(D);[127X[104X
    [4X[28X[ 5 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalClique(D);[127X[104X
    [4X[28X[ 5 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphClique(D, [1, 2]);[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalClique(D, [1, 3]);[127X[104X
    [4X[28X[ 1, 3, 2 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphClique(D, [1], [2]);[127X[104X
    [4X[28X[ 1, 4 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalClique(D, [1], [3, 4]);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDigraphClique(D, [1, 5]);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDigraphClique(D, [], [], 2);[127X[104X
    [4X[28X[ 1, 2 ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph,[127X[104X
    [4X[25X>[125X [27X                [[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);[127X[104X
    [4X[28X<mutable digraph with 5 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XDigraphClique(D);[127X[104X
    [4X[28X[ 5 ][128X[104X
  [4X[32X[104X
  
  [1X8.1-4 DigraphMaximalCliques[101X
  
  [33X[1;0Y[29X[2XDigraphMaximalCliques[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xlimit[103X[, [3Xsize[103X]]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphMaximalCliquesReps[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xlimit[103X[, [3Xsize[103X]]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphCliques[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xlimit[103X[, [3Xsize[103X]]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphCliquesReps[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xlimit[103X[, [3Xsize[103X]]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphMaximalCliquesAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDigraphMaximalCliquesRepsAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YAn immutable list of lists of positive integers.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is digraph, then these functions and attributes use [2XCliquesFinder[102X
  ([14X8.1-2[114X)  to  return  cliques  of  [3Xdigraph[103X. A clique is defined by the set of
  vertices that it contains; see [2XIsClique[102X ([14X8.1-1[114X) and [2XIsMaximalClique[102X ([14X8.1-1[114X).[133X
  
  [33X[0;0YThe  optional  arguments [3Xinclude[103X and [3Xexclude[103X must each be a (possibly empty)
  list  of  vertices  of [3Xdigraph[103X, the optional argument [3Xlimit[103X must be either a
  positive  integer  or  [10Xinfinity[110X,  and  the  optional argument [3Xsize[103X must be a
  positive  integer.  If not specified, then [3Xinclude[103X and [3Xexclude[103X are chosen to
  be empty lists, and [3Xlimit[103X is set to [10Xinfinity[110X.[133X
  
  [33X[0;0YThe  functions  will  return as many suitable cliques as possible, up to the
  number  [3Xlimit[103X.  These  functions  will  find cliques that contain all of the
  vertices  of  [3Xinclude[103X but do not contain any of the vertices of [3Xexclude[103X. The
  argument  [3Xsize[103X  restricts the search to those cliques that contain precisely
  [3Xsize[103X  vertices.  If  the function or attribute has [10XMaximal[110X in its name, then
  only  maximal cliques will be returned; otherwise non-maximal cliques may be
  returned.[133X
  
  [33X[0;0YLet  [10XG[110X  denote  the  automorphism  group  of maximal symmetric subdigraph of
  [3Xdigraph[103X     without     loops    (see    [2XAutomorphismGroup[102X    ([14X7.2-2[114X)    and
  [2XMaximalSymmetricSubdigraphWithoutLoops[102X ([14X3.3-5[114X)).[133X
  
  [8XDistinct cliques[108X
        [33X[0;6Y[10XDigraphMaximalCliques[110X  and [10XDigraphCliques[110X each return a duplicate-free
        list of at most [3Xlimit[103X cliques of [3Xdigraph[103X that satisfy the arguments.[133X
  
        [33X[0;6YThe computation may be significantly faster if [3Xinclude[103X and [3Xexclude[103X are
        invariant under the action of [10XG[110X on sets of vertices.[133X
  
  [8XOrbit representatives of cliques[108X
        [33X[0;6YTo  use [10XDigraphMaximalCliquesReps[110X or [10XDigraphCliquesReps[110X, the arguments
        [3Xinclude[103X  and  [3Xexclude[103X  must each be invariant under the action of [10XG[110X on
        sets of vertices.[133X
  
        [33X[0;6YIf    this   is   the   case,   then   [10XDigraphMaximalCliquesReps[110X   and
        [10XDigraphCliquesReps[110X  each return a duplicate-free list of at most [3Xlimit[103X
        orbits  representatives  (under  the  action of [10XG[110X on sets vertices) of
        cliques of [3Xdigraph[103X that satisfy the arguments.[133X
  
