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 $1$.

翻译：我们朝着刻画满足以下性质的图$H$取得进展：每个连通$H$-自由图都存在大小为$1$的最长路横贯。特别地，我们证明了顶点数不超过$4$且满足该性质的图$H$恰好是线性森林。我们还证明了若连通图$G$的阶数相对于其连通度$\kappa(G)$足够大，且其独立数$\alpha(G)$满足$\alpha(G) \le \kappa(G) + 2$，则每个最大度顶点构成一个大小为$1$的最长路横贯。