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

Authors Bart M. P. Jansen , Jari J. H. de Kroon , Michał Włodarczyk



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Author Details

Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Jari J. H. de Kroon
  • Eindhoven University of Technology, The Netherlands
Michał Włodarczyk
  • University of Warsaw, Poland

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Bart M. P. Jansen, Jari J. H. de Kroon, and Michał Włodarczyk. Single-Exponential FPT Algorithms for Enumerating Secluded ℱ-Free Subgraphs and Deleting to Scattered Graph Classes. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 42:1-42:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.42

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 Π_1, …, 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 Π_i is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • fixed-parameter tractability
  • important separators
  • secluded subgraphs

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