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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>Planar Embedding Concept</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 Embedding Concept</h1>
A planar embedding is an important intermediate representation of a drawing
of a planar graph. Instead of specifying the absolute positions of the vertices
and edges in the plane, a planar embedding specifies their positions relative
to one another. A planar embedding consists of a sequence, for each vertex in
the graph, of all of the edges incident on that vertex in the order in which
they are to be drawn around that vertex.
<p>
A planar embedding is a refinement of
<a href="../../property_map/LvaluePropertyMap.html">LValuePropertyMap</a> that
places additional restrictions the <tt>value_type</tt> used in the property
map.
</p><h3>Notation</h3>
<table>
<tbody>
<tr>
<td> <tt>Embedding</tt> </td>
<td> is a type that models the Planar Embedding concept.</td>
</tr>
<tr>
<td> <tt>embedding</tt> </td>
<td> is an object of type <tt>Embedding</tt>. </td>
</tr>
<tr>
<td> <tt>Graph</tt> </td>
<td> is the type of the underlying graph.</td>
</tr>
<tr>
<td> <tt>e</tt> </td>
<td> is an object of type <tt>graph_traits<Graph>::edge_descriptor</tt>.
</td>
</tr>
<tr>
<td> <tt>v</tt> </td>
<td> is an object of type <tt>graph_traits<Graph>::vertex_descriptor
</tt>.</td>
</tr><tr>
<td>
</td></tr></tbody></table>
<h3>Associated Types</h3>
<table border="1">
<tbody><tr>
<td> Const Iterator </td>
<td> <tt>boost::property_traits<Embedding>::value_type::const_iterator
</tt>
</td>
<td> The iterator type used to iterate over the ordering of the edges in the
planar embedding of a particular vertex
</td>
</tr>
</tbody></table>
<h3>Valid Expressions</h3>
<p>
<table border="1">
<tbody><tr><th>Expression</th><th>Return Type</th><th>Description</th>
</tr><tr>
<td> <tt>embedding[v].begin()</tt> </td>
<td> <tt>boost::property_traits<Embedding>::value_type::const_iterator
</tt></td>
<td> Returns an iterator to the beginning of the range of edges in the
embedding around vertex v</td>
</tr>
<tr>
<td> <tt>embedding[v].end()</tt> </td>
<td> <tt>boost::property_traits<Embedding>::value_type::const_iterator
</tt></td>
<td> Returns an iterator to the end of the range of edges in the
embedding around vertex v</td>
</tr>
<tr>
<td> <tt>embedding[v].clear()</tt> </td>
<td> <tt>void</tt></td>
<td> Clears all edges in the embedding around a vertex <tt>v</tt></td>
</tr>
<tr>
<td> <tt>embedding[v].push_back(e)</tt> </td>
<td> <tt>void</tt></td>
<td> Adds an edge <tt>e</tt> to the end of the sequence of embedded edges
around the vertex <tt>v</tt> </td>
</tr>
</tbody></table>
</p><h3>Complexity Guarantees</h3>
Starting with an empty embedding, any mixed sequence of <i>n</i> calls to a
particular vertex's <tt>push_back</tt> and <tt>clear</tt> should take
<i>O(n)</i> time.
<h3>Models</h3>
Any LValue property map that maps vertices to a <tt>std::vector</tt>,
<tt>std::list</tt>, or <tt>std::deque</tt> models this
concept. Below is an example of using this approach to create a model of
PlanarEmbedding:
<pre>
#include <boost/property_map.hpp>
#include <vector>
...
// Assume a graph type "Graph" defined somewhere above and
// an instance of Graph in a variable g.
// A typedef for the storage - a vector of vectors of edge descriptors
typedef
std::vector< std::vector< graph_traits<Graph>::edge_descriptor > >
planar_embedding_storage_t;
// A typedef for the iterator property map, assuming the graph has
// an interior vertex index map
typedef
boost::iterator_property_map< planar_embedding_storage_t::iterator,
property_map<Graph, vertex_index_t>::type
>
planar_embedding_t;
// Create an instance of the storage and the property map
planar_embedding_storage_t planar_embedding_storage(num_vertices(g));
planar_embedding_t planar_embedding(planar_embedding_storage.begin(),
get(vertex_index, g)
);
// planar_embedding can now be passed to any function expecting a model
// of PlanarEmbedding.
</pre>
<p>
<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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