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 In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs ''G'' and ''H'' are given as input, and one must determine whether ''G'' contains a subgraph that is isomorphic to ''H''. Subgraph isomorp ...

. 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 In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

''V''(''G'') × ''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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

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

*. *. *{{citation , last = Vizing , first = V. G. , authorlink = Vadim G. Vizing , contribution = Reduction of the problem of isomorphism and isomorphic entrance to the task of finding the nondensity of a graph , page = 124 , title = Proc. 3rd All-Union Conf. Problems of Theoretical Cybernetics , year = 1974. Graph products