Transversals of longest paths

Abstract

Let lpt(G) be the minimum cardinality of a transversal of longest paths in G, that is, a set of vertices that intersects all longest paths in a graph G. There are several results in the literature bounding the value of lpt(G) in general or in specific classes of graphs. For instance, lpt(G) = 1 if G is a connected partial 2-tree, and a connected partial 3-tree G is known with lpt(G) = 2. We prove that lpt(G) ≤ 2 for every planar 3-tree G; that lpt(G) ≤ 3 for every connected partial 3-tree G; and that lpt(G) = 1 if G is a connected bipartite permutation graph or a connected full substar graph. Our first two results can be adapted for broader classes, improving slightly some known general results: we prove that lpt(G) ≤ k for every connected partial k-tree G and that lpt(G) ≤ max{1, ω(G) − 2} for every connected chordal graph G, where ω(G) is the cardinality of a maximum clique in G.