On the Parameterized Complexity of Graph Modification to First-Order Logic Properties

TL;DR

This paper investigates the computational complexity of modifying graphs through vertex and edge edits to satisfy properties expressible in first-order logic, identifying conditions for fixed-parameter tractability and kernelization.

Contribution

It provides a comprehensive analysis of the parameterized complexity of graph modification problems for first-order logic properties, establishing new tractability and kernelization results.

Findings

Characterizes when graph modification problems are fixed-parameter tractable.

Identifies conditions for polynomial kernel existence.

Provides complexity classifications based on the quantification pattern of the logic.

Abstract

We consider the problems of deciding whether an input graph can be modified by removing/adding at most k vertices/edges such that the result of the modification satisfies some property definable in first-order logic. We establish a number of sufficient and necessary conditions on the quantification pattern of the first-order formula \phi for the problem to be fixed-parameter tractable or to admit a polynomial kernel.