<Chapter Label="Finding cliques and independent sets">
  <Heading>Finding cliques and independent sets</Heading>

    In &Digraphs;, a <E>clique</E> of a digraph is a set of mutually adjacent
    vertices of the digraph, and an <E>independent set</E> is a set of mutually
    non-adjacent vertices of the digraph. A <E>maximal clique</E> is a clique
    which is not properly contained in another clique, and a <E>maximal
      independent set</E> is an independent set which is not properly contained
    in another independent set.  Using this definition in &Digraphs;, 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 <E>complete
      subgraph</E>.
    <P/>

    &Digraphs; 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 &Digraphs; 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.  &Digraphs; uses several strategies to increase the performance
    of these calculations.
    <P/>

    From 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 <Ref
      Prop="DigraphHasLoops"/> and <Ref Prop="IsMultiDigraph"/> for more
    information about these properties.  Therefore given a digraph
    <A>digraph</A>, the cliques and independent sets of <A>digraph</A> are
    equal to the cliques and independent sets of the digraph:

    <List>
      <Item>
        <C>DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(</C><A>digraph</A><C>))</C>.
      </Item>
    </List>

    See <Ref Oper="DigraphRemoveLoops"/> and <Ref
      Oper="DigraphRemoveAllMultipleEdges"/> 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 <C>[u,v]</C> for which
    the corresponding reverse edge <C>[v,u]</C> is not present.  Therefore,
    overall, the cliques of <A>digraph</A> are equal to the cliques of the
    symmetric digraph:

    <List>
      <Item>
        <C>MaximalSymmetricSubdigraphWithoutLoops(</C><A>digraph</A><C>)</C>.
      </Item>
    </List>

    See <Ref Oper="MaximalSymmetricSubdigraphWithoutLoops"/> for more
    information about this. The <Ref Attr="AutomorphismGroup"
      Label="for a digraph" /> of this symmetric digraph contains the
    automorphism group of <A>digraph</A> 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
    <A>digraph</A> will return orbit representatives of cliques under the
    action of the automorphism group of the <E>maximal symmetric subdigraph
      without loops</E> on sets of vertices.<P/>

    The independent sets of a digraph are equal to the independent sets of the
    <Ref Oper="DigraphSymmetricClosure"/>.  Therefore, overall, the independent
    sets of <A>digraph</A> are equal to the independent sets of the symmetric
    digraph:

    <List>
      <Item>
        <C>DigraphSymmetricClosure(DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(
        </C><A>digraph</A><C>)))</C>.
      </Item>
    </List>

    Again, the automorphism group of this symmetric digraph contains the
    automorphism group of <A>digraph</A>. Since a search for independent sets
    is equal to a search for cliques in the <Ref Oper="DigraphDual"/>, the
    methods used in &Digraphs; 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
    <A>digraph</A> will return orbit representatives of independent sets under
    the action of the automorphism group of the <E>symmetric closure</E> of the
    digraph formed by removing loops and multiple edges.<P/>

    Please note that in &Digraphs;, cliques and independent sets are not
    required to be maximal. Some authors use the word clique to mean
    <E>maximal</E> clique, and some authors use the phrase independent set to
    mean <E>maximal</E> independent set, but please note that &Digraphs; does
    not use this definition.
    <P/>

  <Section><Heading>Finding cliques</Heading>
    <#Include Label="IsClique">
    <#Include Label="CliquesFinder">
    <#Include Label="DigraphClique">
    <#Include Label="DigraphMaximalCliques">
    <#Include Label="CliqueNumber">
  </Section>

  <Section><Heading>Finding independent sets</Heading>
    <#Include Label="IsIndependentSet">
    <#Include Label="DigraphIndependentSet">
    <#Include Label="DigraphMaximalIndependentSets">
  </Section>
</Chapter>
