Non-empty intersection of longest paths in $P_5$-free and claw-free graphs

TL;DR

This paper characterizes specific small graphs H for which (claw, H)-free and (P_5, H)-free graphs have the property that all longest paths share a common vertex, advancing understanding of Gallai families.

Contribution

It provides a complete characterization of small graphs H that ensure (claw, H)-free and (P_5, H)-free graphs form Gallai families, with constructive proofs.

Findings

Characterization of H with up to five vertices for (claw, H)-free graphs.

Identification of H (triangle, paw, diamond) for (P_5, H)-free graphs.

Constructive proofs for the characterizations.

Abstract

A family $\mathcal{F}$ of graphs is a \textit{Gallai family} if for every connected graph $G\in \mathcal{F}$, all longest paths in $G$ have a common vertex. While it is not known whether $P_5$-free graphs are a Gallai family, Long Jr., Milans, and Munaro [The Electronic Journal of Combinatorics, 2023] showed that this is \emph{not} the case for the class of claw-free graphs. We give a complete characterization of the graphs $H$ of size at most five for which $(\text{claw}, H)$-free graphs form a Gallai family. We also show that $(P_5, H)$-free graphs form a Gallai family if $H$ is a triangle, a paw, or a diamond. Both of our results are constructive.