Skeletons, recognition algorithm and distance matrix of quasi-median graphs

Abstract

A quasi-median graph can be characterized as a weak retract of a Cartesian product of complete graphs or equivalently as a graph of finite windex. We derive a new characterization of quasi-median graphs which allows us to recognize these graphs in O(n√n log n+m log n) time. It is shown that skeletons of quasi-median graphs are median graphs. For an arbitrary vertex s an s-skeleton of a graph is obtained from G by deleting all edges uv that satisfy d(u, s) = d(v, s). As a by-product of this approach the size of a maximum complete subgraph of a quasi-median graph can be computed within the same time bound. Furthermore, we show that the distance matrix of a quasi-median graph can be computed in O(n 2).