First-order model-checking characterization by monadic dependence

Establish that first-order model checking is fixed-parameter tractable exactly on hereditary graph classes that are monadically dependent.

Background

Monadic dependence is a broad model-theoretic notion encompassing the graph classes treated in the paper, including nowhere dense, monadically stable, bounded twin-width, and almost bounded merge-width classes. The paper records the central model-checking conjecture that monadic dependence precisely characterizes hereditary classes admitting fixed-parameter tractable first-order model checking.

References

A central conjecture in algorithmic model theory -- which we call the model checking conjecture here -- states that the model checking problem for first-order logic is fixed-parameter tractable for precisely those hereditary graph classes which are monadically dependent.

— Merge-width and First-Order Model Checking  (2502.18065 - Dreier et al., 25 Feb 2025) in Section “Discussion”, paragraph “Model checking and monadic dependence”