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- <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: Planar Canonical Ordering</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>Planar Canonical Ordering</h1>
- <pre>template <typename Graph, typename PlanarEmbedding, typename OutputIterator, typename VertexIndexMap>
- void planar_canonical_ordering(const Graph& g, PlanarEmbedding embedding, OutputIterator ordering, VertexIndexMap vm);
- </pre>
- <p>
- A <i>planar canonical ordering</i> is an ordering <i>v<sub>1</sub>,
- v<sub>2</sub>, ..., v<sub>n</sub></i> of the vertices of a
- <a href="make_maximal_planar.html">maximal</a>
- <a href="planar_graphs.html">planar</a> graph having the property that, for
- each <i>k</i>, <i>3 <= k < n</i>, the graph induced by
- <i>v<sub>1</sub>, v<sub>2</sub>, ..., v<sub>k</sub></i>
- </p><ul>
- <li>is biconnected and contains the edge <i>{v<sub>1</sub>, v<sub>2</sub>}</i>
- on its outer face.
- </li><li>has any vertices in the range <i>v<sub>1</sub>, v<sub>2</sub>, ...,
- v<sub>k</sub></i> that are adjacent to <i>v<sub>(k+1)</sub></i> on its outer
- face, and these vertices form a path along the outer face.
- </li></ul>
- Let <i>G<sub>k</sub></i> be the graph induced by the first <i>k</i> vertices in
- the canonical ordering, along with all edges between any of the first <i>k</i>
- vertices. After <i>G<sub>k</sub></i> has been drawn, the <i>(k+1)</i>st vertex
- can be drawn easily without edge crossings, since it's adjacent only to a
- consecutive sequence of vertices on the outer face of <i>G<sub>k</sub></i>.
- <p>
- </p><blockquote>
- <center>
- <img src="./figs/canonical_ordering.png">
- </center>
- </blockquote>
- A planar canonical ordering exists for every maximal planar graph with at
- least 2 vertices. <tt>planar_canonical_ordering</tt> expects the input graph
- to have at least 2 vertices.
- <p>
- The planar canonical ordering is used as an input in some planar graph drawing
- algorithms, particularly those that create a straight line embedding.
- de Fraysseix, Pach, and Pollack
- [<a href="./bibliography.html#defraysseixpachpollack90">72</a>]
- first proved the
- existence of such an ordering and showed how to compute one in time
- <i>O(n)</i> on a maximal planar graph with <i>n</i> vertices.
- <h3>Complexity</h3>
- If the vertex index map provides constant-time access to indices, this
- function takes time <i>O(n + m)</i> for a planar graph with <i>n</i> vertices
- and <i>m</i> edges. Note that
- in a simple planar graph with <i>f</i> faces, <i>m</i> edges, and <i>n</i>
- vertices, both <i>f</i> and <i>m</i> are <i>O(n)</i>.
- <h3>Where Defined</h3>
- <p>
- <a href="../../../boost/graph/planar_canonical_ordering.hpp">
- <tt>boost/graph/planar_canonical_ordering.hpp</tt></a>
- </p><h3>Parameters</h3>
- IN: <tt>Graph& g</tt>
- <blockquote>
- An undirected graph. The graph type must be a model of
- <a href="VertexAndEdgeListGraph.html">VertexAndEdgeListGraph</a>.
- The graph must:
- <ul>
- <li>Be maximal planar.</li>
- <li>Have at least two vertices.</li>
- </ul>
- </blockquote>
- IN: <tt>PlanarEmbedding</tt>
- <blockquote>
- A model of <a href="PlanarEmbedding.html">PlanarEmbedding</a>.
- </blockquote>
- IN: <tt>OutputIterator</tt>
- <blockquote>
- An OutputIterator with <tt>value_type</tt> equal to
- <tt>graph_traits<Graph>::vertex_descriptor</tt>. The canonical ordering
- will be written to this iterator.
- </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/canonical_ordering.cpp">
- <tt>examples/canonical_ordering.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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