Some classes of graphs that are not PCGs

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 dmin and dmax, dmin≤dmax, such that each node u∈V is uniquely associated to a leaf of T and there is an edge (u,v)∈E if and only if dmin≤dT(u,v)≤dmax, where dT(u,v) is the sum of the weights of the edges on the unique path PT(u,v) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. In this paper we show that some interesting classes of graphs have empty intersection with PCG; they are wheels, strong product of a cycle and P2 and the square of an n node cycle, with n sufficiently large. As a side effect, we show that the smallest planar graph not to be PCG has not 20 nodes, as previously known, but only 8 (it is C82).