Longest Path Transversals in Claw-Free and $$P_5$$-Free Graphs

For a connected graph G, the longest path transversal number of G, denoted by lpt(G), is the minimum cardinality of a set of vertices that intersects all longest paths in G. It is an open problem whether any graph admits a longest path transversal of constant size. This question remains open even when restricted to claw-free graphs and P5-free graphs. In this work, we investigate these two graph classes. We show that, given a connected graph G, lpt(G)=1 if G is a (P5,H)-free graph, when H is a triangle, a paw, or a diamond. We also provide a complete characterization of the graphs H on at most five vertices for which for any (claw, H)-free graph G it holds that lpt(G)=1. Moreover, in each of these cases, we present a polynomial-time algorithm which finds a vertex in G that belongs to all its longest paths.