Intersecting longest paths in chordal graphs

Abstract

We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path transversals. We show that if ω(G) is the clique number of a chordal graph G, then there is a transversal of order at most 4⌈ω(G)5⌉. We also consider the analogous question for longest cycles, and show that if G is a 2-connected chordal graph then there is a transversal intersecting all longest cycles of order at most 2⌈ω(G)3⌉.