Exploring pairwise compatibility graphs

Abstract

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(mm) <= d(max), such that each leaf l(u) of T corresponds to a vertex u is an element of V and there is an edge (u, V) is an element of E if and only if d(min) <= d(T,w)(l(u), l(v)) <= d(max) 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(max) (LPG) or only d(min) (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. (C) 2012 Elsevier B.V. All rights reserved

Pairwise Compatibility Graphs: A Survey

“…Proposition 4.1. [16] The class co-LPG coincides with mLPG and, consequently, the class co-mLPG coincides with LPG.…”

“…• the union of LPG and mLPG does not coincide with the whole class PCG. Indeed, the class C of cycles is in PCG but does not belong either to LPG or to mLPG [16,55]; • the class T of threshold graphs belongs to LPG ∩ mLPG [20];…”

“…In [16] the authors study the closure properties of the classes PCG, mLPG and LPG, under some common graph operations as: adding an isolated or universal node, adding a degree one node, adding a twin, taking the complement of a graph and taking the disjoint union of two graphs. Except for its intrinsic interest, this is also important as it is known that many graph classes can be built by means of recursive applications of particular graph operations.…”

Some reduction operations to pairwise compatibility graphs

“…For false twins, the following lemma has been proven [2]. We show that this can be proven by using normalized PCR.…”

On the Pairwise Compatibility Property of Some Superclasses of Threshold Graphs

“…Moreover, it is proven that all graphs with 7 nodes or less are PCGs [17,6], whereas the smallest example of a graph that is not PCG has 8 nodes [10]. Finally, in [7], the closure properties of the PCG class under some common graph operations are also studied.…”