Modular product of graphs

File:Modular product.png

The modular product of graphs.

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

Definition

The vertex set of the modular product of Template:Mvar and Template:Mvar is the cartesian product V(G) × V(H). Any two vertices (u, v) and (u' , v' ) are adjacent in the modular product of Template:Mvar and Template:Mvar if and only if Template:Mvar is distinct from Template:Mvar, Template:Mvar is distinct from Template:Mvar, and either

• Template:Mvar is adjacent with Template:Mvar and Template:Mvar is adjacent with Template:Mvar, or
• Template:Mvar is not adjacent with Template:Mvar and Template:Mvar is not adjacent with Template:Mvar.

Application to subgraph isomorphism

Cliques in the modular product graph correspond to isomorphisms of induced subgraphs of Template:Mvar and Template:Mvar. 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 Template:Mvar and Template:Mvar 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.

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