The Grid Property and Product-Like Hypergraphs

Equivalence relations on the edge set of a hypergraph that satisfy the "grid property" (a certain restrictive condition on diagonal-free grids that can be seen as a generalization of the more familiar "square property" on graphs) play a crucial role in the theory of Cartesian hypergraph products. In particular, every convex relation with the grid property induces a factorization w.r.t. the Cartesian product. In the class of graphs, even non-convex relations with the square property provide rich structural information on local isomorphisms, local product structures, and product structures of quotient graphs. Here, we examine the grid property in its own right. Vertex partitions derived from these equivalence classes of the edges give rise to equivalence relations on the vertex set. This in turn determines quotient graphs that have non-trivial product structures.