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- <!-- Copyright 2007 Aaron Windsor
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- <Head>
- <Title>Boost Graph Library: Boyer-Myrvold Planarity Testing/Embedding</Title>
- <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
- ALINK="#ff0000">
- <IMG SRC="../../../boost.png"
- ALT="C++ Boost" width="277" height="86">
- <BR Clear>
- <H1>Boyer-Myrvold Planarity Testing/Embedding</H1>
- <p>
- A graph is <a href="./planar_graphs.html#planar"><i>planar</i></a> if it can
- be drawn in two-dimensional space without any of its edges crossing. Such a
- drawing of a planar graph is called a
- <a href="./planar_graphs.html#plane_drawing"><i>plane drawing</i></a>. Each
- plane drawing belongs to an equivalence class called a <i>planar embedding</i>
- <a href="#1">[1]</a> that is defined by the clockwise ordering of adjacent
- edges around each vertex in the graph. A planar embedding is a convenient
- intermediate representation of an actual drawing of a planar graph, and many
- planar graph drawing algorithms are formulated as functions mapping a planar
- embedding to a plane drawing.
- <br>
- <br>
- <table align="center" class="image">
- <caption align="bottom"><h5>A planar graph (top left), along with a planar
- embedding of that graph (bottom left) can be used to create a plane drawing
- (right) by embedding edges around each vertex in the order in which they
- appear in the planar embedding.
- </h5></caption>
- <tr><td>
- <img src="./figs/embedding_illustration.png">
- </td></tr>
- <tr></tr>
- <tr></tr>
- </table>
- <br>
- <p>
- The function <tt>boyer_myrvold_planarity_test</tt> implements the planarity
- testing/embedding algorithm of Boyer and Myrvold
- [<a href="./bibliography.html#boyermyrvold04">70</a>].
- <tt>boyer_myrvold_planarity_test</tt> returns <tt>true</tt> if the input graph
- is planar and <tt>false</tt> otherwise. As a side-effect of this test, a planar
- embedding can be constructed if the graph is planar or a minimal set of edges
- that form a <a href = "./planar_graphs.html#kuratowskisubgraphs">Kuratowski
- subgraph</a> can be found if the graph is not planar.
- <tt>boyer_myrvold_planarity_test</tt> uses named parameter arguments (courtesy
- of the <a href="../../parameter/doc/html/index.html">Boost.Parameter</a>
- library) to specify what the function actually does. Some examples are:
- <ul>
- <li>Testing whether or not a graph is planar:
- <pre>
- bool is_planar = boyer_myrvold_planarity_test(g);
- </pre>
- <li>Computing a planar embedding for a graph if it is planar, otherwise finding
- a set of edges that forms an obstructing Kuratowski subgraph:
- <pre>
- if (boyer_myrvold_planarity_test(boyer_myrvold_params::graph = g,
- boyer_myrvold_params::embedding = embedding_pmap,
- boyer_myrvold_params::kuratowski_subgraph = out_itr
- )
- )
- {
- //do something with the embedding in embedding_pmap
- }
- else
- {
- //do something with the kuratowski subgraph output to out_itr
- }
- </pre>
- </ul>
- <p>
- The parameters passed to <tt>boyer_myrvold_planarity_test</tt> in the examples
- above do more than just carry the data structures used for input and output -
- the algorithm is optimized at compile time based on which parameters are
- present. A complete list of parameters accepted and their interactions are
- described below.
- <p>
- <tt>boyer_myrvold_planarity_test</tt> accepts as input any undirected graph,
- even those with self-loops and multiple edges.
- However, many planar graph drawing algorithms make additional restrictions
- on the structure of the input graph - for example, requiring that the input
- graph is connected, biconnected, or even maximal planar (triangulated.)
- Fortunately, any planar graph on <i>n</i> vertices that lacks one of these
- properties can be augmented with additional edges so that it satisfies that
- property in <i>O(n)</i> time - the functions
- <tt><a href="./make_connected.html">make_connected</a></tt>,
- <tt><a href="./make_biconnected_planar.html">make_biconnected_planar</a></tt>,
- and <tt><a href="./make_maximal_planar.html">make_maximal_planar</a></tt>
- exist for this purpose. If the graph drawing algorithm you're using requires,
- say, a biconnected graph, then you must make your input graph biconnected
- <i>before</i> passing it into <tt>boyer_myrvold_planarity_test</tt> so that the
- computed planar embedding includes these additional edges. This may require
- more than one call to <tt>boyer_myrvold_planarity_test</tt> depending on the
- structure of the graph you begin with, since both
- <tt>make_biconnected_planar</tt> and <tt>make_maximal_planar</tt> require a
- planar embedding of the existing graph as an input parameter.
- <p><p>
- The named parameters accepted by <tt>boyer_myrvold_planarity_test</tt> are:
- <ul>
- <li><b><tt>graph</tt></b> : The input graph - this is the only required
- parameter.
- <li><b><tt>vertex_index_map</tt></b> : A mapping from vertices of the input
- graph to indexes in the range <tt>[0..num_vertices(g))</tt>. If this parameter
- is not provided, the vertex index map is assumed to be available as an interior
- property of the graph, accessible by calling <tt>get(vertex_index, g)</tt>.
- <li><b><tt>edge_index_map</tt></b>: A mapping from the edges of the input graph
- to indexes in the range <tt>[0..num_edges(g))</tt>. This parameter is only
- needed if the <tt>kuratowski_subgraph</tt> argument is provided. If the
- <tt>kuratowski_subgraph</tt> argument is provided and this parameter is not
- provided, the EdgeIndexMap is assumed to be available as an interior property
- accessible by calling <tt>get(edge_index, g)</tt>.
