| <html><head><!-- Copyright 2007 Aaron Windsor |
| |
| Distributed under the Boost Software License, Version 1.0. |
| (See accompanying file LICENSE_1_0.txt or copy at |
| http://www.boost.org/LICENSE_1_0.txt) |
| |
| --> |
| <title>Boost Graph Library: is_kuratowski_subgraph</title> |
| </head> |
| <body alink="#ff0000" |
| bgcolor="#ffffff" |
| link="#0000ee" |
| text="#000000" |
| vlink="#551a8b"> |
| <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> |
