Erdős-Pósa property for induced packings of long $S$-cycles

2026-08-23

AI Summary

本文解决了长S-圈的诱导装包问题，通过引入基于脆弱耳的新耳分解技术，证明了存在多项式函数f(k, l)，使得每个图要么包含k个长度至少为l的S-圈的诱导装包，要么存在至多f(k, l)个顶点的集合与所有长度至少为l的S-圈相交。

Abstract

The Erdős-Pósa theorem states that for every integer $k\geq1$, every graph contains either $k$ vertex-disjoint cycles or a set of $\mathcal{O}(k\log k)$ vertices meeting all cycles. This fundamental min-max duality has been extended to numerous settings, including long cycles, $S$-cycles, that is, cycles containing a vertex in a prescribed set $S$, and cycles satisfying various additional constraints. In contrast, much less is known when the packing itself is required to be induced, namely, when distinct cycles are vertex-disjoint and have no edges between them. We prove that long $S$-cycles admit an induced version of the Erdős-Pósa-type duality. More precisely, we show that there exists a polynomial function $f(k,\ell)$ such that for all integers $k\geq1$ and $\ell\geq3$, every graph contains either an induced packing of $k$ $S$-cycles of length at least $\ell$ or a set of at most $f(k,\ell)$ vertices whose closed neighbourhood intersects all $S$-cycles of length at least $\ell$. The proof introduces a new ear-decomposition technique based on fragile ears and yields a polynomial-time algorithm for every fixed $\ell$.

Problem

• induced packings
• long S-cycles
• Erdős-Pósa property

Innovation

• induced packing
• long S-cycles
• Erdős-Pósa property
• ear-decomposition