Exploring pairwise compatibility graphs

A graph G=(V,E) is called a pairwise compatibility graph (PCG) if there exists a tree T, a positive edge weight function w on T, and two non-negative real numbers d"m"i"[email protected]?d"m"a"x, such that each leaf l"u of T corresponds to a vertex [email protected]?V and there is an edge (u,v)@?E if and only if d"m"i"[email protected]?d"T","w(l"u,l"v)@?d"m"a"x where d"T","w(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 analyze the class of PCGs in relation to two particular subclasses resulting from the cases where the constraints on the distance between the pairs of leaves concern only d"m"a"x (LPG) or only d"m"i"n (mLPG). In particular, we show that the union of LPG and mLPG classes does not coincide with the whole class of PCGs, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, we study the closure properties of the classes PCG, mLPG and LPG, under some common graph operations. In particular, we consider the following operations: adding an isolated or universal vertex, adding a pendant vertex, adding a false or a true twin, taking the complement of a graph and taking the disjoint union of two graphs.