Modular product of graphs

In graph theory, the modular product of graphs $G$ and $H$ is a graph formed by combining $G$ and $H$ that has applications to subgraph isomorphism. It is one of several different kinds of graph products that have been studied, generally using the same vertex set (the Cartesian product of the sets of vertices of the two graphs $G$ and $H$ ) but with different rules for determining which edges to include.

Definition

The vertex set of the modular product of $G$ and $H$ is the cartesian product $V (G) \times V (H)$ . Any two vertices $(u, v)$ and $(u', v')$ are adjacent in the modular product of $G$ and $H$ if and only if $u$ is distinct from $u'$ , $v$ is distinct from $v'$ , and 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 $v'$ .

Application to subgraph isomorphism

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 maximum common 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.