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 $\Pi_1, \ldots, \Pi_d$. We obtain a single-exponential algorithm whenever each class $\Pi_i$ is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.

翻译：重要分离子的经典概念在技术意义上刻画了图中与$S$“最远”的小型$(S,T)$-分离子数量上界。本文引入这一强大算法原语的推广形式，其表述基于$k$-隔离顶点集：即开邻域大小至多为$k$的顶点集。在此术语下，重要分离子的界表明：最多存在$4^k$个包含$S$但与$T$不交的极大$k$-隔离连通顶点集$C$。我们显著推广了这一结论：即使要求$G[C]$避免有限个禁用导出子图$\mathcal{F}$，此类极大子图的数量仍为$2^{O(k)}$，且可被高效枚举。这一结果使我们能够对文献中的两个问题做出重要改进。第一项应用涉及“连通$k$-隔离$\mathcal{F}$-自由子图”问题，其中$\mathcal{F}$为有限个禁用导出子图集合。给定每个顶点具有正整权重的图，该问题要求寻找最大权重的连通$k$-隔离顶点集$C \subseteq V(G)$，使得$G[C]$不含与任意$F \in \mathcal{F}$同构的导出子图。已知基于参数$k$的算法可通过递归理解技术以三指数时间求解，而我们将其改进为单指数时间。第二项应用涉及分散图类的删除问题。该任务要求寻找大小至多为$k$的顶点集，移除后得到的每个连通分量均属于预定图类$\Pi_1, \ldots, \Pi_d$。当每个类$\Pi_i$由有限个禁用导出子图刻画时，我们获得了单指数算法。这一结果推广并改进了文献中的既有成果。