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1 | <html><head><!-- Copyright 2007 Aaron Windsor |
| 2 | ||
| 3 | Distributed under the Boost Software License, Version 1.0. | |
| 4 | (See accompanying file LICENSE_1_0.txt or copy at | |
| 5 | http://www.boost.org/LICENSE_1_0.txt) | |
| 6 | ||
| 7 | --> | |
| 8 | <title>Boost Graph Library: is_kuratowski_subgraph</title> | |
| 9 | </head> | |
| 10 | <body alink="#ff0000" | |
| 11 | bgcolor="#ffffff" | |
| 12 | link="#0000ee" | |
| 13 | text="#000000" | |
| 14 | vlink="#551a8b"> | |
| 15 | <img src="../../../boost.png" alt="C++ Boost" height="86" width="277"> | |
| 16 | ||
| 17 | <br clear=""> | |
| 18 | ||
| 19 | <h1><tt>is_kuratowski_subgraph</tt></h1> | |
| 20 | ||
| 21 | <pre>template <typename Graph, typename ForwardIterator, typename VertexIndexMap> | |
| 22 | bool is_kuratowski_subgraph(const Graph& g, ForwardIterator begin, ForwardIterator end, VertexIndexMap vm) | |
| 23 | </pre> | |
| 24 | ||
| 25 | <p> | |
| 26 | ||
| 27 | <tt>is_kuratowski_subgraph(g, begin, end)</tt> returns <tt>true</tt> exactly | |
| 28 | when the sequence of edges defined by the range <tt>[begin, end)</tt> forms a | |
| 29 | <a href="./planar_graphs.html#kuratowskisubgraphs"> Kuratowski subgraph</a> in | |
| 30 | the graph <tt>g</tt>. If you need to verify that an arbitrary graph has a | |
| 31 | <i>K<sub>5</sub></i> or <i>K<sub>3,3</sub></i> minor, you should use the | |
| 32 | function <tt><a href="boyer_myrvold.html">boyer_myrvold_planarity_test</a></tt> | |
| 33 | to isolate such a minor instead of this function. <tt>is_kuratowski_subgraph | |
| 34 | </tt> exists to aid in testing and verification of the function | |
| 35 | <tt>boyer_myrvold_planarity_test</tt>, and for that reason, it expects its | |
| 36 | input to be a restricted set of edges forming a Kuratowski subgraph, as | |
| 37 | described in detail below. | |
| 38 | <p> | |
| 39 | <tt>is_kuratowski_subgraph</tt> creates a temporary graph out of the sequence | |
| 40 | of edges given and repeatedly contracts edges until it ends up with a graph | |
| 41 | with either all edges of degree 3 or all edges of degree 4. The final | |
| 42 | contracted graph is then checked against <i>K<sub>5</sub></i> or | |
| 43 | <i>K<sub>3,3</sub></i> using the Boost Graph Library's | |
| 44 | <a href="isomorphism.html">isomorphism</a> | |
| 45 | function. The contraction process starts by choosing edges adjacent to a vertex | |
| 46 | of degree 1 and contracting those. When none are left, it moves on to edges | |
| 47 | adjacent to a vertex of degree 2. If only degree 3 vertices are left after this | |
| 48 | stage, the graph is checked against <i>K<sub>3,3</sub></i>. Otherwise, if | |
| 49 | there's at least one degree 4 vertex, edges adjacent to degree 3 vertices are | |
| 50 | contracted as neeeded and the final graph is compared to <i>K<sub>5</sub></i>. | |
| 51 | <p> | |
| 52 | In order for this process to be deterministic, we make the following two | |
| 53 | restrictions on the input graph given to <tt>is_kuratowski_subgraph</tt>: | |
| 54 | <ol> | |
| 55 | <li>No edge contraction needed to produce a kuratowski subgraph results in | |
| 56 | multiple edges between the same pair of vertices (No edge <i>{a,b}</i> will be | |
| 57 | contracted at any point in the contraction process if <i>a</i> and <i>b</i> | |
| 58 | share a common neighbor.) | |
| 59 | </li><li>If the graph contracts to a <i>K<sub>5</sub></i>, once the graph has | |
| 60 | been contracted to only vertices of degree at least 3, no cycles exist that | |
| 61 | contain solely degree 3 vertices. | |
| 62 | </li></ol> | |
| 63 | The second restriction is needed both to discriminate between targeting a | |
| 64 | <i>K<sub>5</sub></i> or a <i>K<sub>3,3</sub></i> and to determinstically | |
| 65 | contract the vertices of degree 4 once the <i>K<sub>5</sub></i> has been | |
| 66 | targeted. The Kuratowski subgraph output by the function <tt> | |
| 67 | <a href="boyer_myrvold.html">boyer_myrvold_planarity_test</a></tt> is | |
| 68 | guaranteed to meet both of the above requirements. | |
| 69 | ||
| 70 | ||
| 71 | <h3>Complexity</h3> | |
| 72 | ||
| 73 | On a graph with <i>n</i> vertices, this function runs in time <i>O(n)</i>. | |
| 74 | ||
| 75 | <h3>Where Defined</h3> | |
| 76 | ||
| 77 | <p> | |
| 78 | <a href="../../../boost/graph/is_kuratowski_subgraph.hpp"> | |
| 79 | <tt>boost/graph/is_kuratowski_subgraph.hpp</tt> | |
| 80 | </a> | |
| 81 | ||
| 82 | </p><h3>Parameters</h3> | |
| 83 | ||
| 84 | IN: <tt>Graph& g</tt> | |
| 85 | ||
| 86 | <blockquote> | |
| 87 | An undirected graph with no self-loops or parallel edges. The graph type must | |
| 88 | be a model of <a href="VertexListGraph.html">Vertex List Graph</a>. | |
| 89 | </blockquote> | |
| 90 | ||
| 91 | IN: <tt>ForwardIterator</tt> | |
| 92 | ||
| 93 | <blockquote> | |
| 94 | A ForwardIterator with value_type | |
| 95 | <tt>graph_traits<Graph>::edge_descriptor</tt>. | |
| 96 | </blockquote> | |
| 97 | ||
| 98 | IN: <tt>VertexIndexMap vm</tt> | |
| 99 | ||
| 100 | <blockquote> | |
| 101 | A <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map | |
| 102 | </a> that maps vertices from <tt>g</tt> to distinct integers in the range | |
| 103 | <tt>[0, num_vertices(g) )</tt><br> | |
| 104 | <b>Default</b>: <tt>get(vertex_index,g)</tt><br> | |
| 105 | </blockquote> | |
| 106 | ||
| 107 | ||
| 108 | <h3>Example</h3> | |
| 109 | ||
| 110 | <p> | |
| 111 | <a href="../example/kuratowski_subgraph.cpp"> | |
| 112 | <tt>examples/kuratowski_subgraph.cpp</tt> | |
| 113 | </a> | |
| 114 | ||
| 115 | </p><h3>See Also</h3> | |
| 116 | ||
| 117 | <p> | |
| 118 | <a href="planar_graphs.html">Planar Graphs in the Boost Graph Library</a> | |
| 119 | ||
| 120 | ||
| 121 | <br> | |
| 122 | </p><hr> | |
| 123 |