A Poset-based Approach to Embedding Median Graphs in Hypercubes and Lattices

Abstract

A median graph G is a graph where, for any three vertices u, v and w, there is a unique node that lies on a shortest path from u to v, from u to w, and from v to w. While not obvious from the definition, median graphs are partial cubes; that is, they can be isometrically embedded in hypercubes and, consequently, in integer lattices. The isometric and lattice dimensions of G, denoted as dim _I (G) and dim _Z (G), are the smallest integers k and r so that G can be isometrically embedded in the k-dimensional hypercube and the r-dimensional lattice respectively. Motivated by recent results on the cover graphs of distributive lattices, we study these parameters through median semilattices, a class of ordered structures related to median graphs. We show that not only does this approach provide new combinatorial characterizations for dim _I (G) and dim _Z (G), they also have nice algorithmic consequences. Assume G has n vertices and m edges. We prove that dim _I (G) can be computed in O(n + m) time, and an isometric embedding of G on a hypercube with dimension dim _I (G) can be constructed in O(n × dim _I (G)) time. The algorithms are extremely simple and the running times are optimal. We also show that dim _Z (G) can be computed and an isometric embedding of G on a lattice with dimension dim _Z (G) can be constructed in $O( n \times dim_I(G) + dim_I(G)^{2.5})$ time by using an existing algorithm of Eppstein’s that performs the same tasks for partial cubes. We are able to speed up his algorithm by using our framework to provide a new “interpretation” to the algorithm. In particular, we note that its main part is essentially a generalization of Fulkerson’s method for finding a smallest-sized chain decomposition of a poset.