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, tailored to undirected graphs, 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 $G\left[C\right]$ avoids a finite set $F$ of forbidden induced subgraphs, the number of such maximal subgraphs is $2^{O\left(k\right)}$ and they can be enumerated efficiently. This enumeration algorithm allows us to give improved parameterized algorithms for Connected k-Secluded $F$-Free Subgraph and for deleting into scattered graph classes.