Weak Modular Product of Bipartite Graphs, Bicliques and Isomorphism

TL;DR

This paper explores the weak modular product of bipartite graphs, identifying cases where graph isomorphism can be efficiently tested using biclique enumeration, and introduces a new graph product called the very weak modular product.

Contribution

It introduces the concept of Isomorphism via Biclique Enumeration (IvBE) and analyzes its computational complexity for bipartite graphs, including the new very weak modular product.

Findings

IvBE is polynomial for bipartite 2K2-free graphs.

IvBE is quasi-polynomial for certain bipartite graph families.

Perfectness of the weak modular product does not generally imply polynomial isomorphism testing.

Abstract

A 1978 theorem of Kozen states that two graphs on $n$ vertices are isomorphic if and only if there is a clique of size $n$ in the weak modular product between the two graphs. Restricting to bipartite graphs and considering complete bipartite subgraphs (bicliques) therein, we study the combinatorics of the weak modular product. We identify cases where isomorphism is tractable using this approach, which we call Isomorphism via Biclique Enumeration (IvBE). We find that IvBE is polynomial for bipartite $2K_2$-free graphs and quasi-polynomial for families of bipartite graphs, where the largest induced matching and largest induced crown graph grows slowly in $n$, that is, $O(\mathrm{polylog }\, n)$. Furthermore, as expected a straightforward corollary of Kozen's theorem and Lov\'{a}sz's sandwich theorem is if the weak modular product between two graphs is perfect, then checking if the graphs are isomorphic is polynomial in $n$. However, we show that for balanced, bipartite graphs this is only true in a few trivial cases. In doing so we define a new graph product on bipartite graphs, the very weak modular product. The results pertaining to bicliques in bipartite graphs proved here may be of independent interest.