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 $\omega (G)$ is the clique number of a chordal graph $G$, then there is a transversal of order at most $4\lceil \frac{\omega (G)}{5}\rceil$. 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 $2\lceil \frac{\omega (G)}{3}\rceil$.

11 pages