Modular product of graphs


In graph theory, the modular product of graphs and is a graph formed by combining and 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 but with different rules for determining which edges to include.

Definition

The vertex set of the modular product of and is the cartesian product.
Any two vertices and are adjacent in the modular product of and if
and only if is distinct from, is distinct from, and either
Cliques in the modular product graph correspond to isomorphisms of induced subgraphs of and. 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 and 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.