Erdős-Pósa 对偶的 Minor 障碍
Obstructions to Erdős-Pósa Dualities for Minors
AI总结：
本文证明对每个 minor 封闭图类 H，其 Erdős-Pósa 反例类有限，并给出完整刻画与构造性算法，从而解决 Thomas 关于半整数 Erdős-Pósa 性质的猜想。
摘要：
设 ${\cal G}$ 和 ${\cal H}$ 是 minor 封闭的图类。若存在函数 $f$，使得对于每个 $k$ 和每个 $G\in{\cal G}$，要么 $G$ 有 $k$ 个两两顶点不相交的子图不属于 ${\cal H}$，要么存在集合 $S\subseteq V(G)$ 满足 $|S|\leq f(k)$ 且 $G-S\in{\cal H}$，则称 $({\cal H},{\cal G})$ 为 Erdős-Pósa 对（EP-对）。Erdős 和 Pósa 的经典结果指出，若 $\mathcal{F}$ 是森林类，则对于每个 ${\cal G}$，$({\cal F},{\cal G})$ 都是 EP-对。若 ${\cal G}$ 是使得 $({\cal H},{\cal G})$ 不是 EP-对的最小图类，则称 ${\cal G}$ 是 ${\cal H}$ 的 EP-反例。我们证明对于每个 ${\cal H}$，所有 EP-反例的集合 $\mathfrak{C}_{\cal H}$ 是有限的。特别地，我们给出了每个 ${\cal H}$ 的 $\mathfrak{C}_{\cal H}$ 的完整刻画，并给出了其大小的构造性上界。每个类 ${\cal G}\in \mathfrak{C}_{\cal H}$ 可以描述为一系列网格状图 $\langle \mathscr{W}_{k} \rangle_{k\in \mathbb{N}}$ 的所有 minor。此外，每个 $\mathscr{W}_{k}$ 允许半整数打包：$k$ 个某个 $H\not\in{\cal H}$ 的副本，其中每个顶点至多被使用两次。这给出了 minor 的 Erdős-Pósa 性质的半整数阈值的完整描绘，并为 Thomas 关于 minor 的半整数 Erdős-Pósa 性质的猜想（最近由 Liu 非构造性地证实）提供了构造性证明。设 $h$ 为 ${\cal H}$ 中图的最大大小。对于每个类 ${\cal H}$，我们构造一个算法，给定图 $G$ 和 $k$，要么输出 $k$ 个某个 $H \not\in {\cal H}$ 的副本的半整数打包，要么输出至多 ${2^{k^{\cal O}_h(1)}}$ 个顶点的集合，其删除后得到 ${\cal H}$ 中的图，时间复杂度为 $2^{2^{k^{{\cal O}_h(1)}}}\cdot |G|^4\log |G|$。
英文摘要：
Let ${\cal G}$ and ${\cal H}$ be minor-closed graph classes. The pair $({\cal H},{\cal G})$ is an Erdős-Pósa pair (EP-pair) if there is a function $f$ where, for every $k$ and every $G\in{\cal G},$ either $G$ has $k$ pairwise vertex-disjoint subgraphs not belonging to ${\cal H},$ or there is a set $S\subseteq V(G)$ where $|S|\leq f(k)$ and $G-S\in{\cal H}.$ The classic result of Erdős and Pósa says that if $\mathcal{F}$ is the class of forests, then $({\cal F},{\cal G})$ is an EP-pair for every ${\cal G}$. The class ${\cal G}$ is an EP-counterexample for ${\cal H}$ if ${\cal G}$ is minimal with the property that $({\cal H},{\cal G})$ is not an EP-pair. We prove that for every ${\cal H}$ the set $\mathfrak{C}_{\cal H}$ of all EP-counterexamples for ${\cal H}$ is finite. In particular, we provide a complete characterization of $\mathfrak{C}_{\cal H}$ for every ${\cal H}$ and give a constructive upper bound on its size. Each class ${\cal G}\in \mathfrak{C}_{\cal H}$ can be described as all minors of a sequence of grid-like graphs $\langle \mathscr{W}_{k} \rangle_{k\in \mathbb{N}}.$ Moreover, each $\mathscr{W}_{k}$ admits a half-integral packing: $k$ copies of some $H\not\in{\cal H}$ where no vertex is used more than twice. This gives a complete delineation of the half-integrality threshold of the Erdős-Pósa property for minors and yields a constructive proof of Thomas' conjecture on the half-integral Erdős-Pósa property for minors (recently confirmed, non-constructively, by Liu). Let $h$ be the maximum size of a graph in ${\cal H}.$ For every class ${\cal H},$ we construct an algorithm that, given a graph $G$ and a $k,$ either outputs a half-integral packing of $k$ copies of some $H \not\in {\cal H}$ or outputs a set of at most ${2^{k^{\cal O}_h(1)}}$ vertices whose deletion creates a graph in ${\cal H}$ in time $2^{2^{k^{{\cal O}_h(1)}}}\cdot |G|^4\log |G|.$
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。
