Modular product of graphs

The modular product of graphs.

In graph theory, the modular product of graphs G and H is a graph such that

• the vertex set of the modular product of G and H is the cartesian product V(G) ×  V(H); and
• any two vertices (u, v) and (u' , v' ) are adjacent in the modular product of G and H if and only if either
  • u is adjacent with u' and v is adjacent with v' , or
  • u is not adjacent with u' and v is not adjacent with v' .

Cliques in the modular product graph correspond to isomorphisms of induced subgraphs of G and H. Therefore, the modular product graph can be used to reduce problems of induced subgraph isomorphism to problems of finding cliques in graphs. Specifically, the largest graph that is an induced subgraph of both G and H corresponds to the maximum clique in their modular product. Although the problems of finding largest common induced subgraphs and of finding maximum cliques are both NP-complete, this reduction allows clique-finding algorithms to be applied to the common subgraph problem.

References

• Barrow, H.; Burstall, R. (1976), "Subgraph isomorphism, matching relational structures and maximal cliques", Information Processing Letters 4 (4): 83–84, doi:10.1016/0020-0190(76)90049-1.
• Levi, G. (1973), "A note on the derivation of maximal common subgraphs of two directed or undirected graphs", Calcolo 9 (4): 341–352, doi:10.1007/BF02575586.
• Vizing, V. G. (1974), "Reduction of the problem of isomorphism and isomorphic entrance to the task of finding the nondensity of a graph", Proc. 3rd All-Union Conf. Problems of Theoretical Cybernetics, p. 124.