Peano partial cubes

Abstract: Peano partial cubes are the bipartite graphs whose geodesic interval spaces are (closed) join spaces. They are the partial cubes all of whose finite convex subgraphs have a pre-hull number which is at most 1. Special Peano partial cubes are median graphs, cellular bipartite graphs and netlike partial cubes. Analogous properties of these graphs are satisfied by Peano partial cubes. In particular the convex hull of any isometric cycle of such a graph is a gated quasi-hypertori (i.e., the Cartesian product of copies of K_2 and even cycles). Moreover, for any Peano partial cubes G that contains no isometric rays, there exists a finite qasi-hypertorus which is fixed by all automorphisms of G, and any self-contraction of G fixes some finite quasi-hypertorus. A Peano partial cube G is called a hyper-median partial cube if any triple of vertices of has either a median or a hyper-median, that is, a quasi-median whose convex-hull induces a hypertorus (i.e., the Cartesian product of even cycles such that at least one of them has length greater than 4). These graphs have several properties similar to that of median graphs. In particular a graph is a hyper-median partial cube if and only if all its finite convex subgraphs are obtained by successive gated amalgamations from finite quasi-hypertori. Also a finite graph is a hyper-median partial cube if and only if it can be obtained from K_1 by a sequence of special expansions. The class of Peano partial cubes and that of hyper-median partial cubes are closed under convex subgraphs, retracts, Cartesian products and gated amalgamations. We study two convex invariants: the Helly number of a Peano partial cube, and the depth of a hyper-median partial cube that contains no isometric rays. Finally, for a finite Peano partial cube G, we prove an Euler-type formula, and a similar formula giving the isometric dimension of G.