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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 | --><title>Boost Graph Library: Planar Canonical Ordering</title> | |
| 8 | </head> | |
| 9 | <body alink="#ff0000" | |
| 10 | bgcolor="#ffffff" | |
| 11 | link="#0000ee" | |
| 12 | text="#000000" | |
| 13 | vlink="#551a8b"> | |
| 14 | <img src="../../../boost.png" alt="C++ Boost" height="86" width="277"> | |
| 15 | ||
| 16 | <br clear=""> | |
| 17 | ||
| 18 | <h1>Planar Canonical Ordering</h1> | |
| 19 | ||
| 20 | <pre>template <typename Graph, typename PlanarEmbedding, typename OutputIterator, typename VertexIndexMap> | |
| 21 | void planar_canonical_ordering(const Graph& g, PlanarEmbedding embedding, OutputIterator ordering, VertexIndexMap vm); | |
| 22 | </pre> | |
| 23 | ||
| 24 | <p> | |
| 25 | A <i>planar canonical ordering</i> is an ordering <i>v<sub>1</sub>, | |
| 26 | v<sub>2</sub>, ..., v<sub>n</sub></i> of the vertices of a | |
| 27 | <a href="make_maximal_planar.html">maximal</a> | |
| 28 | <a href="planar_graphs.html">planar</a> graph having the property that, for | |
| 29 | each <i>k</i>, <i>3 <= k < n</i>, the graph induced by | |
| 30 | <i>v<sub>1</sub>, v<sub>2</sub>, ..., v<sub>k</sub></i> | |
| 31 | </p><ul> | |
| 32 | <li>is biconnected and contains the edge <i>{v<sub>1</sub>, v<sub>2</sub>}</i> | |
| 33 | on its outer face. | |
| 34 | </li><li>has any vertices in the range <i>v<sub>1</sub>, v<sub>2</sub>, ..., | |
| 35 | v<sub>k</sub></i> that are adjacent to <i>v<sub>(k+1)</sub></i> on its outer | |
| 36 | face, and these vertices form a path along the outer face. | |
| 37 | </li></ul> | |
| 38 | ||
| 39 | Let <i>G<sub>k</sub></i> be the graph induced by the first <i>k</i> vertices in | |
| 40 | the canonical ordering, along with all edges between any of the first <i>k</i> | |
| 41 | vertices. After <i>G<sub>k</sub></i> has been drawn, the <i>(k+1)</i>st vertex | |
| 42 | can be drawn easily without edge crossings, since it's adjacent only to a | |
| 43 | consecutive sequence of vertices on the outer face of <i>G<sub>k</sub></i>. | |
| 44 | <p> | |
| 45 | </p><blockquote> | |
| 46 | <center> | |
| 47 | <img src="./figs/canonical_ordering.png"> | |
| 48 | </center> | |
| 49 | </blockquote> | |
| 50 | ||
| 51 | A planar canonical ordering exists for every maximal planar graph with at | |
| 52 | least 2 vertices. <tt>planar_canonical_ordering</tt> expects the input graph | |
| 53 | to have at least 2 vertices. | |
| 54 | <p> | |
| 55 | ||
| 56 | The planar canonical ordering is used as an input in some planar graph drawing | |
| 57 | algorithms, particularly those that create a straight line embedding. | |
| 58 | de Fraysseix, Pach, and Pollack | |
| 59 | [<a href="./bibliography.html#defraysseixpachpollack90">72</a>] | |
| 60 | first proved the | |
| 61 | existence of such an ordering and showed how to compute one in time | |
| 62 | <i>O(n)</i> on a maximal planar graph with <i>n</i> vertices. | |
| 63 | ||
| 64 | ||
| 65 | <h3>Complexity</h3> | |
| 66 | If the vertex index map provides constant-time access to indices, this | |
| 67 | function takes time <i>O(n + m)</i> for a planar graph with <i>n</i> vertices | |
| 68 | and <i>m</i> edges. Note that | |
| 69 | in a simple planar graph with <i>f</i> faces, <i>m</i> edges, and <i>n</i> | |
| 70 | vertices, both <i>f</i> and <i>m</i> are <i>O(n)</i>. | |
| 71 | ||
| 72 | <h3>Where Defined</h3> | |
| 73 | ||
| 74 | <p> | |
| 75 | <a href="../../../boost/graph/planar_canonical_ordering.hpp"> | |
| 76 | <tt>boost/graph/planar_canonical_ordering.hpp</tt></a> | |
| 77 | ||
| 78 | </p><h3>Parameters</h3> | |
| 79 | ||
| 80 | IN: <tt>Graph& g</tt> | |
| 81 | ||
| 82 | <blockquote> | |
| 83 | An undirected graph. The graph type must be a model of | |
| 84 | <a href="VertexAndEdgeListGraph.html">VertexAndEdgeListGraph</a>. | |
| 85 | The graph must: | |
| 86 | <ul> | |
| 87 | <li>Be maximal planar.</li> | |
| 88 | <li>Have at least two vertices.</li> | |
| 89 | </ul> | |
| 90 | </blockquote> | |
| 91 | ||
| 92 | IN: <tt>PlanarEmbedding</tt> | |
| 93 | ||
| 94 | <blockquote> | |
| 95 | A model of <a href="PlanarEmbedding.html">PlanarEmbedding</a>. | |
| 96 | </blockquote> | |
| 97 | ||
| 98 | IN: <tt>OutputIterator</tt> | |
| 99 | ||
| 100 | <blockquote> | |
| 101 | An OutputIterator with <tt>value_type</tt> equal to | |
| 102 | <tt>graph_traits<Graph>::vertex_descriptor</tt>. The canonical ordering | |
| 103 | will be written to this iterator. | |
| 104 | </blockquote> | |
| 105 | ||
| 106 | IN: <tt>VertexIndexMap vm</tt> | |
| 107 | ||
| 108 | <blockquote> | |
| 109 | A <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map | |
| 110 | </a> that maps vertices from <tt>g</tt> to distinct integers in the range | |
| 111 | <tt>[0, num_vertices(g) )</tt><br> | |
| 112 | <b>Default</b>: <tt>get(vertex_index,g)</tt><br> | |
| 113 | </blockquote> | |
| 114 | ||
| 115 | <h3>Example</h3> | |
| 116 | ||
| 117 | <p> | |
| 118 | <a href="../example/canonical_ordering.cpp"> | |
| 119 | <tt>examples/canonical_ordering.cpp</tt></a> | |
| 120 | ||
| 121 | </p><h3>See Also</h3> | |
| 122 | ||
| 123 | <p> | |
| 124 | <a href="planar_graphs.html">Planar Graphs in the Boost Graph Library</a> | |
| 125 | ||
| 126 | <br> | |
| 127 | </p><hr> | |
| 128 |