On first-order model checking parameterized by the number of variables

TL;DR

This paper investigates the fixed-parameter tractability of first-order model checking on graph classes when parameterized by the number of variables, providing characterizations in monotone and hereditary cases.

Contribution

It characterizes graph classes where FO model checking is FPT when parameterized by variables, extending known results to broader classes.

Findings

Characterizes classes with FPT model checking in the monotone setting.

Proves a weaker characterization in the hereditary setting.

Addresses complexity under variable-based parameterization.

Abstract

The first-order (FO) model checking problem asks, given an FO sentence $\phi$ and a graph $G$, whether $G$ is a model of $\phi$. This problem is known to be $\mathsf{AW[*]}$-hard when parameterized by the quantifier rank of the formula. A classical algorithm decides this problem in XP-time parameterized by the number of variables in the formula. Due to $\mathsf{AW[*]}$-hardness, it is natural to ask about the complexity of the problem when restricted to some well-behaved class of graphs. There are many results describing graph classes $\mathcal{C}$ such that the FO model checking problem restricted to $\mathcal{C}$ admits an $\mathsf{FPT}$-time algorithm when parameterized by the quantifier rank of the formula. Parameterization by the quantifier rank is significantly more restrictive than parameterization by the number of variables. We investigate the graph classes $\mathcal{C}$ for which the FO model checking problem restricted to $\mathcal{C}$ admits an $\mathsf{FPT}$-time algorithm when parameterized by the number of variables in the formula. We characterize these classes in the monotone setting, and prove a slightly weaker result in the hereditary setting.