An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL
Authors: 
Fedor V. Fomin, Petr A. Golovach, Giannos Stamoulis, Dimitrios M. ThilikosAuthors Info & Claims
Fedor V. Fomin, Petr A. Golovach, Giannos Stamoulis, Dimitrios M. ThilikosAuthors Info & ClaimsArticle No.: 13, Pages 1 - 29
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 deletion, edge deletion, 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 the operation ⊠ k times on G. This problem has been extensively studied for particular instantiations of ⊠ and 𝒫. In this article, we consider the general property 𝒫𝛗 of being planar and, additionally, being a model of some First-Order Logic (FOL) 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.
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Index Terms
- An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL
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December 2022
122 pages
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Publication History
Published: 01 February 2023
Online AM: 23 November 2022
Accepted: 10 November 2022
Revised: 04 November 2022
Received: 07 June 2021
Published in TOCT Volume 14, Issue 3-4
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- Research Council of Norway via the project BWCA
- ANR projects DEMOGRAPH
- ESIGMA
- French-German Collaboration ANR/DFG project UTMA
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