Given an $H$-minor-free graph $G$ and an integer $k$, our main technical contribution is sampling in randomized polynomial time an induced subgraph $G'$ of $G$ and a tree decomposition of $G'$ of width $\widetilde{O}(k)$ such that for every $Z\subseteq V(G)$ of size $k$, with probability at least $\left(2^{\widetilde{O}(\sqrt{k})}|V(G)|^{O(1)}\right)^{-1}$, we have $Z \subseteq V(G')$ and every bag of the tree decomposition contains at most $\widetilde{O}(\sqrt{k})$ vertices of $Z$. Having such a tree decomposition allows us to solve a wide range of problems in (randomized) time $2^{\widetilde{O}(\sqrt{k})}n^{O(1)}$ where the solution is a pattern $Z$ of size $k$, e.g., Directed $k$-Path, $H$-Packing, etc. In particular, our result recovers all the algorithmic applications of the pattern-covering result of Fomin et al. [SIAM J. Computing 2022] (which requires the pattern to be connected) and the planar subgraph-finding algorithms of Nederlof [STOC 2020]. Furthermore, for $K_{h,3}$-free graphs (which include bounded-genus graphs) and for a fixed constant $d$, we signficantly strengthen the result by ensuring that not only $Z$ has intersection $\widetilde{O}(\sqrt{k})$ with each bag, but even the distance-$d$ neighborhood $N^d_{G}[Z]$ as well. This extension makes it possible to handle a wider range of problems where the neighborhood of the pattern also plays a role in the solution, such as partial domination problems and problems involving distance constraints.

翻译：给定一个不含$H$作为子式缩并的图$G$和一个整数$k$，我们的主要技术贡献在于：在随机多项式时间内采样得到一个诱导子图$G'\subseteq G$及其宽度为$\widetilde{O}(k)$的树分解，使得对于每个大小为$k$的子集$Z\subseteq V(G)$，以至少$\left(2^{\widetilde{O}(\sqrt{k})}|V(G)|^{O(1)}\right)^{-1}$的概率，$Z\subseteq V(G')$且该树分解的每个袋中至多包含$\widetilde{O}(\sqrt{k})$个$Z$中的顶点。拥有这样的树分解使我们能够在$2^{\widetilde{O}(\sqrt{k})}n^{O(1)}$的（随机化）时间内解决广泛的问题，其中解是一个大小为$k$的模式$Z$，例如有向$k$路径问题、$H$打包问题等。特别地，我们的结果涵盖了Fomin等人[SIAM J. Computing 2022]的模式覆盖结果（该结果要求模式为连通结构）以及Nederlof [STOC 2020]的平面子图查找算法的所有算法应用。此外，对于$K_{h,3}$-free图（包括有界亏格图）和固定常数$d$，我们显著加强了该结果：不仅确保$Z$与每个袋的交集大小为$\widetilde{O}(\sqrt{k})$，甚至其距离-$d$邻域$N^d_{G}[Z]$也具有相同性质。这一扩展使得处理更广泛的、模式邻域在解中起作用的问 题成为可能，例如部分支配问题以及涉及距离约束的问题。