Parameterized Complexity of Elimination Distance to First-Order Logic Properties

Abstract

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 φ ∈ 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 φ ∈ Π3, for which computing elimination distance is W[2]-hard.