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Maximum Independent Set When Excluding an Induced Minor: K₁ + tK₂ and tC₃ ⊎ C₄

Authors Édouard Bonnet , Julien Duron, Colin Geniet , Stéphan Thomassé, Alexandra Wesolek

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Maximum Independent Set
  • forbidden induced minors
  • quasipolynomial-time algorithms

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Abstract

Dallard, Milanič, and Štorgel [arXiv '22] ask if, for every class excluding a fixed planar graph H as an induced minor, Maximum Independent Set can be solved in polynomial time, and show that this is indeed the case when H is any planar complete bipartite graph, or the 5-vertex clique minus one edge, or minus two disjoint edges. A positive answer would constitute a far-reaching generalization of the state-of-the-art, when we currently do not know if a polynomial-time algorithm exists when H is the 7-vertex path. Relaxing tractability to the existence of a quasipolynomial-time algorithm, we know substantially more. Indeed, quasipolynomial-time algorithms were recently obtained for the t-vertex cycle, C_t [Gartland et al., STOC '21], and the disjoint union of t triangles, tC₃ [Bonamy et al., SODA '23].
We give, for every integer t, a polynomial-time algorithm running in n^O(t⁵) when H is the friendship graph K₁ + tK₂ (t disjoint edges plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm running in n^{O(t² log n) + f(t)}, with f a single-exponential function, when H is tC₃ ⊎ C₄ (the disjoint union of t triangles and a 4-vertex cycle). The former generalizes the algorithm readily obtained from Alekseev’s structural result on graphs excluding tK₂ as an induced subgraph [Alekseev, DAM '07], while the latter extends Bonamy et al.’s result.

Cite As Get BibTex

Édouard Bonnet, Julien Duron, Colin Geniet, Stéphan Thomassé, and Alexandra Wesolek. Maximum Independent Set When Excluding an Induced Minor: K₁ + tK₂ and tC₃ ⊎ C₄. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.23

Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Julien Duron
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Colin Geniet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Stéphan Thomassé
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Alexandra Wesolek
  • Simon Fraser University, Burnaby, Canada

Funding

This work was supported by the ANR projects TWIN-WIDTH (ANR-21-CE48-0014) and Digraphs (ANR-19-CE48-0013).
  • Wesolek, Alexandra: Supported by the Vanier Canada Scholarship Program.

Acknowledgements

We would like to thank Dibyayan Chakraborty for useful discussions, and anonymous reviewers for their helpful comments, and in particular a nice simplification in the proof of Theorem 2.

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