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 d min ≤ d max such that G has an edge uv ∈ E if and only if the distance between u and v in the weighted tree (T, w) is in the interval [d min , d max ]. PCG is a new graph class motivated from bioinformatics. In this paper, we give some necessary and sufficient conditions for PCG based on cut-vertices and twins, which provide reductions among PCGs. Our results imply that complete k-partite graph, cactus, and some other graph classes are subsets of PCG.

Characterizing Star-PCGs

“…It is also known that a PCG with a witness tree being a caterpillar also allows a witness tree being a centipede [4]. A method for constructing PCGs is derived [17], where it is shown that a graph G consisting two graphs G 1 and G 2 that share a vertex as a cut-vertex in G is a PCG if and only both G 1 and G 2 are PCGs.…”