On relaxing the constraints in pairwise compatibility graphs

TL;DR

This paper investigates the class of pairwise compatibility graphs (PCG) and its subclasses LPG and mLPG, analyzing their relationships and exploring their membership within split matrogenic graphs.

Contribution

It demonstrates that LPG and mLPG are proper subclasses of PCG, explores their intersection, and examines their relation to split matrogenic graphs.

Findings

LPG and mLPG do not cover all PCGs.

LPG and mLPG intersect but are not contained within each other.

The paper analyzes PCG membership in split matrogenic graphs.

Abstract

A graph $G$ is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$ such that each leaf $l_u$ of $T$ corresponds to a vertex $u \in V$ and there is an edge $(u,v) \in E$ if and only if $d_{min} \leq d_T (l_u, l_v) \leq d_{max}$ where $d_T (l_u, l_v)$ 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 PCG in relation with two particular subclasses resulting from the the cases where $\dmin=0$ (LPG) and $\dmax=+\infty$ (mLPG). In particular, we show that the union of LPG and mLPG does not coincide with the whole class PCG, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, as the graphs we deal with belong to the more general class of split matrogenic graphs, we focus on this class of graphs for which we try to establish the membership to the PCG class.