# Model-Checking for Successor-Invariant First-Order Formulas on Graph Classes of Bounded Expansion

## TL;DR

This paper proves that model-checking successor-invariant first-order formulas is fixed-parameter tractable on classes of graphs with bounded expansion, extending previous results and approaching the best known results for nowhere dense classes.

## Contribution

It generalizes earlier fixed-parameter tractability results for successor-invariant formulas to all classes of graphs with bounded expansion.

## Key findings

- Model-checking successor-invariant formulas is fixed-parameter tractable on bounded expansion classes.
- Extends previous results from planar and minor-free graphs to broader classes.
- Approaches the tractability results known for nowhere dense graph classes.

## Abstract

A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula. It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [Engelmann et al., 2012], excludes a fixed minor [Eickmeyer et al., 2013] or a fixed topological minor [Eickmeyer and Kawarabayashi, 2016; Kreutzer et al., 2016]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic.