We investigate polynomial-time preprocessing for the problem of hitting forbidden minors in a graph, using the framework of kernelization. For a fixed finite set of connected graphs $\mathcal{F}$, the $\mathcal{F}$-Deletion problem is the following: given a graph G and integer k, is it possible to delete k vertices from G to ensure the resulting graph does not contain any graph from $\mathcal{F}$ as a minor? Earlier work by Fomin, Lokshtanov, Misra, and Saurabh [FOCS'12] showed that when $\mathcal{F}$ contains a planar graph, an instance $(G,k)$ can be reduced in polynomial time to an equivalent one of size $k^{\mathcal{O}(1)}$. In this work we focus on structural measures of the complexity of an instance, with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large. Motivated by several impossibility results, we parameterize the $\mathcal{F}$-Deletion problem by the size of a vertex modulator whose removal results in a graph of constant treedepth η.

We prove that for each set $\mathcal{F}$ of connected graphs and constant η, the $\mathcal{F}$-Deletion problem parameterized by the size of a treedepth-η modulator has a polynomial kernel. Our kernelization is fully explicit and does not depend on protrusion reduction or well-quasi-ordering, which are sources of algorithmic non-constructivity in earlier works on $\mathcal{F}$-Deletion. Our main technical contribution is to analyze how models of a forbidden minor in a graph G with modulator X, interact with the various connected components of $G-X$. Using the language of labeled minors, we analyze the fragments of potential forbidden minor models that can remain after removing an optimal $\mathcal{F}$-Deletion solution from a single connected component of $G-X$. By bounding the number of different types of behavior that can occur by a polynomial in $|X|$, we obtain a polynomial kernel using a recursive preprocessing strategy. Our results extend earlier work for specific instances of $\mathcal{F}$-Deletion such as Vertex Cover and Feedback Vertex Set. It also generalizes earlier preprocessing results for $\mathcal{F}$-Deletion parameterized by a vertex cover, which is a treedepth-one modulator.

我们使用核化框架研究多项式时间预处理，以解决图中未成年人未成年人的问题。对于一组固定的有限连通图$\mathcal{F}$， $\mathcal{F}$-删除问题如下：给定一个图G和整数k，是否有可能从G中删除k个顶点以确保生成的图不包含任何图。$\mathcal{F}$作为未成年人？Fomin，Lokshtanov，Misra和Saurabh [FOCS'12]的早期工作表明，$\mathcal{F}$ 包含一个平面图，一个实例 $（G，ķ）$ 可以将多项式时间减少到大小的等分之一 ${ķ}^{\mathcal{Ø}（\mathrm{1个}）}$。在这项工作中，我们专注于实例复杂性的结构度量，目的是为解决方案较大的实例提供非平凡的预处理保证。受几个不可能结果的启发，我们将$\mathcal{F}$-由顶点调制器的大小引起的删除问题，该顶点调制器的去除导致恒定树深η的图。

我们证明每一组 $\mathcal{F}$连接图形和恒定的η，该$\mathcal{F}$-Deletion通过的treedepth-尺寸参数化问题η调制器具有多项式内核。我们的内核化是完全明确的，并且不依赖于凸出约简或良好拟序化，这是早期关于的算法非构造性的来源$\mathcal{F}$-删除。我们的主要技术贡献是分析带有调制器X的图形G中的禁止未成年人模型与模型的各个连接组件之间的相互作用$G-X$。使用标记的未成年人的语言，我们分析了去除最佳状态后可能保留的潜在禁止的未成年人模型的片段$\mathcal{F}$-从单个连接组件删除解决方案$G-X$。通过限制多项式可能发生的不同类型行为的数量$|X|$，我们使用递归预处理策略获得多项式内核。我们的结果将早期工作扩展到了$\mathcal{F}$-删除，例如“顶点覆盖”和“反馈顶点集”。它还概括了之前的预处理结果$\mathcal{F}$-由顶点覆盖参数化的删除，顶点覆盖是树深一调制器。