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arXiv 2607.07697math.COcs.DM
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诱导 Erdős–Pósa 性质用于长孔、长 theta 以及更广泛的结构

Induced Erdős--Pósa property for long holes, long thetas, and beyond

University of Warsaw（华沙大学）；Warsaw University of Technology（华沙理工大学）

Jadwiga Czyżewska, Tomáš Masařík, Marcin Pilipczuk, Amadeus Reinald, Paweł Rzążewski

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35 pages, 6 figures
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AI总结：

本文研究了长环和长 theta 的诱导 Erdős–Pósa 性质，并由此得出排除 $k$ 个不相交的 $\Theta_t$ 的图具有平衡分离器，进而得到最大权重独立集的 QPTAS。

摘要：

诱导 Erdős–Pósa 性质在图中将最大数量的成对反邻接的副本与击中所有副本所需的最小邻域数联系起来。在本文中，我们考虑的物体是长环和长 theta，作为诱导极小图。令 $C_t$ 表示具有 $t$ 个顶点的环，令 $\Theta_t$ 是由三个内部不相交且反邻接的路径组成，每个路径有 $t$ 个内部顶点，连接相同的两个不同顶点的图。我们证明，对于每个固定的 $t$，$C_t$ 和 $\Theta_t$ 都具有诱导 Erdős–Pósa 性质。更具体地说，对于每一个整数 $k$ 和每一个图 $G$，以下两种情况之一会发生：(i) $G$ 包含 $k$ 个互不相交且反邻接的 $C_t$ (resp., $\Theta_t$) 作为诱导极小图，或者 (ii) 存在一个子集 $X \subseteq V(G)$，其大小为 $\mathcal{O}(tk \log k)$，使得集合 $N[X]$，由 $X$ 和其邻居组成，击中所有 $C_t$ (resp., 所有 $\Theta_t$) 诱导极小图。这在 Ahn、Gollin、Huynh 和 Kwon [SODA 2025] 的猜想的一个特例中得到强有力的解决。从这些结果中我们得出，排除 $k$ 个不相交的 $\Theta_t$ 作为诱导极小图的图具有平衡分离器，由 $\mathcal{O}(tk \log k)$ 个顶点的邻域组成。这反过来解决了 Gartland 和 Lokshtanov 的猜想的一个特例，并结合已知技术，得出最大权重独立集以及其多个推广的 QPTAS。

英文摘要：

The induced Erdős--Pósa property in graphs relates the maximum number of pairwise anti-adjacent copies of an object with the minimum number of neighborhoods required to hit all copies. In this paper, the objects we consider are long cycles and long thetas, both as induced minors. Let $C_t$ denote the cycle with $t$ vertices and let $Θ_t$ be the graph consisting of three internally disjoint and anti-adjacent paths, each with $t$ internal vertices, connecting the same pair of distinct vertices. We show that for every fixed $t$, both $C_t$ and $Θ_t$ have the induced Erdős--Pósa property with respect to the induced minor relation. More precisely, for every integer $k$ and every graph $G$, one of the following two outcomes occurs: (i) $G$ contains $k$ pairwise vertex-disjoint and anti-adjacent copies of $C_t$ (resp., $Θ_t$) as induced minors, or (ii) there is a set $X \subseteq V(G)$ of size $\mathcal{O}(tk \log k)$ such that the set $N[X]$, consisting of $X$ and its neighbors, hits all $C_t$ (resp., all $Θ_t$) induced minors in $G$. This resolves in a strong form a special case of a conjecture of Ahn, Gollin, Huynh, and Kwon [SODA 2025]. From these results we derive that graphs that exclude $k$ disjoint copies of $Θ_t$ as an induced minor admit balanced separators consisting of the neighborhood of $\mathcal{O}(tk \log k)$ vertices. This in turn resolves a special case of a conjecture of Gartland and Lokshtanov and, combined with known techniques, yields a QPTAS for Maximum Weight Independent Set and a number of its generalizations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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