DISCOVERING PAIRWISE COMPATIBILITY GRAPHS

Let T be an edge weighted tree, let d(T) (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 dmax be two non-negative real numbers such that d(min) <= d(max). Then a pairwise compatibility graph of T for dmin and dmax is a graph G = (V, E), where each vertex u' is an element of V corresponds to a leaf u of T and there is an edge (u', v') is an element of E if and only if d(min) <= d(T) (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 dmin 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 PCGs. 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.