Pairwise compatibility graphs

Given an edge weighted tree T and two non-negative real numbers dmin and dmax, a pairwise compatibility graph (PCG) of T is a graph G = (V,E), where each vertex u ∈ V corresponds to a leaf u of T and an edge (u, v) ∈ E if and only if dmin ≤ distance(u, v) ≤ dmax in T. In this paper we give some properties of these graphs. We establish a relationship between pairwise compatibility graphs and chordal graphs. We show that all chordless cycles and single chord cycles are pairwise compatibility graphs.We also provide a linear-time algorithm for constructing trees that can generate graphs having cycles as their maximal biconnected subgraphs as PCGs. The techniques that we used to identify various types of pairwise compatibility graphs are quite generic and may be useful to discover other properties of these graphs.