The Relaxed Square Property

TL;DR

This paper explores RSP-relations, a generalization of graph product characterizations via square properties, revealing their computational aspects and connections to graph bundles and coverings.

Contribution

It introduces RSP-relations as a broader framework, analyzes their properties, and shows how they relate to graph products and coverings, including polynomial-time computability in certain cases.

Findings

RSP-relations generalize graph product characterizations.

Finest RSP-relations can be computed efficiently for K_23-free graphs.

RSP-relations behave well under graph products, derived from prime factors.

Abstract

Graph products are characterized by the existence of non-trivial equivalence relations on the edge set of a graph that satisfy a so-called square property. We investigate here a generalization, termed RSP-relations. The class of graphs with non-trivial RSP-relations in particular includes graph bundles. Furthermore, RSP-relations are intimately related with covering graph constructions. For K_23-free graphs finest RSP-relations can be computed in polynomial-time. In general, however, they are not unique and their number may even grow exponentially. They behave well for graph products, however, in sense that a finest RSP-relations can be obtained easily from finest RSP-relations on the prime factors.