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

Abstract

In general, a {\sl 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 {\sf vertex removal}, {\sf edge removal}, {\sf edge contraction}, or {\sf 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 \mnb{\sc Graph $\boxtimes$-Modification to Planarity and $\phi$} and prove the following algorithmic meta-theorem: there exists a function $f:\Bbb{N}^{2}\to\Bbb{N}$ such that, for every $\boxtimes$ and every FOL sentence $\phi$, the \mnb{\sc 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 {\em 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 {\sl not} FOL-expressible) and the second is the use of {\em Gaifman's Locality Theorem} that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.