DISCOVERING PAIRWISE COMPATIBILITY GRAPHS

Abstract

Let T be an edge weighted tree, let dT(u, v) be the sum of the weights of the edges on the path from u to v in T, and let d min and d max be two non-negative real numbers such that d min ≤ d max . Then a pairwise compatibility graph of T for d min and d max is a graph G = (V, E), where each vertex u' ∈ V corresponds to a leaf u of T and there is an edge (u', v') ∈ E if and only if d min ≤ dT(u, v) ≤ d max . A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers d min and d max such that G is a pairwise compatibility graph of T for d min and d max . Kearney et al. conjectured that every graph is a PCG [3]. In this paper, we refute the conjecture by showing that not all graphs are PCG s . Moreover, we recognize several classes of graphs as pairwise compatibility graphs. We identify two restricted classes of bipartite graphs as PCG. We also show that the well known tree power graphs and some of their extensions are PCGs.