长S-圈诱导填充的Erdős-Pósa性质
Erdős-Pósa property for induced packings of long $S$-cycles
AI总结：
该研究证明长S-圈具有Erdős-Pósa型对偶性的诱导版本，引入基于脆弱耳的耳分解技术，得到多项式时间算法。
AI中文摘要：
Erdős-Pósa定理指出，对于每个整数k≥1，任意图要么包含k个顶点不相交的圈，要么包含一个大小为𝒪(k log k)的顶点集，该集与所有圈相交。这一基本的最小-最大对偶性已被扩展到多种情形，包括长圈、S-圈（即包含指定集合S中一个顶点的圈）以及满足各种附加约束的圈。相比之下，当要求填充本身是诱导的时，即不同圈顶点不相交且它们之间没有边，人们的了解要少得多。我们证明长S-圈具有Erdős-Pósa型对偶性的诱导版本。更确切地说，我们证明存在一个多项式函数f(k,ℓ)，使得对于所有整数k≥1和ℓ≥3，任意图要么包含k个长度至少为ℓ的S-圈的诱导填充，要么包含一个大小不超过f(k,ℓ)的顶点集，其闭邻域与所有长度至少为ℓ的S-圈相交。该证明引入了一种基于脆弱耳的新型耳分解技术，并为每个固定的ℓ得到了一个多项式时间算法。
英文摘要：
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 $\ell$.
