<HTML> <!-- Copyright (c) Jeremy Siek 2000 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) --> <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 IsoMap, typename VertexInvariant1, typename VertexInvariant2, typename V1Map, typename V2Map> bool isomorphism(const Graph1& g1, const Graph2& g2, IsoMap f, VertexInvariant1 invariant2, VertexInvariant2 invariant2, std::size_t max_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/doc/ReadWritePropertyMap.html">Read/Write Property Map</a>.<br> <b>Default:</b> an <a href="../../property_map/doc/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_invariant1(VertexInvariant1 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 graph 1. <br> <b>Default:</b> <tt>degree_vertex_invariant</tt><br> <b>Python</b>: Unsupported parameter. </blockquote> IN: <tt>vertex_invariant2(VertexInvariant2 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 2. <br> <b>Default:</b> <tt>degree_vertex_invariant</tt><br> <b>Python</b>: Unsupported parameter. </blockquote> IN: <tt>vertex_max_invariant(std::size_t max_invariant)</tt> <blockquote> An upper bound on the possible values returned from either vertex_invariant1 or vertex_invariant2. <br> <b>Default:</b> <tt>vertex_invariant2.max()</tt>. The default <tt>vertex_invariant2</tt> parameter, an instance of <tt>degree_vertex_invariant</tt>, defines this function to return <tt>num_vertices(g2) * (num_vertices(g2)+1)</tt>.<br> <b>Python</b>: Unsupported parameter. </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/doc/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> Note: if you use this default, make sure your graph has an internal <tt>vertex_index</tt> property. For example, <tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does not have an internal <tt>vertex_index</tt> property. <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/doc/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> Note: if you use this default, make sure your graph has an internal <tt>vertex_index</tt> property. For example, <tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does not have an internal <tt>vertex_index</tt> property. <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> <HR> <TABLE> <TR valign=top> <TD nowrap>Copyright © 2000-2001</TD><TD> <A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>) </TD></TR></TABLE> </BODY> </HTML>