        [33X[0;6YThe  representatives  are  not  guaranteed  to  be in distinct orbits.
        However, if fewer than [3Xlim[103X results are returned, then there will be at
        least one representative from each orbit of maximal cliques.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[127X[104X
    [4X[25X>[125X [27X[2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]]);[127X[104X
    [4X[28X<immutable digraph with 6 vertices, 14 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XG := AutomorphismGroup(D);[127X[104X
    [4X[28XGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])[128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalCliques(D);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalCliquesReps(D);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XOrbit(G, [1, 2, 3], OnSets);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 4, 5, 6 ] ][128X[104X
    [4X[25Xgap>[125X [27XOrbit(G, [3, 4], OnSets);[127X[104X
    [4X[28X[ [ 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalCliquesReps(D, [3, 4], [], 1);[127X[104X
    [4X[28X[ [ 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalCliques(D, [1, 2], [5, 6], 1, 2);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XDigraphCliques(D, [1], [5, 6], infinity, 2);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XD := Digraph(IsMutableDigraph, [[127X[104X
    [4X[25X>[125X [27X[2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]]);[127X[104X
    [4X[28X<mutable digraph with 6 vertices, 14 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XG := AutomorphismGroup(D);[127X[104X
    [4X[28XGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])[128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalCliques(D);[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ][128X[104X
  [4X[32X[104X
  
  [1X8.1-5 CliqueNumber[101X
  
  [33X[1;0Y[29X[2XCliqueNumber[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA non-negative integer.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph,  then [10XCliqueNumber([3Xdigraph[103X[10X)[110X returns the largest
  integer  [10Xn[110X such that [3Xdigraph[103X contains a clique with [10Xn[110X vertices as an induced
  subdigraph.[133X
  
  [33X[0;0YA [13Xclique[113X of a digraph is a set of mutually adjacent vertices of the digraph.
  Loops  and  multiple  edges  are  ignored for the purpose of determining the
  clique number of a digraph.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(4);;[127X[104X
    [4X[25Xgap>[125X [27XCliqueNumber(D);[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XD := Digraph([[1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);[127X[104X
    [4X[28X<immutable multidigraph with 4 vertices, 11 edges>[128X[104X
    [4X[25Xgap>[125X [27XCliqueNumber(D);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XD := CompleteDigraph(IsMutableDigraph, 4);;[127X[104X
    [4X[25Xgap>[125X [27XCliqueNumber(D);[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  
  [1X8.2 [33X[0;0YFinding independent sets[133X[101X
  
  [1X8.2-1 IsIndependentSet[101X
  
  [33X[1;0Y[29X[2XIsIndependentSet[102X( [3Xdigraph[103X, [3Xl[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsMaximalIndependentSet[102X( [3Xdigraph[103X, [3Xl[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X  is  a  digraph  and  [3Xl[103X  is a duplicate-free list of vertices of
  [3Xdigraph[103X,   then   [10XIsIndependentSet([110X[3Xdigraph[103X[10X,[110X[3Xl[103X[10X)[110X   returns  [9Xtrue[109X  if  [3Xl[103X  is  an
  [13Xindependent   set[113X   of   [3Xdigraph[103X   and   [9Xfalse[109X  if  it  is  not.  Similarly,
  [10XIsMaximalIndependentSet([110X[3Xdigraph[103X[10X,[110X[3Xl[103X[10X)[110X   returns   [9Xtrue[109X   if   [3Xl[103X  is  a  [13Xmaximal
  independent set[113X of [3Xdigraph[103X and [9Xfalse[109X if it is not.[133X
  
  [33X[0;0YAn  [13Xindependent  set[113X of a digraph is a set of mutually non-adjacent vertices
  of  the digraph. A [13Xmaximal independent set[113X is an independent set that is not
  properly  contained  in  another  independent  set.  An  independent  set is
  permitted, but not required, to contain vertices at which there is a loop.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(4);;[127X[104X
    [4X[25Xgap>[125X [27XIsIndependentSet(D, [1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsMaximalIndependentSet(D, [1]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsIndependentSet(D, [1, 4, 3]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsIndependentSet(D, [2, 4]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsMaximalIndependentSet(D, [2, 4]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(IsMutableDigraph, 4);;[127X[104X
    [4X[25Xgap>[125X [27XIsIndependentSet(D, [1]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X8.2-2 DigraphIndependentSet[101X
  
  [33X[1;0Y[29X[2XDigraphIndependentSet[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xsize[103X]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphMaximalIndependentSet[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xsize[103X]]] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn immutable list of positive integers, or [9Xfail[109X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is a digraph, then these functions returns an independent set of
  [3Xdigraph[103X if one exists that satisfies the arguments, else it returns [9Xfail[109X. An
  independent  set  is  defined  by  the set of vertices that it contains; see
  [2XIsIndependentSet[102X ([14X8.2-1[114X) and [2XIsMaximalIndependentSet[102X ([14X8.2-1[114X).[133X
  
