All graphs with at most seven vertices are Pairwise Compatibility Graphs

TL;DR

This paper proves that all graphs with up to seven vertices are pairwise compatibility graphs (PCGs), except for the wheel graph on seven vertices, and most are representable by a centipede tree structure.

Contribution

It establishes that all small graphs (up to seven vertices) are PCGs, identifying the wheel graph on seven vertices as a notable exception, and characterizes the tree structures representing these graphs.

Findings

All graphs with ≤7 vertices are PCGs.

The wheel graph W7 is not a PCG.

Most graphs are represented by centipede trees.

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 $d_{min} \leq d_{T,w} (l_u, l_v) \leq d_{max}$ where $d_{T,w} (l_u, l_v)$ is the sum of the weights of the edges on the unique path from $l_u$ to $l_v$ in $T$. In this note, we show that all the graphs with at most seven vertices are PCGs. In particular all these graphs except for the wheel on 7 vertices $W_7$ are PCGs of a particular structure of a tree: a centipede.