Parameterized Complexity of Elimination Distance to First-Order Logic Properties

The elimination distance to some target graph property 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 P expressible by a first order-logic formula \phi\in \Sigma_3, that is, of the form \phi=\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 P does not exceed k, is 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 \phi\in \Pi_3, for which computing elimination distance is W[2]-hard.