Triangle-Free Outerplanar 3-Graphs Are Pairwise Compatibility Graphs

Abstract

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. It is known that not all graphs are PCGs. Thus it is interesting to know which classes of graphs are PCGs. In this paper we show that triangle-free outerplanar graphs with the maximum degree 3 are PCGs.