Longest Path and Cycle Transversals in Chordal Graphs

TL;DR

This paper proves that connected chordal graphs have small transversals for longest paths and cycles, with sizes depending on the graph's size and leafage, advancing understanding of their structural properties.

Contribution

It establishes bounds on the size of longest path and cycle transversals in chordal graphs, linking these bounds to graph parameters like leafage and connectivity.

Findings

Longest path transversal size is O(log^2 n) in connected chordal graphs.

Longest cycle transversal size is O(log n) in 2-connected chordal graphs.

Transversals are bounded by the leafage of the graph.

Abstract

We show that if $G$ is a $n$-vertex connected chordal graph, then it admits a longest path transversal of size $O(\log^2 n)$. Under the stronger assumption of 2-connectivity, we show $G$ admits a longest cycle transversal of size $O(\log n)$. We also provide longest path and longest cycle transversals which are bounded by the leafage of the chordal graph.