On the Pairwise Compatibility Property of some Superclasses of Threshold Graphs

A graph G is called a pairwise compatibility graph (PCG) if there exists a positive edge weighted tree T and two non-negative real numbers dmin and dmax such that each leaf lu of T corresponds to a node u ∈ V and there is an edge (u, v) ∈ E if and only if dmin ≤ dT (lu, lv) ≤ dmax, where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T . In this paper we study the relations between the pairwise compatibility property and superclasses of threshold graphs, i.e. graphs where the neighborhoods of any couple of nodes either coincide or are included one into the other. Namely, we prove that some of these superclasses belong to the PCG class. Moreover, we tackle the problem of characterizing the class of graphs that are PCGs of a star, deducing that also these graphs are a generalization of threshold graphs.