127 lines
4.6 KiB
HTML
127 lines
4.6 KiB
HTML
<html><head><!-- Copyright 2007 Aaron Windsor
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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-->
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<title>Boost Graph Library: is_kuratowski_subgraph</title>
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</head>
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<body alink="#ff0000"
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bgcolor="#ffffff"
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link="#0000ee"
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text="#000000"
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vlink="#551a8b">
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<img src="../../../boost.png" alt="C++ Boost" height="86" width="277">
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<br clear="">
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<h1><tt>is_kuratowski_subgraph</tt></h1>
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<pre>template <typename Graph, typename ForwardIterator, typename VertexIndexMap>
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bool is_kuratowski_subgraph(const Graph& g, ForwardIterator begin, ForwardIterator end, VertexIndexMap vm)
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</pre>
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<p>
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<tt>is_kuratowski_subgraph(g, begin, end)</tt> returns <tt>true</tt> exactly
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when the sequence of edges defined by the range <tt>[begin, end)</tt> forms a
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<a href="./planar_graphs.html#kuratowskisubgraphs"> Kuratowski subgraph</a> in
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the graph <tt>g</tt>. If you need to verify that an arbitrary graph has a
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<i>K<sub>5</sub></i> or <i>K<sub>3,3</sub></i> minor, you should use the
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function <tt><a href="boyer_myrvold.html">boyer_myrvold_planarity_test</a></tt>
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to isolate such a minor instead of this function. <tt>is_kuratowski_subgraph
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</tt> exists to aid in testing and verification of the function
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<tt>boyer_myrvold_planarity_test</tt>, and for that reason, it expects its
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input to be a restricted set of edges forming a Kuratowski subgraph, as
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described in detail below.
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<p>
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<tt>is_kuratowski_subgraph</tt> creates a temporary graph out of the sequence
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of edges given and repeatedly contracts edges until it ends up with a graph
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with either all edges of degree 3 or all edges of degree 4. The final
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contracted graph is then checked against <i>K<sub>5</sub></i> or
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<i>K<sub>3,3</sub></i> using the Boost Graph Library's
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<a href="isomorphism.html">isomorphism</a>
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function. The contraction process starts by choosing edges adjacent to a vertex
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of degree 1 and contracting those. When none are left, it moves on to edges
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adjacent to a vertex of degree 2. If only degree 3 vertices are left after this
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stage, the graph is checked against <i>K<sub>3,3</sub></i>. Otherwise, if
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there's at least one degree 4 vertex, edges adjacent to degree 3 vertices are
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contracted as neeeded and the final graph is compared to <i>K<sub>5</sub></i>.
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<p>
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In order for this process to be deterministic, we make the following two
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restrictions on the input graph given to <tt>is_kuratowski_subgraph</tt>:
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<ol>
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<li>No edge contraction needed to produce a kuratowski subgraph results in
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multiple edges between the same pair of vertices (No edge <i>{a,b}</i> will be
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contracted at any point in the contraction process if <i>a</i> and <i>b</i>
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share a common neighbor.)
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</li><li>If the graph contracts to a <i>K<sub>5</sub></i>, once the graph has
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been contracted to only vertices of degree at least 3, no cycles exist that
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contain solely degree 3 vertices.
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</li></ol>
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The second restriction is needed both to discriminate between targeting a
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<i>K<sub>5</sub></i> or a <i>K<sub>3,3</sub></i> and to determinstically
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contract the vertices of degree 4 once the <i>K<sub>5</sub></i> has been
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targeted. The Kuratowski subgraph output by the function <tt>
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<a href="boyer_myrvold.html">boyer_myrvold_planarity_test</a></tt> is
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guaranteed to meet both of the above requirements.
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<h3>Complexity</h3>
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On a graph with <i>n</i> vertices, this function runs in time <i>O(n)</i>.
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<h3>Where Defined</h3>
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<p>
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<a href="../../../boost/graph/is_kuratowski_subgraph.hpp">
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<tt>boost/graph/is_kuratowski_subgraph.hpp</tt>
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</a>
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</p><h3>Parameters</h3>
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IN: <tt>Graph& g</tt>
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<blockquote>
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An undirected graph with no self-loops or parallel edges. The graph type must
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be a model of <a href="VertexListGraph.html">Vertex List Graph</a>.
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</blockquote>
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IN: <tt>ForwardIterator</tt>
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<blockquote>
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A ForwardIterator with value_type
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<tt>graph_traits<Graph>::edge_descriptor</tt>.
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</blockquote>
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IN: <tt>VertexIndexMap vm</tt>
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<blockquote>
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A <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map
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</a> that maps vertices from <tt>g</tt> to distinct integers in the range
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<tt>[0, num_vertices(g) )</tt><br>
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<b>Default</b>: <tt>get(vertex_index,g)</tt><br>
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</blockquote>
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<h3>Example</h3>
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<p>
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<a href="../example/kuratowski_subgraph.cpp">
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<tt>examples/kuratowski_subgraph.cpp</tt>
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</a>
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</p><h3>See Also</h3>
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<p>
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<a href="planar_graphs.html">Planar Graphs in the Boost Graph Library</a>
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<br>
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</p><hr>
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Copyright <20> 2007 Aaron Windsor (<a href="mailto:aaron.windsor@gmail.com">
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aaron.windsor@gmail.com</a>)
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</body></html>
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