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

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.