Discovering Pairwise Compatibility Graphs

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 dmin and dmax be two nonnegative real numbers such that dmin ≤ dmax. Then 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. 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 dmax such that G is a pairwise compatibility graph of T for dmin and dmax. 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. We also show that the well known tree power graphs and some of their extensions are PCGs.