Cited By
View all- Schirrmacher NSiebertz SVigny A(2023)First-order Logic with Connectivity OperatorsACM Transactions on Computational Logic10.1145/359592224:4(1-23)Online publication date: 25-Jul-2023
Parameterized Complexity of Elimination Distance to First-Order Logic Properties
Article No.: 17, Pages 1 - 35
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 formwhere \(\psi\) 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.
\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*}
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Publication History
Published: 06 April 2022
Online AM: 29 March 2022
Accepted: 01 February 2022
Revised: 01 October 2021
Received: 01 April 2021
Published in TOCL Volume 23, Issue 3
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- Research Council of Norway via the project BWCA
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- French-German Collaboration ANR/DFG
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View all- Schirrmacher NSiebertz SVigny A(2023)First-order Logic with Connectivity OperatorsACM Transactions on Computational Logic10.1145/359592224:4(1-23)Online publication date: 25-Jul-2023
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