On bipartite graphs whose interval space is a closed join space

Abstract

We characterize the bipartite graphs whose geodesic interval spaces are (closed) join spaces, i.e. which share a number of geometrical properties with Euclidean spaces. We prove that the geodesic interval space \((V(G),I_G)\) of a bipartite graph G is a join space if and only if G is a partial cube all of whose finite convex subgraphs have a pre-hull number which is at most 1. Such a partial cube is a called a Peano graph. Also we study several fundamental notions related to the interval spaces of connected graphs viewed as commutative quasihypergroups, namely, homomorphisms, subhypergroupoids, direct and weak direct products, and retracts. We show that the interval space of an interval monotone bipartite graph is a join space if and only if all its cosets are subhypergroupoids, and we prove that the class of interval spaces of a Peano graph is closed under subhypergroups, weak direct products and retracts, but generally not for homomorphic images.