  [33X[0;0YThe  optional  arguments [3Xinclude[103X and [3Xexclude[103X must each be a (possibly empty)
  duplicate-free  list  of vertices of [3Xdigraph[103X, and the optional argument [3Xsize[103X
  must be a positive integer. By default, [3Xinclude[103X and [3Xexclude[103X are empty. These
  functions  will  search  for an independent set of [3Xdigraph[103X that includes the
  vertices  of  [3Xinclude[103X  but  does not include any vertices in [3Xexclude[103X; if the
  argument  [3Xsize[103X  is  supplied,  then additionally the independent set will be
  required to contain precisely [3Xsize[103X vertices.[133X
  
  [33X[0;0YIf  [3Xinclude[103X  is  not  an  independent set, then these functions return [9Xfail[109X.
  Otherwise,  the  functions  behave  in  the  following way, depending on the
  number of arguments:[133X
  
  [8XOne or two arguments[108X
        [33X[0;6YIf  one  or two arguments are supplied, then [10XDigraphIndependentSet[110X and
        [10XDigraphMaximalIndependentSet[110X  greedily  enlarge  the  independent  set
        [3Xinclude[103X  until  it  cannot  continue. The result is guaranteed to be a
        maximal  independent  set.  This  may or may not return an answer more
        quickly  than using [2XDigraphMaximalIndependentSets[102X ([14X8.2-3[114X) with a limit
        of 1.[133X
  
  [8XThree arguments[108X
        [33X[0;6YIf  three  arguments are supplied, then [10XDigraphIndependentSet[110X greedily
        enlarges  the  independent  set  [3Xinclude[103X  until  it  cannot  continue,
        although this independent set may not be maximal.[133X
  
        [33X[0;6YGiven   three   arguments,  [10XDigraphMaximalIndependentSet[110X  returns  the
        maximal          independent          set          returned         by
        [10XDigraphMaximalIndependentSets([110X[3Xdigraph[103X[10X,  [110X[3Xinclude[103X[10X,  [110X[3Xexclude[103X[10X,  1)[110X  if one
        exists, else [9Xfail[109X.[133X
  
  [8XFour arguments[108X
        [33X[0;6YIf four arguments are supplied, then [10XDigraphIndependentSet[110X returns the
        independent  set  returned by [10XDigraphIndependentSets([110X[3Xdigraph[103X[10X, [110X[3Xinclude[103X[10X,
        [110X[3Xexclude[103X[10X,  1,  [110X[3Xsize[103X[10X)[110X if one exists, else [9Xfail[109X. This independent set may
        not be maximal.[133X
  
        [33X[0;6YGiven four arguments, [10XDigraphMaximalIndependentSet[110X returns the maximal
        independent  set  returned  by  [10XDigraphMaximalIndependentSets([110X[3Xdigraph[103X[10X,
        [110X[3Xinclude[103X[10X, [110X[3Xexclude[103X[10X, 1, [110X[3Xsize[103X[10X)[110X if one exists, else [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := ChainDigraph(6);[127X[104X
    [4X[28X<immutable chain digraph with 6 vertices>[128X[104X
    [4X[25Xgap>[125X [27XDigraphIndependentSet(D);[127X[104X
    [4X[28X[ 6, 4, 2 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D);[127X[104X
    [4X[28X[ 6, 4, 2 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphIndependentSet(D, [2, 4]);[127X[104X
    [4X[28X[ 2, 4, 6 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D, [1, 3]);[127X[104X
    [4X[28X[ 1, 3, 6 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphIndependentSet(D, [2, 4], [6]);[127X[104X
    [4X[28X[ 2, 4 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D, [2, 4], [6]);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDigraphIndependentSet(D, [1], [], 2);[127X[104X
    [4X[28X[ 1, 3 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D, [1], [], 2);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D, [1], [], 3);[127X[104X
    [4X[28X[ 1, 3, 5 ][128X[104X
    [4X[25Xgap>[125X [27XD := ChainDigraph(IsMutableDigraph, 6);[127X[104X
    [4X[28X<mutable digraph with 6 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphIndependentSet(D);[127X[104X
    [4X[28X[ 6, 4, 2 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D);[127X[104X
    [4X[28X[ 6, 4, 2 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphIndependentSet(D, [2, 4]);[127X[104X
    [4X[28X[ 2, 4, 6 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D, [1, 3]);[127X[104X
    [4X[28X[ 1, 3, 6 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphIndependentSet(D, [2, 4], [6]);[127X[104X
    [4X[28X[ 2, 4 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D, [2, 4], [6]);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDigraphIndependentSet(D, [1], [], 2);[127X[104X
    [4X[28X[ 1, 3 ][128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D, [1], [], 2);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSet(D, [1], [], 3);[127X[104X
    [4X[28X[ 1, 3, 5 ][128X[104X
  [4X[32X[104X
  
