Parameterized Complexity of Elimination Distance to First-Order Logic Properties

Abstract

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 \( \varphi \in \Sigma _3 \) , that is, of the form \( \begin{equation*} \varphi =\exists x_1\exists x_2\cdots \exists x_r\ \ \forall y_{1}\forall y_{2}\cdots \forall y_{s}\ \ \exists z_1\exists z_2\cdots \exists z_t~~ \psi ,\end{equation*} \) 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 \( \varphi \in \Pi _3 \) , for which computing elimination distance is \( {\sf W}[2] \) -hard.