Non-empty intersection of longest paths in $H$-free graphs

TL;DR

This paper investigates the conditions under which all longest paths in certain $H$-free graphs share a common vertex, identifying specific graph classes where this property holds.

Contribution

It characterizes the graphs $H$ on up to 4 vertices for which every connected $H$-free graph has a single-vertex longest path transversal.

Findings

Graphs $H$ on at most 4 vertices are linear forests for the property to hold.

Large connected graphs with bounded independence number have a maximum degree vertex as a longest path transversal.

Abstract

We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $\kappa(G)$, and its independence number $\alpha(G)$ satisfies $\alpha(G) \le \kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.