  [1X8.2-3 DigraphMaximalIndependentSets[101X
  
  [33X[1;0Y[29X[2XDigraphMaximalIndependentSets[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xlimit[103X[, [3Xsize[103X]]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphMaximalIndependentSetsReps[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xlimit[103X[, [3Xsize[103X]]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphIndependentSets[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xlimit[103X[, [3Xsize[103X]]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphIndependentSetsReps[102X( [3Xdigraph[103X[, [3Xinclude[103X[, [3Xexclude[103X[, [3Xlimit[103X[, [3Xsize[103X]]]] ) [32X function[133X
  [33X[1;0Y[29X[2XDigraphMaximalIndependentSetsAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDigraphMaximalIndependentSetsRepsAttr[102X( [3Xdigraph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YAn immutable list of lists of positive integers.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is digraph, then these functions and attributes use [2XCliquesFinder[102X
  ([14X8.1-2[114X) to return independent sets of [3Xdigraph[103X. An independent set is defined
  by the set of vertices that it contains; see [2XIsMaximalIndependentSet[102X ([14X8.2-1[114X)
  and [2XIsIndependentSet[102X ([14X8.2-1[114X).[133X
  
  [33X[0;0YThe  optional  arguments [3Xinclude[103X and [3Xexclude[103X must each be a (possibly empty)
  list  of  vertices  of [3Xdigraph[103X, the optional argument [3Xlimit[103X must be either a
  positive  integer  or  [10Xinfinity[110X,  and  the  optional argument [3Xsize[103X must be a
  positive  integer.  If not specified, then [3Xinclude[103X and [3Xexclude[103X are chosen to
  be empty lists, and [3Xlimit[103X is set to [10Xinfinity[110X.[133X
  
  [33X[0;0YThe  functions will return as many suitable independent sets as possible, up
  to the number [3Xlimit[103X. These functions will find independent sets that contain
  all  of  the  vertices  of [3Xinclude[103X but do not contain any of the vertices of
  [3Xexclude[103X The argument [3Xsize[103X restricts the search to those cliques that contain
  precisely  [3Xsize[103X  vertices.  If  the function or attribute has [10XMaximal[110X in its
  name,  then  only  maximal  independent  sets  will  be  returned; otherwise
  non-maximal independent sets may be returned.[133X
  
  [33X[0;0YLet  [10XG[110X  denote  the [2XAutomorphismGroup[102X ([14X7.2-2[114X) of the [2XDigraphSymmetricClosure[102X
  ([14X3.3-12[114X)  of  the digraph formed from [3Xdigraph[103X by removing loops and ignoring
  the multiplicity of edges.[133X
  
  [8XDistinct independent sets[108X
        [33X[0;6Y[10XDigraphMaximalIndependentSets[110X and [10XDigraphIndependentSets[110X each return a
        duplicate-free  list of at most [3Xlimit[103X independent sets of [3Xdigraph[103X that
        satisfy the arguments.[133X
  
        [33X[0;6YThe computation may be significantly faster if [3Xinclude[103X and [3Xexclude[103X are
        invariant under the action of [10XG[110X on sets of vertices.[133X
  
  [8XRepresentatives of distinct orbits of independent sets[108X
        [33X[0;6YTo          use          [10XDigraphMaximalIndependentSetsReps[110X          or
        [10XDigraphIndependentSetsReps[110X,  the  arguments  [3Xinclude[103X  and [3Xexclude[103X must
        each be invariant under the action of [10XG[110X on sets of vertices.[133X
  
        [33X[0;6YIf  this  is  the  case,  then  [10XDigraphMaximalIndependentSetsReps[110X  and
        [10XDigraphIndependentSetsReps[110X  each return a list of at most [3Xlimit[103X orbits
        representatives  (under  the  action  of  [10XG[110X  on  sets  of vertices) of
        independent sets of [3Xdigraph[103X that satisfy the arguments.[133X
  
        [33X[0;6YThe  representatives  are  not  guaranteed  to  be in distinct orbits.
        However,  if  [3Xlim[103X  is  not  specified,  or  fewer than [3Xlim[103X results are
        returned,  then  there  will  be at least one representative from each
        orbit of maximal independent sets.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := CycleDigraph(5);[127X[104X
    [4X[28X<immutable cycle digraph with 5 vertices>[128X[104X
    [4X[25Xgap>[125X [27XDigraphMaximalIndependentSetsReps(D);[127X[104X
    [4X[28X[ [ 1, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphIndependentSetsReps(D);[127X[104X
    [4X[28X[ [ 1 ], [ 1, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XSet(DigraphMaximalIndependentSets(D));[127X[104X
    [4X[28X[ [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 2, 5 ], [ 3, 5 ] ][128X[104X
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