Abstract
The \emph{elimination distance} to some target graph property $\mathcal{P}$ is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem's fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: For every graph property $\mathcal{P}$ expressible by a first order-logic formula $\varphi\in \Sigma_3$, that is, of the form \[\varphi=\exists x_1\exists x_2\cdots \exists x_r\ \ \forall y_{1}\forall y_{2}\cdots \forall y_{s}\ \ \exists z_1\exists z_2\cdots \exists z_t~~ \psi,\]where $\psi$ is a quantifier-free first-order formula, checking whether the elimination distance of a graph to $\mathcal{P}$ does not exceed $k$, is \emph{fixed-parameter tractable} parameterized by $k$.Properties of graphs expressible by formulas from $\Sigma_3$ include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: There are formulas $\varphi\in \Pi_3$, for which computing elimination distance is ${\sf W}[2]$-hard.