  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 $1$.
