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An FPT-Algorithm for Recognizing k-Apices of Minor-Closed Graph Classes

Authors Ignasi Sau , Giannos Stamoulis, Dimitrios M. Thilikos

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Author Details

Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France
Giannos Stamoulis
  • Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Greece
  • Inter-university Postgraduate Programme "Algorithms, Logic, and Discrete Mathematics" (ALMA), Athens, Greece
Dimitrios M. Thilikos
  • LIRMM, Université de Montpellier, CNRS, France

Funding

  • Sau, Ignasi: Supported by projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010).
  • Thilikos, Dimitrios M.: Supported by projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010). Supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs, via the Franco-Norwegian project PHC AURORA 2019.

Acknowledgements

We wish to thank the anonymous reviewers for their comments and remarks that improved the presentation of this paper. Moreover, we wish to thank Dániel Marx for his valuable comments and advise.

Cite As Get BibTex

Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. An FPT-Algorithm for Recognizing k-Apices of Minor-Closed Graph Classes. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 95:1-95:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.95

Abstract

Let G be a graph class. We say that a graph G is a k-apex of G if G contains a set S of at most k vertices such that G⧵S belongs to G. We prove that if G is minor-closed, then there is an algorithm that either returns a set S certifying that G is a k-apex of G or reports that such a set does not exist, in 2^{poly(k)}n³ time. Here poly is a polynomial function whose degree depends on the maximum size of a minor-obstruction of G, i.e., the minor-minimal set of graphs not belonging to G. In the special case where G excludes some apex graph as a minor, we give an alternative algorithm running in 2^{poly(k)}n² time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Graph modification problems
  • irrelevant vertex technique
  • graph minors
  • parameterized algorithms

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