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A Cubic Vertex-Kernel for Trivially Perfect Editing

Authors Maël Dumas, Anthony Perez, Ioan Todinca

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Maël Dumas
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France
Anthony Perez
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France
Ioan Todinca
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France

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Maël Dumas, Anthony Perez, and Ioan Todinca. A Cubic Vertex-Kernel for Trivially Perfect Editing. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.MFCS.2021.45

Abstract

We consider the Trivially Perfect Editing problem, where one is given an undirected graph G = (V,E) and a parameter k ∈ ℕ and seeks to edit (add or delete) at most k edges from G to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing edge additions or edge deletions, respectively. Trivially perfect graphs are both chordal and cographs, and have applications related to the tree-depth width parameter and to social network analysis. All variants of the problem are known to be NP-complete [Burzyn et al., 2006; James Nastos and Yong Gao, 2013] and to admit so-called polynomial kernels [Pål Grønås Drange and Michał Pilipczuk, 2018; Jiong Guo, 2007]. More precisely, the existence of an O(k³) vertex-kernel for Trivially Perfect Completion was announced by Guo [Jiong Guo, 2007] but without a stand-alone proof. More recently, Drange and Pilipczuk [Pål Grønås Drange and Michał Pilipczuk, 2018] provided O(k⁷) vertex-kernels for these problems and left open the existence of cubic vertex-kernels. In this work, we answer positively to this question for all three variants of the problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized complexity
  • kernelization algorithms
  • graph modification
  • trivially perfect graphs

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References

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