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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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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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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.

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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

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

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 803421, ReduceSearch).

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