- <li><b><tt>embedding</tt></b> : If the graph is planar, this will be populated
- with a mapping from vertices to the clockwise order of neighbors in the planar
- embedding.
- <li><b><tt>kuratowski_subgraph</tt></b> : If the graph is not planar, a minimal
- set of edges that form the obstructing Kuratowski subgraph will be written to
- this iterator.
- </ul>
- These named parameters all belong to the namespace
- <tt>boyer_myrvold_params</tt>. See below for more information on the concepts
- required for these arguments.
- <H3>Verifying the output</H3>
- Whether or not the input graph is planar, <tt>boyer_myrvold_planarity_test</tt>
- can produce a certificate that can be automatically checked to verify that the
- function is working properly.
- <p>
- If the graph is planar, a planar embedding can be produced. The
- planar embedding can be verified by passing it to a plane drawing routine
- (such as <tt><a href="straight_line_drawing.html">
- chrobak_payne_straight_line_drawing</a></tt>) and using the function
- <tt><a href="is_straight_line_drawing.html">is_straight_line_drawing</a></tt>
- to verify that the resulting graph is planar.
- <p>
- If the graph is not planar, a set of edges that forms a Kuratowski subgraph in
- the original graph can be produced. This set of edges can be passed to the
- function <tt><a href="is_kuratowski_subgraph.html">is_kuratowski_subgraph</a>
- </tt> to verify that they can be contracted into a <i>K<sub>5</sub></i> or
- <i>K<sub>3,3</sub></i>. <tt>boyer_myrvold_planarity_test</tt> chooses the set
- of edges forming the Kuratowski subgraph in such a way that the contraction to
- a <i>K<sub>5</sub></i> or <i>K<sub>3,3</sub></i> can be done by a simple
- deterministic process which is described in the documentation to
- <tt>is_kuratowski_subgraph</tt>.
- <H3>Where Defined</H3>
- <P>
- <a href="../../../boost/graph/boyer_myrvold_planar_test.hpp">
- <TT>boost/graph/boyer_myrvold_planar_test.hpp</TT>
- </a>
- <H3>Parameters</H3>
- IN: <tt>Graph& g</tt>
- <blockquote>
- Any undirected graph. The graph type must be a model of
- <a href="VertexAndEdgeListGraph.html">VertexAndEdgeListGraph</a> and
- <a href="IncidenceGraph.html">IncidenceGraph</a>.
- </blockquote>
- OUT <tt>PlanarEmbedding embedding</tt>
- <blockquote>
- Must model the <a href="PlanarEmbedding.html">PlanarEmbedding</a> concept.
- </blockquote>
- IN <tt>OutputIterator kuratowski_subgraph</tt>
- <blockquote>
- An OutputIterator which accepts values of the type
- <tt>graph_traits<Graph>::edge_descriptor</tt>
- </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>
- IN <tt>EdgeIndexMap em</tt>
- <blockquote>
- A <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map
- </a> that maps edges from <tt>g</tt> to distinct integers in the range
- <tt>[0, num_edges(g) )</tt><br>
- <b>Default</b>: <tt>get(edge_index,g)</tt>, but this parameter is only used if
- the <tt>kuratowski_subgraph_iterator</tt> is provided.<br>
- </blockquote>
- <H3>Complexity</H3>
- Assuming that both the vertex index and edge index supplied take time
- <i>O(1)</i> to return an index and there are <i>O(n)</i>
- total self-loops and parallel edges in the graph, most combinations of
- arguments given to
- <tt>boyer_myrvold_planarity_test</tt> result in an algorithm that runs in time
- <i>O(n)</i> for a graph with <i>n</i> vertices and <i>m</i> edges. The only
- exception is when Kuratowski subgraph isolation is requested for a dense graph
- (a graph with <i>n = o(m)</i>) - the running time will be <i>O(n+m)</i>
- <a href = "#2">[2]</a>.
- <H3>Examples</H3>
- <P>
- <ul>
- <li><a href="../example/simple_planarity_test.cpp">A simple planarity test</a>
- <li><a href="../example/kuratowski_subgraph.cpp">Isolating a Kuratowski
- Subgraph</a>
- <li><a href="../example/straight_line_drawing.cpp">Using a planar embedding to
- create a straight line drawing</a>
- </ul>
- <h3>See Also</h3>
- <a href="./planar_graphs.html">Planar Graphs in the Boost Graph Library</a>
- <h3>Notes</h3>
- <p><a name="1">[1] A planar embedding is also called a <i>combinatorial
- embedding</i>.
- <p><a name="2">[2] The algorithm can still be made to run in time <i>O(n)</i>
- for this case, if needed. <a href="planar_graphs.html#EulersFormula">Euler's
- formula</a> implies that a planar graph with <i>n</i> vertices can have no more
- than <i>3n - 6</i> edges, which means that any non-planar graph on <i>n</i>
- vertices has a subgraph of only <i>3n - 5</i> edges that contains a Kuratowski
- subgraph. So, if you need to find a Kuratowski subgraph of a graph with more
- than <i>3n - 5</i> edges in time <i>O(n)</i>, you can create a subgraph of the
- original graph consisting of any arbitrary <i>3n - 5</i> edges and pass that
- graph to <tt>boyer_myrvold_planarity_test</tt>.
- <br>
- <HR>
- Copyright © 2007 Aaron Windsor (<a href="mailto:aaron.windsor@gmail.com">
- aaron.windsor@gmail.com</a>)
- </BODY>
- </HTML>
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