Pairwise compatibility graphs: complete characterization for wheels

Abstract

A simple graph $G$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ with positive weights and nonnegative numbers $d_{min}$ and $d_{max}$ such that the leaves of $T$ are exactly the vertices of $G$, and $uv$ is an edge in $G$ if and only if the sum of weights of edges on the unique path between $u$ and $v$ in $T$ is at least $d_{min}$ and at most $d_{max}$. We show that a wheel on $n$ vertices is a PCG if and only if $n\le 8$, settling an open problem proposed by Calamoneri and Sinaimeri (SIAM Review 58:3 (2016), 445–460). Our approach is based on unavoidable binary classifications of the edges in the complement of wheels that are PCGs. (Note: during the review process of our work, we learned that the same result has been obtained independently with an alternative proof.)