Some Reduction Operations to Pairwise Compatibility Graphs

TL;DR

This paper investigates the properties of pairwise compatibility graphs (PCGs), providing reduction techniques and characterizing certain graph classes as subsets of PCGs, with applications in bioinformatics.

Contribution

It introduces necessary and sufficient conditions for PCGs based on cut-vertices and twins, enabling reductions and identifying subclasses within PCGs.

Findings

Complete k-partite graphs are subsets of PCGs.

Cactus graphs are subsets of PCGs.

Reduction techniques for PCGs based on graph properties.

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 $d_{\min}\leq d_{\max}$ such that $G$ has an edge $uv\in E$ 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.