Not All Graphs are Pairwise Compatibility Graphs

Given an edge weighted tree T and two non-negative real numbers dmin and dmax, a pairwise compatibility graph of T for dmin and dmax 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 dmin ≤ dT (u, v) ≤ dmax in T . Here, dT (u, v) denotes the distance between u and v in T , which is the sum of the weights of the edges on the path from u to v. We call T a pairwise compatibility tree of G. We call a graph G a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers dmin and dmax such that G is a pairwise compatibility graph of T for dmin and dmax. Since the introduction of PCGs it remains an open problem: whether or not all undirected graphs are PCGs; in other words, is there always a pairwise compatibility tree T for any arbitrary graph G? In this paper we give a negative answer to the open problem by showing that not all undirected graphs are PCGs.