A coarse Erd\H{o}s-P\'{o}sa theorem

TL;DR

This paper extends the Erdős-Pósa theorem to induced cycle packings, providing bounds, algorithms, and implications for graph structure and computational complexity.

Contribution

It introduces a polynomial-time method for finding induced cycle packings or vertex sets that simplify the graph's cycle structure, generalizing classic results.

Findings

Existence of functions bounding the size of vertex sets for cycle packings

Polynomial-time algorithms for constant ll

Bounded tree-independence number for certain graph classes

Abstract

An induced packing of cycles in a graph is a set of vertex-disjoint cycles with no edges between them. We generalise the classic Erd\H{o}s-P\'osa theorem to induced packings of cycles. More specifically, we show that there exist functions $f(k,\ell)=\mathcal{O}(\ell k\log k)$ and $g(k)=\mathcal{O}(k\log k)$ such that for all integers $k\geq1$ and $\ell\geq3$, every graph $G$ contains either an induced packing of $k$ cycles of length at least $\ell$, not necessarily induced cycles, or sets $X_1$ and $X_2$ of vertices with $|X_1|\leq f(k,\ell)$ and $|X_2|\leq g(k)$ such that, after removing the closed neighbourhood of $X_1$ or the ball of radius $\ell$ around $X_2$, the resulting graph has no cycle of length at least $\ell$ in $G$. Our proof is constructive and yields a polynomial-time algorithm finding either the induced packing or the sets $X_1$ and $X_2$ when $\ell$ is a constant. Furthermore, we show that for every positive integer $d$, if a graph $G$ does not contain two cycles at distance more than $d$, then $G$ contains sets $X_1$ and $X_2$ of vertices with $|X_1|\leq12(d+1)$ and $|X_2|\leq12$ such that, after removing the ball of radius $2d$ around $X_1$ or the ball of radius $3d$ around $X_2$, the resulting graphs are forests. As a corollary, we prove that every graph with no $K_{1,t}$ induced subgraph and no induced packing of $k$ cycles of length at least $\ell$ has tree-independence number at most $\mathcal{O}(t\ell k\log k)$, and one can construct a corresponding tree-decomposition in polynomial time when $\ell$ is a constant. This resolves a special case of a conjecture of Dallard et al. (arXiv:2402.11222), and implies that on such graphs, many NP-hard problems, are solvable in polynomial time. On the other hand, we show that the class of all graphs with no $K_{1,3}$ induced subgraph and no two cycles at distance more than $2$ has unbounded tree-independence number.