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							<html><head><!-- Copyright 2007 Aaron Windsor
     
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<title>Planar Embedding Concept</title>
</head>
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<img src="../../../boost.png" alt="C++ Boost" height="86" width="277"> 

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<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/doc/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&lt;Graph&gt;::edge_descriptor</tt>. 
</td>
</tr>

<tr>
<td> <tt>v</tt> </td>
<td> is an object of type <tt>graph_traits&lt;Graph&gt;::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&lt;Embedding&gt;::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&lt;Embedding&gt;::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&lt;Embedding&gt;::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 &lt;boost/property_map/property_map.hpp&gt;
#include &lt;vector&gt;

...

// 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&lt; std::vector&lt; graph_traits&lt;Graph&gt;::edge_descriptor &gt; &gt;
    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&lt; planar_embedding_storage_t::iterator,
                                  property_map&lt;Graph, vertex_index_t&gt;::type
                                &gt;
    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>) 

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