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An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL

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

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Author Details

Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
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, Univ. Montpellier, CNRS, Montpellier, France

Funding

The research has been supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs via the Franco-Norwegian project PHC AURORA 2019.
  • Fomin, Fedor V.: Research Council of Norway via the project "MULTIVAL".
  • Golovach, Petr A.: Research Council of Norway via the project "MULTIVAL".
  • Thilikos, Dimitrios M.: DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010).

Cite AsGet BibTex

Fedor V. Fomin, Petr A. Golovach, Giannos Stamoulis, and Dimitrios M. Thilikos. An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.51

Abstract

In general, a graph modification problem is defined by a graph modification operation ⊠ and a target graph property 𝒫. Typically, the modification operation ⊠ may be vertex removal, edge removal, edge contraction, or edge addition and the question is, given a graph G and an integer k, whether it is possible to transform G to a graph in 𝒫 after applying k times the operation ⊠ on G. This problem has been extensively studied for particilar instantiations of ⊠ and 𝒫. In this paper we consider the general property 𝒫_ϕ of being planar and, moreover, being a model of some First-Order Logic sentence ϕ (an FOL-sentence). We call the corresponding meta-problem Graph ⊠-Modification to Planarity and ϕ and prove the following algorithmic meta-theorem: there exists a function f: ℕ² → ℕ such that, for every ⊠ and every FOL sentence ϕ, the Graph ⊠-Modification to Planarity and ϕ is solvable in f(k,|ϕ|)⋅n² time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman’s Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Graph modification Problems
  • Algorithmic meta-theorems
  • First Order Logic
  • Irrelevant vertex technique
  • Planar graphs
  • Surface embeddable graphs

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