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Modification to Planarity is Fixed Parameter Tractable

Authors Fedor V. Fomin , Petr A. Golovach , Dimitrios M. Thilikos

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Modification problems
  • Planar graphs
  • Irrelevant vertex technique

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Abstract

A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G|^{2}) steps. We also present several applications of our approach to related problems.

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Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Modification to Planarity is Fixed Parameter Tractable. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.STACS.2019.28

Author Details

Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
Dimitrios M. Thilikos
  • AlGCo project-team, LIRMM, Université de Montpellier, CNRS, Montpellier, France

Funding

  • Fomin, Fedor V.: Supported by the Research Council of Norway via the projects "CLASSIS" and "MULTIVAL". Supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs, via the Franco-Norwegian project PHC AURORA 2019.
  • Golovach, Petr A.: Supported by the Research Council of Norway via the projects "CLASSIS" and "MULTIVAL". Supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs, via the Franco-Norwegian project PHC AURORA 2019.
  • 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.

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