# `isomorphism`

    // 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)

    // non-named parameter 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 invariant2, VertexInvariant2 invariant2,
    		 std::size_t max_invariant, VertexIndex1Map i1_map, VertexIndex2Map i2_map)

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. Another 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 a 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 [46,48]. The file isomorphism-impl.pdf contains a (somewhat out-of-date) "literate" description of the implementation. The algorithm used is simple but slow. A more efficient (and much more complex) algorithm is described in [47]. When (and if) a version of this algorithm is ported to the BGL interface it should replace the current algorithm.

### Where Defined

`boost/graph/isomorphism.hpp`

### Parameters

IN: `const Graph1& g1`

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

IN: `const Graph2& g2`

> A directed or undirected graph. The graph type must model Bidirectional Graph and Vertex List Graph.

### Named Parameters

OUT: `isomorphism_map(IsoMap f)`

> The mapping from vertices in graph 1 to vertices in graph 2. This 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.
> Python: Must be a `vertex_vertex_map` for the first graph.

IN: `vertex_invariant1(VertexInvariant1 i1)` IN: `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 UnaryFunction, 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 having integral result types. The values returned by these two functions must be in the range [0, `vertex_max_invariant`).
> Default: `degree_vertex_invariant` for both arguments
> Python: Unsupported parameter.

IN: `vertex_max_invariant(std::size_t max_invariant)`

> An upper bound on the possible values returned from either vertex_invariant1 or vertex_invariant2.
> Default: `vertex_invariant2.max()`. The default `vertex_invariant2` parameter, an instance of `degree_vertex_invariant`, defines this function to return `num_vertices(g2) * (num_vertices(g2)+1)`.
> Python: Unsupported parameter.

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.
> Python: Unsupported parameter.

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.
> Python: Unsupported parameter.

### Complexity

The worst-case time complexity is O(|V|!).

### Example

See `libs/graph/example/isomorphism.cpp`.

Copyright © 2000-2001 | Jeremy Siek, Indiana University (jsiek@osl.iu.edu)