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<html><head><!-- Copyright 2007 Aaron Windsor
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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2018-01-12 21:47:58 +01:00
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-->
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<title>Planar Embedding Concept</title>
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</head>
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<body alink="#ff0000"
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bgcolor="#ffffff"
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link="#0000ee"
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text="#000000"
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vlink="#551a8b">
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<img src="../../../boost.png" alt="C++ Boost" height="86" width="277">
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<br clear="">
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<h1>Planar Embedding Concept</h1>
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A planar embedding is an important intermediate representation of a drawing
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of a planar graph. Instead of specifying the absolute positions of the vertices
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and edges in the plane, a planar embedding specifies their positions relative
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to one another. A planar embedding consists of a sequence, for each vertex in
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the graph, of all of the edges incident on that vertex in the order in which
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they are to be drawn around that vertex.
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<p>
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A planar embedding is a refinement of
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<a href="../../property_map/doc/LvaluePropertyMap.html">LValuePropertyMap</a> that
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places additional restrictions the <tt>value_type</tt> used in the property
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map.
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</p><h3>Notation</h3>
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<table>
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<tbody>
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<tr>
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<td> <tt>Embedding</tt> </td>
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<td> is a type that models the Planar Embedding concept.</td>
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</tr>
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<tr>
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<td> <tt>embedding</tt> </td>
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<td> is an object of type <tt>Embedding</tt>. </td>
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</tr>
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<tr>
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<td> <tt>Graph</tt> </td>
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<td> is the type of the underlying graph.</td>
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</tr>
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<tr>
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<td> <tt>e</tt> </td>
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<td> is an object of type <tt>graph_traits<Graph>::edge_descriptor</tt>.
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</td>
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</tr>
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<tr>
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<td> <tt>v</tt> </td>
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<td> is an object of type <tt>graph_traits<Graph>::vertex_descriptor
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</tt>.</td>
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</tr><tr>
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<td>
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</td></tr></tbody></table>
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<h3>Associated Types</h3>
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<table border="1">
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<tbody><tr>
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<td> Const Iterator </td>
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<td> <tt>boost::property_traits<Embedding>::value_type::const_iterator
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</tt>
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</td>
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<td> The iterator type used to iterate over the ordering of the edges in the
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planar embedding of a particular vertex
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</td>
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</tr>
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</tbody></table>
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<h3>Valid Expressions</h3>
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<p>
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<table border="1">
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<tbody><tr><th>Expression</th><th>Return Type</th><th>Description</th>
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</tr><tr>
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<td> <tt>embedding[v].begin()</tt> </td>
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<td> <tt>boost::property_traits<Embedding>::value_type::const_iterator
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</tt></td>
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<td> Returns an iterator to the beginning of the range of edges in the
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embedding around vertex v</td>
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</tr>
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<tr>
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<td> <tt>embedding[v].end()</tt> </td>
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<td> <tt>boost::property_traits<Embedding>::value_type::const_iterator
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</tt></td>
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<td> Returns an iterator to the end of the range of edges in the
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embedding around vertex v</td>
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</tr>
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<tr>
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<td> <tt>embedding[v].clear()</tt> </td>
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<td> <tt>void</tt></td>
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<td> Clears all edges in the embedding around a vertex <tt>v</tt></td>
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</tr>
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<tr>
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<td> <tt>embedding[v].push_back(e)</tt> </td>
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<td> <tt>void</tt></td>
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<td> Adds an edge <tt>e</tt> to the end of the sequence of embedded edges
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around the vertex <tt>v</tt> </td>
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</tr>
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</tbody></table>
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</p><h3>Complexity Guarantees</h3>
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Starting with an empty embedding, any mixed sequence of <i>n</i> calls to a
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particular vertex's <tt>push_back</tt> and <tt>clear</tt> should take
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<i>O(n)</i> time.
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<h3>Models</h3>
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Any LValue property map that maps vertices to a <tt>std::vector</tt>,
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<tt>std::list</tt>, or <tt>std::deque</tt> models this
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concept. Below is an example of using this approach to create a model of
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PlanarEmbedding:
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<pre>
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#include <boost/property_map/property_map.hpp>
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#include <vector>
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...
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// Assume a graph type "Graph" defined somewhere above and
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// an instance of Graph in a variable g.
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// A typedef for the storage - a vector of vectors of edge descriptors
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typedef
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std::vector< std::vector< graph_traits<Graph>::edge_descriptor > >
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planar_embedding_storage_t;
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// A typedef for the iterator property map, assuming the graph has
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// an interior vertex index map
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typedef
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boost::iterator_property_map< planar_embedding_storage_t::iterator,
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property_map<Graph, vertex_index_t>::type
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>
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planar_embedding_t;
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// Create an instance of the storage and the property map
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planar_embedding_storage_t planar_embedding_storage(num_vertices(g));
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planar_embedding_t planar_embedding(planar_embedding_storage.begin(),
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get(vertex_index, g)
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);
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// planar_embedding can now be passed to any function expecting a model
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// of PlanarEmbedding.
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</pre>
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<p>
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<br>
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</p><hr>
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Copyright <20> 2007 Aaron Windsor (<a href="mailto:aaron.windsor@gmail.com">
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aaron.windsor@gmail.com</a>)
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</body></html>
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