Transversals of Longest Cycles in Partial $k$-Trees and Chordal Graphs

TL;DR

This paper investigates the minimum vertex set intersecting all longest cycles in specific graph classes, establishing bounds for partial k-trees and chordal graphs, with implications for series parallel graphs and 3-trees.

Contribution

It provides new upper bounds on the size of transversals of longest cycles in partial k-trees and chordal graphs, extending understanding of cycle structure in these classes.

Findings

For partial k-trees, the transversal size is at most k-1.

In chordal graphs, the transversal size is at most max{1, ω(G)-3}.

All longest cycles intersect in 2-connected series parallel graphs and 3-trees.

Abstract

Let $lct(G)$ be the minimum cardinality of a set of vertices that intersects every longest cycle of a 2-connected graph $G$. We show that $lct(G)\leq k-1$ if $G$ is a partial $k$-tree and that $lct(G)\leq \max \{1, {\omega(G){-}3}\}$ if $G$ is chordal, where $\omega(G)$ is the cardinality of a maximum clique in $G$. Those results imply that all longest cycles intersect in 2-connected series parallel graphs and in 3-trees.