Parameterized Complexity of Elimination Distance to First-Order Logic Properties

The 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 Σ3, that is, of the form\begin equation*\varphi = \exists x_1\exists x_2 \cdots \exists x_r\forall y_1\forall y_2 \cdots \forall y_s\quad \exists z_1\exists z_2 \cdots \exists z_t\,\psi ,\end equation*where ψ is a quantifier-free first-order formula, checking whether the elimination distance of a graph to \mathcal P does not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from Σ3 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 Π3, for which computing elimination distance is W[2]-hard.