Kernelization Dichotomies for Hitting Subgraphs Under Structural Parameterizations

Authors Marin Bougeret , Bart M. P. Jansen , Ignasi Sau



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Marin Bougeret
  • LIRMM, Université de Montpellier, CNRS, France
Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France

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Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Kernelization Dichotomies for Hitting Subgraphs Under Structural Parameterizations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.33

Abstract

For a fixed graph H, the H-Subgraph Hitting problem consists in deleting the minimum number of vertices from an input graph to obtain a graph without any occurrence of H as a subgraph. This problem can be seen as a generalization of Vertex Cover, which corresponds to the case H = K₂. We initiate a study of H-Subgraph Hitting from the point of view of characterizing structural parameterizations that allow for polynomial kernels, within the recently active framework of taking as the parameter the number of vertex deletions to obtain a graph in a "simple" class 𝒞. Our main contribution is to identify graph parameters that, when H-Subgraph Hitting is parameterized by the vertex-deletion distance to a class 𝒞 where any of these parameters is bounded, and assuming standard complexity assumptions and that H is biconnected, allow us to prove the following sharp dichotomy: the problem admits a polynomial kernel if and only if H is a clique. These new graph parameters are inspired by the notion of 𝒞-elimination distance introduced by Bulian and Dawar [Algorithmica 2016], and generalize it in two directions. Our results also apply to the version of the problem where one wants to hit H as an induced subgraph, and imply in particular, that the problems of hitting minors and hitting (induced) subgraphs have a substantially different behavior with respect to the existence of polynomial kernels under structural parameterizations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • hitting subgraphs
  • hitting induced subgraphs
  • parameterized complexity
  • polynomial kernel
  • complexity dichotomy
  • elimination distance

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