# Single-Exponential FPT Algorithms for Enumerating Secluded ℱ-Free Subgraphs and Deleting to Scattered Graph Classes

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 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 ℱ-free subgraph' problem, where 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 ⊆ V(G) such that G[C] does not contain an induced subgraph isomorphic to any F ∈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, …, 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.

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