Transversals of Longest Paths

Let lpt(G) be the minimum cardinality of a set of vertices that intersects all longest paths in a graph G. Let ω(G) be the size of a maximum clique in G, and tw(G) be the treewidth of G. We prove that lpt(G) ≤ max{1,ω(G)− 2} when G is a connected chordal graph; that lpt(G) = 1 when G is a connected bipartite permutation graph or a connected full substar graph; and that lpt(G) ≤ tw(G) for any connected graph G.