带根树子式的Erdős-Pósa性质
Erdős-Pósa property of rooted tree minors
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中文总结 AI 辅助
本文证明了以顶点子集S为根的树子式仍满足带线性界的Erdős-Pósa性质，改进了此前的O(k²)上界，是Gallai经典S-路径定理的推广。
中文摘要 AI 辅助
Fiorini、Joret和Wood（2013）证明了树子式满足所谓带线性界的Erdős-Pósa性质：对每棵树T，存在常数c≥1，使得对每个图G和整数k≥0，要么G包含k个顶点不交的子图，每个子图都包含一个T-子式；要么G有一个大小至多为ck的顶点集合X，使得G-X不含T-子式。本文证明，若给定G的顶点子集S，仅考虑G中以S为根的T-子式时，上述结论仍然成立。这里，若T的子式模型的每个分支集都包含S中的一个顶点，则称该T-子式以S为根。该结果可视为Gallai经典S-路径定理的推广，对应T=K₂的情形。X大小的上界在常数c的取值范围内是最优的，且改进了Hodor、La、Micek和Rambaud（2026）之前的O(k²)上界。
英文摘要
Fiorini, Joret, and Wood (2013) showed that tree minors satisfy the so-called Erdős-Pósa property with a linear bound: For every tree $T$ there exists a constant $c \geq 1$ such that, for every graph $G$ and integer $k\geq 0$, either $G$ contains $k$ vertex-disjoint subgraphs each containing a $T$-minor, or $G$ has a set $X$ of at most $c k$ vertices such that $G-X$ has no $T$-minor. In this paper, we prove that the same result remains true if, given a subset $S$ of vertices of $G$, one only considers $T$-minors of $G$ that are rooted in $S$. Here, a $T$-minor is rooted in $S$ if there is a minor-model of $T$ where each branch set contains a vertex from $S$. This result can be seen as a generalization of the classical $S$-Path Theorem of Gallai, which corresponds to the case $T=K_2$. The upper bound on the size of $X$ is best possible up to the value of the constant $c$, and improves on an earlier $O(k^2)$ bound due to Hodor, La, Micek, and Rambaud (2026).
