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

In general, a graph modification problem is defined by a graph modification operation $boxtimes$ and a target graph property ${cal P}$. Typically, the modification operation $boxtimes$ 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 ${cal P}$ after applying $k$ times the operation $boxtimes$ on $G$. This problem has been extensively studied for particilar instantiations of $boxtimes$ and ${cal P}$. In this paper we consider the general property ${cal P}_{{phi}}$ of being planar and, moreover, being a model of some First-Order Logic sentence ${phi}$ (an FOL-sentence). We call the corresponding meta-problem Graph $boxtimes$-Modification to Planarity and ${phi}$ and prove the following algorithmic meta-theorem: there exists a function $f:Bbb{N}^{2}toBbb{N}$ such that, for every $boxtimes$ and every FOL sentence ${phi}$, the Graph $boxtimes$-Modification to Planarity and ${phi}$ is solvable in $f(k,|{phi}|)cdot n^2$ 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 Gaifmans Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.