枚举隔离的 $\mathcal{F}$-无关于图与删除至分散图类的单指数 FPT 算法
Single-Exponential FPT Algorithms for Enumerating Secluded $\mathcal{F}$-Free Subgraphs and Deleting to Scattered Graph Classes
AI总结：
本文推广重要分隔符概念至 $k$-隔离顶点集，证明在有限禁用诱导子图约束下极大子图数量为 $2^{O(k)}$ 且可高效枚举，并将连通 $k$-隔离 $\mathcal{F}$-无关于图问题和删除至分散图类问题从三重指数时间改进为单指数时间。
摘要：
著名的重要分隔符概念在技术意义上限制了图中“离 $S$ 最远”的小型 $(S,T)$-分隔符的数量。本文引入这一强大算法原语的推广，其表述基于 $k$-隔离顶点集：即开放邻域大小至多为 $k$ 的集合。在此术语下，关于重要分隔符的界表明，至多有 $4^k$ 个极大的 $k$-隔离连通顶点集 $C$，它们包含 $S$ 且与 $T$ 不相交。我们显著推广了这一陈述：即使要求 $G[C]$ 避开一个有限的禁用诱导子图集合 $\mathcal{F}$，此类极大子图的数量仍为 $2^{O(k)}$，并且可以被高效枚举。这使我们能够对文献中的两个问题做出显著改进。我们的第一个应用涉及“连通 $k$-隔离 $\mathcal{F}$-无关于图”问题，其中 $\mathcal{F}$ 是一个有限的禁用诱导子图集合。给定一个每个顶点具有正整数权重的图，该问题要求找到一个最大权重的连通 $k$-隔离顶点集 $C \subseteq V(G)$，使得 $G[C]$ 不包含与任何 $F \in \mathcal{F}$ 同构的诱导子图。以 $k$ 为参数的问题已知可通过递归理解技术在三重指数时间内求解，我们将其改进为单指数时间。我们的第二个应用涉及删除至分散图类的问题。这里，任务是找到一个大小至多为 $k$ 的顶点集，其删除后所得图的每个连通分量都属于指定的图类 $\Pi_1, \ldots, \Pi_d$ 之一。当每个类 $\Pi_i$ 由有限数量的禁用诱导子图刻画时，我们得到了一个单指数算法。这推广并改进了文献中的早期结果。
英文摘要：
The celebrated notion of important separators bounds the number of small $(S,T)$-separators in a graph which are 'farthest from $S$' in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive that is phrased in terms of $k$-secluded vertex sets: sets with an open neighborhood of size at most $k$. In this terminology, the bound on important separators says that there are at most $4^k$ maximal $k$-secluded connected vertex sets $C$ containing $S$ but disjoint from $T$. We generalize this statement significantly: even when we demand that $G[C]$ avoids a finite set $\mathcal{F}$ of forbidden induced subgraphs, the number of such maximal subgraphs is $2^{O(k)}$ and they can be enumerated efficiently. This allows us to make significant improvements for two problems from the literature. Our first application concerns the 'Connected $k$-Secluded $\mathcal{F}$-free subgraph' problem, where $\mathcal{F}$ is a finite set of forbidden induced subgraphs. Given a graph in which each vertex has a positive integer weight, the problem asks to find a maximum-weight connected $k$-secluded vertex set $C \subseteq V(G)$ such that $G[C]$ does not contain an induced subgraph isomorphic to any $F \in \mathcal{F}$. The parameterization by $k$ is known to be solvable in triple-exponential time via the technique of recursive understanding, which we improve to single-exponential. Our second application concerns the deletion problem to scattered graph classes. Here, the task is to find a vertex set of size at most $k$ whose removal yields a graph whose each connected component belongs to one of the prescribed graph classes $Π_1, \ldots, Π_d$. We obtain a single-exponential algorithm whenever each class $Π_i$ is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.
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。
