Parameterized Complexity of Elimination Distance to First-Order Logic Properties

The elimination distance to some target graph property P is a general modification parameter introduced by Bulian and Dawar. We initiate the study of distances properties expressible in first-order logic. delimit problem’s fixed-parameter tractability identifying sufficient necessary conditions on structure prefixes logic formulas. Our main result following meta-theorem: For every first order-logic formula \( \varphi \in \Sigma _3 \) , that is, form \begin{equation*} =\exists x_1\exists x_2\cdots \exists x_r\ \ \forall y_{1}\forall y_{2}\cdots y_{s}\ z_1\exists z_2\cdots z_t~~ \psi ,\end{equation*} where quantifier-free formula, checking whether does not exceed k tractable parameterized . Properties graphs formulas from include being bounded degree, excluding forbidden subgraph, or containing dominating set. complement this theorem showing such statement hold for with even slightly more expressive prefix structure: There are \Pi which computing {\sf W}[2] -hard.