On Pairwise Compatibility of Some Graph (Super)Classes

A graph G=(V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers `d' and `D' such that each leaf `u' of T is a node of V and the edge `(u,v) belongs to E' iff `d <= d_T(u, v) <= D' where d_T(u, v) is the sum of weights of the edges on the unique path from `u' to `v' in T. The main issue on these graphs consists in characterizing them. In this note we prove the inclusion in the PCG class of threshold tolerance graphs and the non-inclusion of a number of intersection graphs, such as disk and grid intersection graphs, circular arc and tolerance graphs. The non-inclusion of some superclasses (trapezoid, permutation and rectangle intersection graphs) follows.