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

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

Research output: Contribution to journalArticlepeer-review


In general, a graph modification problem is defined by a graph modification operation and a target graph property . 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 after applying the operation k times on G. This problem has been extensively studied for particular instantiations of and . In this article, we consider the general property of being planar and, additionally, being a model of some First-Order Logic (FOL) sentence (an FOL-sentence). We call the corresponding meta-problem Graph -Modification to Planarity and and prove the following algorithmic meta-theorem: there exists a function f : ĝ.,•2 → ĝ.,• such that, for every and every FOL-sentence , the Graph -Modification to Planarity and is solvable in f(k,||)g 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.

Original languageEnglish (US)
Article number13
JournalACM Transactions on Computation Theory
Issue number3-4
StatePublished - Feb 1 2023


  • algorithmic meta-theorems
  • First-Order Logic
  • Graph modification problems
  • irrelevant vertex technique
  • planar graphs

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics


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