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<Head>
<Title>Boost Graph Library: Isomorphism</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>
<img src="figs/python.gif" alt="(Python)"/>
<TT>isomorphism</TT>
</H1>
<PRE>
<i>// named parameter version</i>
template <class Graph1, class Graph2, class P, class T, class R>
bool isomorphism(const Graph1& g1, const Graph2& g2,
const bgl_named_params<P,T,R>& params = <i>all defaults</i>)
<i>// non-named parameter version</i>
template <typename Graph1, typename Graph2,
typename IndexMapping, typename VertexInvariant,
typename V1Map, typename V2Map>
bool isomorphism(const Graph1& g1, const Graph2& g2,
IsoMap f, VertexInvariant invariant,
VertexIndex1Map i1_map, VertexIndex2Map i2_map)
</pre>
<p>
An <b><i>isomorphism</i></b> is a 1-to-1 mapping of the vertices in
one graph to the vertices of another graph such that adjacency is
preserved. Another words, given graphs <i>G<sub>1</sub> =
(V<sub>1</sub>,E<sub>1</sub>)</i> and <i>G<sub>2</sub> =
(V<sub>2</sub>,E<sub>2</sub>)</i> an isomorphism is a function
<i>f</i> such that for all pairs of vertices <i>a,b</i> in
<i>V<sub>1</sub></i>, edge <i>(a,b)</i> is in <i>E<sub>1</sub></i> if
and only if edge <i>(f(a),f(b))</i> is in <i>E<sub>2</sub></i>.
</p>
<p>
This function returns <tt>true</tt> if there exists an isomorphism
between graph 1 and graph 2 and <tt>false</tt> otherwise. Also, if a
isomorphism map named parameter is provided then an isomorphism is
recorded in the map.
</p>
<p>
The current implementation is based on descriptions of a backtracking
algorithm in [<a
href="./bibliography.html#fortin96:_graph_iso_prob">46</a>,<a
href="./bibliography.html#reingold77:_combin_algo">48</a>]. The file
<a href="./isomorphism-impl.pdf">isomorphism-impl.pdf</a> contains a
"literate" description of the implementation. The algorithm
used is simple but slow. A more efficient (and much more complex)
algorithm is described in [<a
href="./bibliography.html#mckay81:_pract_graph_iso">47</a>]. When (and
if) a version of this algorithm is ported to the BGL interface it
should replace the current algorithm.
</p>
<H3>Where Defined</H3>
<a href="../../../boost/graph/isomorphism.hpp"><TT>boost/graph/isomorphism.hpp</TT></a>
<h3>Parameters</h3>
IN: <tt>const Graph1& g1</tt>
<blockquote>
A directed or undirected graph. The graph type must model of <a
href="./VertexListGraph.html">Vertex List Graph</a> and <a
href="./EdgeListGraph.html">Edge List Graph</a>.
</blockquote>
IN: <tt>const Graph2& g2</tt>
<blockquote>
A directed or undirected graph. The graph type must model <a
href="./BidirectionalGraph.html">Bidirectional Graph</a> and <a
href="./VertexListGraph.html">Vertex List Graph</a>.
</blockquote>
<h3>Named Parameters</h3>
OUT: <tt>isomorphism_map(IsoMap f)</tt>
<blockquote>
The mapping from vertices in graph 1 to vertices in graph 2. This must
be a <a href="../../property_map/ReadWritePropertyMap.html">Read/Write
Property Map</a>.<br> <b>Default:</b> an <a
href="../../property_map/iterator_property_map.html"><tt>iterator_property_map</tt></a>
constructed from a <tt>std::vector</tt> of graph 2's vertex
descriptor type and the vertex index map for graph 1.<br>
<b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the first graph.
</blockquote>
IN: <tt>vertex_invariant(VertexInvariant i)</tt>
<blockquote>
A mapping <i>i</i> from vertices to integers such that if there is
some isomorphism that maps <i>v</i> onto <i>v'</i> then <i>i(v) ==
i(v')</i>. The <tt>VertexInvariant</tt> type must be a <a
href="http://www.sgi.com/tech/stl/BinaryFunction.html">BinaryFunction</a>
where the first argument is a vertex descriptor, the second argument is a
graph, and the result type is an integer. The vertex invariant must
work with the types for both graph 1 and graph 2 and therefore may need
to have a templated <tt>operator()</tt>.
<br>
<b>Default:</b> <tt>degree_vertex_invariant</tt><br>
<b>Python</b>: Any callable Python object will suffice.
</blockquote>
IN: <tt>vertex_index1_map(VertexIndex1Map i1_map)</tt>
<blockquote>
This maps each vertex to an integer in the range <tt>[0,
num_vertices(g))</tt>. This is necessary for efficient updates of the
heap data structure when an edge is relaxed. The type
<tt>VertexIndex1Map</tt> must be a model of <a
href="../../property_map/ReadablePropertyMap.html">Readable Property
Map</a>. The value type of the map must be an integer type. The vertex
descriptor type of graph 1 needs to be usable as the key type of the
map.<br>
<b>Default:</b> <tt>get(vertex_index, g1)</tt><br>
<b>Python</b>: Unsupported parameter.
</blockquote>
IN: <tt>vertex_index2_map(VertexIndex2Map i2_map)</tt>
<blockquote>
This maps each vertex to an integer in the range <tt>[0,
num_vertices(g))</tt>. This is necessary for efficient updates of the
heap data structure when an edge is relaxed. The type
<tt>VertexIndex2Map</tt> must be a model of <a
href="../../property_map/ReadablePropertyMap.html">Readable Property
Map</a>. The value type of the map must be an integer type. The vertex
descriptor type of graph 2 needs to be usable as the key type of the
map.<br>
<b>Default:</b> <tt>get(vertex_index, g2)</tt><br>
<b>Python</b>: Unsupported parameter.
</blockquote>
<h3>Complexity</h3>
The worst-case time complexity is <i>O(|V|!)</i>.
<h3>Example</h3>
See <a href="../example/isomorphism.cpp"><tt>libs/graph/example/isomorphism.cpp</tt></a>.
<br>
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<TABLE>
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<TD nowrap>Copyright © 2000-2001</TD><TD>
<A HREF="../../../people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
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