Some classes of graphs that are not PCGs

TL;DR

This paper introduces a new proof technique to identify certain graph classes, such as wheels and specific strong product graphs, that do not belong to the class of pairwise compatibility graphs (PCGs).

Contribution

The paper presents a novel proof method to demonstrate that some graph classes are outside the PCG class, expanding understanding of PCG boundaries.

Findings

Wheels are not PCGs.

Graphs from the strong product of a cycle and P2 are not PCGs.

New proof technique for classifying non-PCG graphs.

Abstract

A graph $G=(V,E)$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$, $d_{min} \leq d_{max}$, such that each node $u \in V$ is uniquely associated to a leaf of $T$ and there is an edge $(u,v) \in E$ if and only if $d_{min} \leq d_{T} (u, v) \leq d_{max}$, where $d_{T} (u, v)$ is the sum of the weights of the edges on the unique path $P_{T}(u,v)$ from $u$ to $v$ in $T$. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. As an example, we use this technique to show that wheels and graphs obtained as strong product between a cycle and $P_2$ are not PCGs.