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- <html><head><!-- Copyright 2007 Aaron Windsor
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- (See accompanying file LICENSE_1_0.txt or copy at
- http://www.boost.org/LICENSE_1_0.txt)
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- <title>Boost Graph Library: is_kuratowski_subgraph</title>
- </head>
- <body alink="#ff0000"
- bgcolor="#ffffff"
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- <img src="../../../boost.png" alt="C++ Boost" height="86" width="277">
- <br clear="">
- <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>)
-
- </body></html>
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