A coarse Erdős-Pósa theorem

TL;DR

This work presents a coarse Erdős–Pósa theorem for induced cycle packings, showing that for all ${k\ge1}$ and ${\ell\ge3}$ a graph either contains an induced packing of ${k}$ cycles of length at least ${\ell}$ or admits small hitting sets ${X_1},{X_2}$ with ${|X_1|=\mathcal{O}(\ell k\log k)}$ and ${|X_2|=\mathcal{O}(k\log k)}$ such that removing their radius-1 or radius-${\ell}$ neighborhoods eliminates all ${\ell}$-cycles; moreover, the result is constructive with a polynomial-time algorithm for fixed ${\ell}$. The authors develop ${\ell}$-coarse ear-decompositions as a central tool, build two auxiliary graphs ${O_\mathcal{H}}$ and ${U_\mathcal{H}}$ to certify either a large induced packing or the hitting sets, and derive tight asymptotics on the bounds. They extend the theory to distance-${d}$ packings of two cycles, derive tighter planar-bound results, and obtain corollaries bounding the tree-independence number for ${K_{1,t}}$-free graphs, enabling polynomial-time solutions to several NP-hard problems on such classes; they also show limitations via unbounded tree-independence numbers for certain ${K_{1,3}}$-free graphs. The paper culminates with discussions of open problems and conjectures about generalizations to induced minors and induced topological minors, highlighting the broader impact on structural graph theory and algorithmic graph problems.

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.

Figures (3)

• Figure 6.1: Possible appendages of $H$. Every appendage of $H$ contains one vertex of $C$.
• Figure 6.2: When $u_{\eta-1}\in V(F_{\beta})$ in Case 2. The dashed path denotes the path $Q$. If this happens, then in the $\beta$-th step, we had to select a subtree whose root has level larger than the root of $F_{\beta}$.
• Figure 6.3: When $u_{\eta-1}\notin V(F_{\beta})$ and $u_{\eta-1}\in B_{G}\left(\bigcup_{i\in [\beta-1]}\{w_i, z_i\}, d\right)$ in Case 2. The dashed path denotes the path $Q$. If this happens, then the ball $B_{G}(z_j,d)$ contains two vertices $z_j$ and $u_{\eta-1}$ having neighbours in $B_G(C, d)$ which makes a $C$-path of length at most $2d+2$.

Theorems & Definitions (78)

• Theorem 1.0
• Corollary 1.0
• Corollary 1.1
• Theorem 1.2
• Theorem 1.2
• Theorem 1.2
• Corollary 1.2
• Theorem 1.2
• Conjecture 1.3
• Lemma 2.1