Some reduction operations to pairwise compatibility graphs

Abstract

A graph G=(V,E) with a vertex set V and an edge set E is called a pairwise compatibility graph (PCG, for short) if there are a tree T whose leaf set is V, a non-negative edge weight w in T, and two non-negative reals dmin≤dmax such that G has an edge uv∈E if and only if the sum of the edge weights in the unique path between u and v in the weighted tree (T,w) is in the interval [dmin,dmax]. PCG is a relatively new graph class motivated from bioinformatics. In this paper, we give some necessary conditions and sufficient conditions for a graph to be in PCG based on cut-vertices and twins, which provide reductions among PCGs. Our results imply that complete k-partite graphs, cacti, and some other graph classes are subsets of PCG.