Erd\H{o}s--P\'{o}sa property of cycles that are far apart

TL;DR

This paper establishes a generalized Erdős–Pósa property for cycles that are pairwise far apart, linking the existence of multiple such cycles to a small vertex set that simplifies the graph structure.

Contribution

It proves the existence of functions characterizing when a graph contains many distant cycles or a small vertex set that reduces the graph to a forest.

Findings

Either the graph contains k cycles far apart from each other.

Or there exists a small vertex set whose removal makes the graph a forest.

The functions f and g depend only on k and d, not on the graph size.

Abstract

We prove that there exist functions $f,g:\mathbb{N}\to\mathbb{N}$ such that for all nonnegative integers $k$ and $d$, for every graph $G$, either $G$ contains $k$ cycles such that vertices of different cycles have distance greater than $d$ in $G$, or there exists a subset $X$ of vertices of $G$ with $|X|\leq f(k)$ such that $G-B_G(X,g(d))$ is a forest, where $B_G(X,r)$ denotes the set of vertices of $G$ having distance at most $r$ from a vertex of $X$.