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<title>Boost Graph Library: is_kuratowski_subgraph</title>
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<h1><tt>is_kuratowski_subgraph</tt></h1>
<pre>template <typename Graph, typename ForwardIterator, typename VertexIndexMap>
bool is_kuratowski_subgraph(const Graph& g, ForwardIterator begin, ForwardIterator end, VertexIndexMap vm)
</pre>
<p>
<tt>is_kuratowski_subgraph(g, begin, end)</tt> returns <tt>true</tt> exactly
when the sequence of edges defined by the range <tt>[begin, end)</tt> forms a
<a href="./planar_graphs.html#kuratowskisubgraphs"> Kuratowski subgraph</a> in
the graph <tt>g</tt>. If you need to verify that an arbitrary graph has a
<i>K<sub>5</sub></i> or <i>K<sub>3,3</sub></i> minor, you should use the
function <tt><a href="boyer_myrvold.html">boyer_myrvold_planarity_test</a></tt>
to isolate such a minor instead of this function. <tt>is_kuratowski_subgraph
</tt> exists to aid in testing and verification of the function
<tt>boyer_myrvold_planarity_test</tt>, and for that reason, it expects its
input to be a restricted set of edges forming a Kuratowski subgraph, as
described in detail below.
<p>
<tt>is_kuratowski_subgraph</tt> creates a temporary graph out of the sequence
of edges given and repeatedly contracts edges until it ends up with a graph
with either all edges of degree 3 or all edges of degree 4. The final
contracted graph is then checked against <i>K<sub>5</sub></i> or
<i>K<sub>3,3</sub></i> using the Boost Graph Library's
<a href="isomorphism.html">isomorphism</a>
function. The contraction process starts by choosing edges adjacent to a vertex
of degree 1 and contracting those. When none are left, it moves on to edges
adjacent to a vertex of degree 2. If only degree 3 vertices are left after this
stage, the graph is checked against <i>K<sub>3,3</sub></i>. Otherwise, if
there's at least one degree 4 vertex, edges adjacent to degree 3 vertices are
contracted as neeeded and the final graph is compared to <i>K<sub>5</sub></i>.
<p>
In order for this process to be deterministic, we make the following two
restrictions on the input graph given to <tt>is_kuratowski_subgraph</tt>:
<ol>
<li>No edge contraction needed to produce a kuratowski subgraph results in
multiple edges between the same pair of vertices (No edge <i>{a,b}</i> will be
contracted at any point in the contraction process if <i>a</i> and <i>b</i>
share a common neighbor.)
</li><li>If the graph contracts to a <i>K<sub>5</sub></i>, once the graph has
been contracted to only vertices of degree at least 3, no cycles exist that
contain solely degree 3 vertices.
</li></ol>
The second restriction is needed both to discriminate between targeting a
<i>K<sub>5</sub></i> or a <i>K<sub>3,3</sub></i> and to determinstically
contract the vertices of degree 4 once the <i>K<sub>5</sub></i> has been
targeted. The Kuratowski subgraph output by the function <tt>
<a href="boyer_myrvold.html">boyer_myrvold_planarity_test</a></tt> is
guaranteed to meet both of the above requirements.
<h3>Complexity</h3>
On a graph with <i>n</i> vertices, this function runs in time <i>O(n)</i>.
<h3>Where Defined</h3>
<p>
<a href="../../../boost/graph/is_kuratowski_subgraph.hpp">
<tt>boost/graph/is_kuratowski_subgraph.hpp</tt>
</a>
</p><h3>Parameters</h3>
IN: <tt>Graph& g</tt>
<blockquote>
An undirected graph with no self-loops or parallel edges. The graph type must
be a model of <a href="VertexListGraph.html">Vertex List Graph</a>.
</blockquote>
IN: <tt>ForwardIterator</tt>
<blockquote>
A ForwardIterator with value_type
<tt>graph_traits<Graph>::edge_descriptor</tt>.
</blockquote>
IN: <tt>VertexIndexMap vm</tt>
<blockquote>
A <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map
</a> that maps vertices from <tt>g</tt> to distinct integers in the range
<tt>[0, num_vertices(g) )</tt><br>
<b>Default</b>: <tt>get(vertex_index,g)</tt><br>
</blockquote>
<h3>Example</h3>
<p>
<a href="../example/kuratowski_subgraph.cpp">
<tt>examples/kuratowski_subgraph.cpp</tt>
</a>
</p><h3>See Also</h3>
<p>
<a href="planar_graphs.html">Planar Graphs in the Boost Graph Library</a>
<br>
</p><hr>
Copyright � 2007 Aaron Windsor (<a href="mailto:aaron.windsor@gmail.com">
aaron.windsor@gmail.com</a>)
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