FO model-checking on monadically dependent classes

Prove that First-Order model-checking is fixed-parameter tractable on every monadically dependent graph class.

Background

Monadically dependent classes are defined by forbidding a fixed FO transduction from producing all graphs. Nowhere dense classes form a sparse subclass of monadically dependent classes, and fixed-parameter tractability is known for nowhere dense and monadically stable classes. The survey identifies the extension to all monadically dependent classes as its principal algorithmic conjecture.