A coarse Erdős-Pósa theorem

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 Erdős-Pósa 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.