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# graphisomorphism

Find isomorphism between two graphs

## Syntax

[Isomorphic, Map] = graphisomorphism(G1, G2)
[Isomorphic, Map] = graphisomorphism(G1, G2,'Directed', DirectedValue)

## Arguments

 G1 N-by-N sparse matrix that represents a directed or undirected graph. Nonzero entries in matrix G1 indicate the presence of an edge. G2 N-by-N sparse matrix that represents a directed or undirected graph. G2 must be the same (directed or undirected) as G1. DirectedValue Property that indicates whether the graphs are directed or undirected. Enter false when both G1 and G2 are undirected graphs. In this case, the upper triangles of the sparse matrices G1 and G2 are ignored. Default is true, meaning that both graphs are directed.

## Description

 Tip   For introductory information on graph theory functions, see Graph Theory Functions.

[Isomorphic, Map] = graphisomorphism(G1, G2) returns logical 1 (true) in Isomorphic if G1 and G2 are isomorphic graphs, and logical 0 (false) otherwise. A graph isomorphism is a 1-to-1 mapping of the nodes in the graph G1 and the nodes in the graph G2 such that adjacencies are preserved. G1 and G2 are both N-by-N sparse matrices that represent directed or undirected graphs. Return value Isomorphic is Boolean. When Isomorphic is true, Map is a row vector containing the node indices that map from G2 to G1 for one possible isomorphism. When Isomorphic is false, Map is empty. The worst-case time complexity is O(N!), where N is the number of nodes.

[Isomorphic, Map] = graphisomorphism(G1, G2,'Directed', DirectedValue) indicates whether the graphs are directed or undirected. Set DirectedValue to false when both G1 and G2 are undirected graphs. In this case, the upper triangles of the sparse matrices G1 and G2 are ignored. Default is true, meaning that both graphs are directed.

## Examples

1. Create and view a directed graph with 8 nodes and 11 edges.

m('ABCDEFGH') = [1 2 3 4 5 6 7 8];
g1 = sparse(m('ABDCDCGEFFG'),m('BCBDGEEFHGH'),true,8,8)

g1 =

(1,2)        1
(4,2)        1
(2,3)        1
(3,4)        1
(3,5)        1
(7,5)        1
(5,6)        1
(4,7)        1
(6,7)        1
(6,8)        1
(7,8)        1

view(biograph(g1,'ABCDEFGH'))

2. Set a random permutation vector and then create and view a new permuted graph.

p = randperm(8)

p =

7     8     2     3     6     4     1     5

g2 = g1(p,p);
view(biograph(g2,'12345678'))

3. Check if the two graphs are isomorphic.

[F,Map] = graphisomorphism(g2,g1)

F =

1

Map =

7     8     2     3     6     4     1     5

Note that the Map row vector containing the node indices that map from g2 to g1 is the same as the permutation vector you created in step 2.

4. Reverse the direction of the D-G edge in the first graph, and then check for isomorphism again.

g1(m('DG'),m('GD')) = g1(m('GD'),m('DG'));
view(biograph(g1,'ABCDEFGH'))

[F,M] = graphisomorphism(g2,g1)

F =

0

M =

[]
5. Convert the graphs to undirected graphs, and then check for isomorphism.

[F,M] = graphisomorphism(g2+g2',g1+g1','directed',false)

F =

1

M =

7     8     2     3     6     4     1     5

## References

[1] Fortin, S. (1996). The Graph Isomorphism Problem. Technical Report, 96-20, Dept. of Computer Science, University of Alberta, Edomonton, Alberta, Canada.

[2] McKay, B.D. (1981). Practical Graph Isomorphism. Congressus Numerantium 30, 45-87.

[3] Siek, J.G., Lee, L-Q, and Lumsdaine, A. (2002). The Boost Graph Library User Guide and Reference Manual, (Upper Saddle River, NJ:Pearson Education).