1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
|
<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>.
</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/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>
|
