Non-empty intersection of longest paths in \(H\)-free graphs (Q2692169)

Summary: 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 \(\alpha(G) \leqslant \kappa(G) + 2\), then each vertex of maximum degree forms a longest path transversal of size 1.