Intersecting longest paths in chordal graphs

TL;DR

This paper investigates the minimum vertex sets needed to intersect all longest paths and cycles in chordal graphs, providing bounds related to the clique number.

Contribution

It establishes new bounds on the size of longest path and cycle transversals in chordal graphs based on their clique number.

Findings

Existence of a vertex set of size at most 4 times the ceiling of clique number over 5 for longest path transversals.

Existence of a vertex set of size at most 2 times the ceiling of clique number over 3 for longest cycle transversals in 2-connected 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 $2\lceil\frac{\omega(G)}{3}\rceil$.