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%A  special.tex               GRAPE documentation              Leonard Soicher
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%
%
\def\GRAPE{\sf GRAPE}
\def\nauty{\it nauty}
\def\G{\Gamma}
\def\Aut{{\rm Aut}\,}
\def\x{\times}
\Chapter{Some special vertex subsets of a graph}

This chapter describes functions to determine certain special vertex
subsets of a graph.

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\Section{ConnectedComponent}

\>ConnectedComponent( <gamma>, <v> )

This function  returns the set  of all vertices  in <gamma> which  can be
reached by a path starting at the vertex  <v>.  The graph <gamma> must be
simple.

See also "ConnectedComponents".

\beginexample
gap> ConnectedComponent( NullGraph( Group((1,2)) ), 2 );
[ 2 ]
gap> ConnectedComponent( JohnsonGraph(4,2), 2 );
[ 1, 2, 3, 4, 5, 6 ] 
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{ConnectedComponents}

\>ConnectedComponents( <gamma> )

This  function returns  a  list  of  the  vertex  sets  of  the connected
components of <gamma>, which must be a simple graph.

See also "ConnectedComponent".

\beginexample
gap> ConnectedComponents( NullGraph( Group((1,2,3,4)) ) );
[ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
gap> ConnectedComponents( JohnsonGraph(4,2) );
[ [ 1, 2, 3, 4, 5, 6 ] ] 
\endexample

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\Section{Bicomponents}

\>Bicomponents( <gamma> )

If the graph  <gamma>, which must be simple,  is bipartite, this function
returns  a length 2 list  of  bicomponents or parts of <gamma>, otherwise
the empty list is returned.

*Note* If <gamma> is bipartite but not connected, then its set of
bicomponents is not uniquely determined.  

See also "IsBipartite".

\beginexample
gap> Bicomponents( NullGraph(SymmetricGroup(4)) );
[ [ 1 .. 3 ], [ 4 ] ]
gap> Bicomponents( JohnsonGraph(4,2) );
[  ]
gap> Bicomponents( BipartiteDouble( JohnsonGraph(4,2) ) );
[ [ 1, 2, 3, 4, 5, 6 ], [ 7, 8, 9, 10, 11, 12 ] ]
\endexample

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\Section{DistanceSet}

\>DistanceSet( <gamma>, <distances>, <V> )
\>DistanceSet( <gamma>, <distances>, <V>, <G> )

Let <V> be a vertex or a nonempty list of vertices of <gamma>.
This function returns the set of vertices $w$ of <gamma>, such that
$d(<V>,w)$ is in <distances> (a list or singleton distance).

The optional  parameter <G>, if  present, is assumed to be  a subgroup of
$\Aut(<gamma>)$  fixing <V> setwise.  Including  such a <G> can speed  up
the function.

See also "Distance" and "DistanceSetInduced".

\beginexample
gap> DistanceSet( JohnsonGraph(4,2), 1, [1,6] );
[ 2, 3, 4, 5 ] 
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Layers}

\>Layers( <gamma>, <V> )
\>Layers( <gamma>, <V>, <G> )


Let <V> be a vertex or a nonempty list of vertices of <gamma>.  This
function returns a list whose $i$-th element is the set of vertices of
<gamma> at distance $i-1$ from <V>.

The optional  parameter  <G>, if present, is assumed to  be a subgroup of
$\Aut(<gamma>)$ which fixes <V>  setwise.  Including such a <G> can speed
up the function.

See also "Distance".

\beginexample
gap> Layers( JohnsonGraph(4,2), 6 );
[ [ 6 ], [ 2, 3, 4, 5 ], [ 1 ] ] 
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{IndependentSet}

\>IndependentSet( <gamma> )
\>IndependentSet( <gamma>, <indset> )
\>IndependentSet( <gamma>, <indset>, <forbidden> )

Returns a (hopefully large) independent set of the graph <gamma>, which
must be simple.  An *independent set* of <gamma> is a set of vertices
of <gamma>, no two of which are joined by an edge.  At present, a
greedy algorithm is used.  The returned independent set will contain
the (assumed) independent set <indset> (default: `[]'), and not contain
any element of <forbidden> (default: `[]', in which case the returned
independent set is maximal).

An error is signalled if <indset> and <forbidden> have non-trivial
intersection.

See also "CompleteSubgraphs" and "CompleteSubgraphsOfGivenSize", which
can be used on the complement graph of <gamma> to look seriously for
independent sets.

\beginexample
gap> IndependentSet( JohnsonGraph(4,2), [3] );
[ 3, 4 ] 
\endexample