Long cycles through prescribed vertices have the Erd\H{o}s-P\'osa property

TL;DR

This paper proves a generalized Erdős-Pósa property for long cycles through prescribed vertices, providing both a theoretical result and an efficient algorithm for finding such cycles or hitting sets.

Contribution

It extends previous Erdős-Pósa results to long cycles through a specified vertex set and offers an algorithm with explicit runtime bounds.

Findings

Existence of $k$ disjoint long cycles through $S$ or a small hitting set

Generalization of prior Erdős-Pósa results

Efficient algorithm with $O(k \, ext{log} \, k \, s^2 \, (f(\, ext{ell}) \, n+m))$ runtime

Abstract

We prove that for every graph, any vertex subset $S$, and given integers $k,\ell$: there are $k$ disjoint cycles of length at least $\ell$ that each contain at least one vertex from $S$, or a vertex set of size $O(\ell \cdot k \log k)$ that meets all such cycles. This generalises previous results of Fiorini and Hendrickx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in $O(k \log k \cdot s^2 \cdot (f(\ell) \cdot n+m))$ time, where $s$ denotes the cardinality of $S$.