## Abstract

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[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 enumeration algorithm 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. A scattered graph class is defined by demanding that every connected component is contained in at least one of the prescribed graph classes Π1, . . ., Π_{d}. The deletion problem to a scattered graph class is to find a vertex set of size at most k whose removal yields a graph from the class. We obtain a single-exponential algorithm whenever each class Πi is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.

Original language | English |
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Title of host publication | 34th International Symposium on Algorithms and Computation, ISAAC 2023 |

Editors | Satoru Iwata, Satoru Iwata, Naonori Kakimura |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772891 |

DOIs | |

State | Published - 1 Dec 2023 |

Externally published | Yes |

Event | 34th International Symposium on Algorithms and Computation, ISAAC 2023 - Kyoto, Japan Duration: 3 Dec 2023 → 6 Dec 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 283 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 34th International Symposium on Algorithms and Computation, ISAAC 2023 |
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Country/Territory | Japan |

City | Kyoto |

Period | 3/12/23 → 6/12/23 |

## Keywords

- fixed-parameter tractability
- important separators
- secluded subgraphs

## ASJC Scopus subject areas

- Software