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<h1>5 Some special vertex subsets of a graph</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP005.htm#SECT001">ConnectedComponent</a>
<li> <A HREF="CHAP005.htm#SECT002">ConnectedComponents</a>
<li> <A HREF="CHAP005.htm#SECT003">Bicomponents</a>
<li> <A HREF="CHAP005.htm#SECT004">DistanceSet</a>
<li> <A HREF="CHAP005.htm#SECT005">Layers</a>
<li> <A HREF="CHAP005.htm#SECT006">IndependentSet</a>
</ol><p>
<p>
This chapter describes functions to determine certain special vertex
subsets of a graph.
<p>
<p>
<h2><a name="SECT001">5.1 ConnectedComponent</a></h2>
<p><p>
<a name = "SSEC001.1"></a>
<li><code>ConnectedComponent( </code><var>gamma</var><code>, </code><var>v</var><code> )</code>
<p>
This function returns the set of all vertices in <var>gamma</var> which can be
reached by a path starting at the vertex <var>v</var>. The graph <var>gamma</var> must be
simple.
<p>
See also <a href="CHAP005.htm#SSEC002.1">ConnectedComponents</a>.
<p>
<pre>
gap> ConnectedComponent( NullGraph( Group((1,2)) ), 2 );
[ 2 ]
gap> ConnectedComponent( JohnsonGraph(4,2), 2 );
[ 1, 2, 3, 4, 5, 6 ]
</pre>
<p>
<p>
<h2><a name="SECT002">5.2 ConnectedComponents</a></h2>
<p><p>
<a name = "SSEC002.1"></a>
<li><code>ConnectedComponents( </code><var>gamma</var><code> )</code>
<p>
This function returns a list of the vertex sets of the connected
components of <var>gamma</var>, which must be a simple graph.
<p>
See also <a href="CHAP005.htm#SSEC001.1">ConnectedComponent</a>.
<p>
<pre>
gap> ConnectedComponents( NullGraph( Group((1,2,3,4)) ) );
[ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
gap> ConnectedComponents( JohnsonGraph(4,2) );
[ [ 1, 2, 3, 4, 5, 6 ] ]
</pre>
<p>
<p>
<h2><a name="SECT003">5.3 Bicomponents</a></h2>
<p><p>
<a name = "SSEC003.1"></a>
<li><code>Bicomponents( </code><var>gamma</var><code> )</code>
<p>
If the graph <var>gamma</var>, which must be simple, is bipartite, this function
returns a length 2 list of bicomponents or parts of <var>gamma</var>, otherwise
the empty list is returned.
<p>
<strong>Note</strong> If <var>gamma</var> is bipartite but not connected, then its set of
bicomponents is not uniquely determined.
<p>
See also <a href="CHAP003.htm#SSEC019.1">IsBipartite</a>.
<p>
<pre>
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 ] ]
</pre>
<p>
<p>
<h2><a name="SECT004">5.4 DistanceSet</a></h2>
<p><p>
<a name = "SSEC004.1"></a>
<li><code>DistanceSet( </code><var>gamma</var><code>, </code><var>distances</var><code>, </code><var>V</var><code> )</code>
<li><code>DistanceSet( </code><var>gamma</var><code>, </code><var>distances</var><code>, </code><var>V</var><code>, </code><var>G</var><code> )</code>
<p>
Let <var>V</var> be a vertex or a nonempty list of vertices of <var>gamma</var>.
This function returns the set of vertices <i>w</i> of <var>gamma</var>, such that
<i>d</i>(<i>V</i> ,<i>w</i>) is in <var>distances</var> (a list or singleton distance).
<p>
The optional parameter <var>G</var>, if present, is assumed to be a subgroup of
\Aut(<i>gamma</i> ) fixing <var>V</var> setwise. Including such a <var>G</var> can speed up
the function.
<p>
See also <a href="CHAP003.htm#SSEC015.1">Distance</a> and <a href="CHAP006.htm#SSEC002.1">DistanceSetInduced</a>.
<p>
<pre>
gap> DistanceSet( JohnsonGraph(4,2), 1, [1,6] );
[ 2, 3, 4, 5 ]
</pre>
<p>
<p>
<h2><a name="SECT005">5.5 Layers</a></h2>
<p><p>
<a name = "SSEC005.1"></a>
<li><code>Layers( </code><var>gamma</var><code>, </code><var>V</var><code> )</code>
<li><code>Layers( </code><var>gamma</var><code>, </code><var>V</var><code>, </code><var>G</var><code> )</code>
<p>
Let <var>V</var> be a vertex or a nonempty list of vertices of <var>gamma</var>. This
function returns a list whose <i>i</i>-th element is the set of vertices of
<var>gamma</var> at distance <i>i</i>−1 from <var>V</var>.
<p>
The optional parameter <var>G</var>, if present, is assumed to be a subgroup of
\Aut(<i>gamma</i> ) which fixes <var>V</var> setwise. Including such a <var>G</var> can speed
up the function.
<p>
See also <a href="CHAP003.htm#SSEC015.1">Distance</a>.
<p>
<pre>
gap> Layers( JohnsonGraph(4,2), 6 );
[ [ 6 ], [ 2, 3, 4, 5 ], [ 1 ] ]
</pre>
<p>
<p>
<h2><a name="SECT006">5.6 IndependentSet</a></h2>
<p><p>
<a name = "SSEC006.1"></a>
<li><code>IndependentSet( </code><var>gamma</var><code> )</code>
<li><code>IndependentSet( </code><var>gamma</var><code>, </code><var>indset</var><code> )</code>
<li><code>IndependentSet( </code><var>gamma</var><code>, </code><var>indset</var><code>, </code><var>forbidden</var><code> )</code>
<p>
Returns a (hopefully large) independent set of the graph <var>gamma</var>, which
must be simple. An <strong>independent set</strong> of <var>gamma</var> is a set of vertices
of <var>gamma</var>, 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 <var>indset</var> (default: <code>[]</code>), and not contain
any element of <var>forbidden</var> (default: <code>[]</code>, in which case the returned
independent set is maximal).
<p>
An error is signalled if <var>indset</var> and <var>forbidden</var> have non-trivial
intersection.
<p>
See also <a href="CHAP007.htm#SSEC002.1">CompleteSubgraphs</a> and <a href="CHAP007.htm#SSEC003.1">CompleteSubgraphsOfGivenSize</a>, which
can be used on the complement graph of <var>gamma</var> to look seriously for
independent sets.
<p>
<pre>
gap> IndependentSet( JohnsonGraph(4,2), [3] );
[ 3, 4 ]
</pre>
<p>
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<P>
<address>grape manual<br>January 2016 (Debian 4r7+ds-3)
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