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Isomorphism

Determines whether an isomorphism exists between two graphs.

Complexity: O(|V|!)
Defined in: <boost/graph/isomorphism.hpp>

Example

#include <iostream>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/isomorphism.hpp>

int main() {
    using Graph = boost::adjacency_list<boost::vecS, boost::vecS, boost::bidirectionalS>;
    Graph g1, g2;
    // Triangle: 0->1->2->0
    boost::add_edge(0, 1, g1); boost::add_edge(1, 2, g1); boost::add_edge(2, 0, g1);
    // Same triangle, relabeled: 2->0->1->2
    boost::add_edge(2, 0, g2); boost::add_edge(0, 1, g2); boost::add_edge(1, 2, g2);

    bool result = boost::isomorphism(g1, g2);
    std::cout << "Graphs are " << (result ? "isomorphic" : "not isomorphic") << "\n";
}
Graphs are isomorphic

(1) Positional version

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 invariant1, VertexInvariant2 invariant2,
    std::size_t max_invariant, VertexIndex1Map i1_map, VertexIndex2Map i2_map);
Direction Parameter Description

IN

const Graph1& g1

A directed or undirected graph. The graph type must be a model of Vertex List Graph and Edge List Graph.

IN

const Graph2& g2

A directed or undirected graph. The graph type must be a model of Bidirectional Graph and Vertex List Graph.

OUT

IsoMap f

The mapping from vertices in graph 1 to vertices in graph 2. IsoMap must be a Read/Write Property Map.

IN

VertexInvariant1 invariant1

Mapping from vertices of graph 1 to integers which restricts which vertices may be considered isomorphic. See the named parameter version for details.

IN

VertexInvariant2 invariant2

Mapping from vertices of graph 2 to integers which restricts which vertices may be considered isomorphic. See the named parameter version for details.

IN

std::size_t max_invariant

This parameter is ignored as it is no longer necessary, but kept for backwards compatibility.

IN

VertexIndex1Map i1_map

This maps each vertex of graph 1 to an integer in the range [0, num_vertices(g1)). The type VertexIndex1Map must be a model of Readable Property Map. 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.

IN

VertexIndex2Map i2_map

This maps each vertex of graph 2 to an integer in the range [0, num_vertices(g2)). The type VertexIndex2Map must be a model of Readable Property Map. 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.

(2) Named parameter version

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 = all defaults);
Direction Parameter Description

IN

const Graph1& g1

A directed or undirected graph. The graph type must be a model of Vertex List Graph and Edge List Graph.

IN

const Graph2& g2

A directed or undirected graph. The graph type must be a model of Bidirectional Graph and Vertex List Graph.

IN

params

Named parameters passed via bgl_named_params. The following are accepted:
Direction Named Parameter Description / Default

OUT

isomorphism_map(IsoMap f)

The mapping from vertices in graph 1 to vertices in graph 2. IsoMap must be a Read/Write Property Map.
Default: an iterator_property_map constructed from a std::vector of graph 2’s vertex descriptor type and the vertex index map for graph 1.

IN

vertex_invariant1(VertexInvariant1 i1) / vertex_invariant2(VertexInvariant2 i2)

Mappings from vertices to integers which restrict which vertices may be considered isomorphic. If a candidate isomorphism maps v1 to v2 but i1(v1) != i2(v2), that candidate is rejected. This mapping can be used either to speed up the search (as is done by the default value, which requires that the degrees of v1 and v2 are equal) or to impose extra conditions on the result. The VertexInvariant1 and VertexInvariant2 types must model AdaptableUnaryFunction, with the argument type of vertex_invariant1 being Graph1’s vertex descriptor type, the argument type of `vertex_invariant2 being Graph2’s vertex descriptor type, and both functions sharing a result type that is totally ordered and hashable, such as an integer.
Default: `degree_vertex_invariant for both arguments

IN

vertex_max_invariant(std::size_t max_invariant)

This parameter is ignored as it is no longer necessary, but kept for backwards compatibility.

IN

vertex_index1_map(VertexIndex1Map i1_map)

This maps each vertex to an integer in the range [0, num_vertices(g)). This is necessary for efficient updates of the heap data structure when an edge is relaxed. The type VertexIndex1Map must be a model of Readable Property Map. 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.
Default: get(vertex_index, g1). Note: if you use this default, make sure your graph has an internal vertex_index property. For example, adjacency_list with VertexList=listS does not have an internal vertex_index property.

IN

vertex_index2_map(VertexIndex2Map i2_map)

This maps each vertex to an integer in the range [0, num_vertices(g)). This is necessary for efficient updates of the heap data structure when an edge is relaxed. The type VertexIndex2Map must be a model of Readable Property Map. 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.
Default: get(vertex_index, g2). Note: if you use this default, make sure your graph has an internal vertex_index property. For example, adjacency_list with VertexList=listS does not have an internal vertex_index property.

Description

An isomorphism is a 1-to-1 mapping of the vertices in one graph to the vertices of another graph such that adjacency is preserved. In other words, given graphs G1 = (V1,E1) and G2 = (V2,E2) an isomorphism is a function f such that for all pairs of vertices a,b in V1, edge (a,b) is in E1 if and only if edge (f(a),f(b)) is in E2.

This function returns true if there exists an isomorphism between graph 1 and graph 2 and false otherwise. Also, if an isomorphism map named parameter is provided then an isomorphism is recorded in the map.

The current implementation is based on descriptions of a backtracking algorithm in [41,43]. The algorithm used is simple but slow. A more efficient (and much more complex) algorithm is described in [42]. When (and if) a version of this algorithm is ported to the BGL interface it should replace the current algorithm.

Returns

Returns true if there exists an isomorphism between graph 1 and graph 2, false otherwise.

Example

See libs/graph/example/isomorphism.cpp.

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