Unique square property, equitable partitions, and product-like graphs

Equivalence relations on the edge set of a graph G that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of G , which in turn give rise to quotient graphs that can have a rich product structure even if G itself is prime.