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 d_min and d_max such that each leaf l_u of T corresponds to a node u ∈ V and there is an edge (u, v) ∈ E if and only if d_min ≤ d_T (l_u, l_v) ≤ d_max, where d_T (l_u, l_v) is the sum of the weights of the edges on the unique path from l_u to l_v 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.