The monadic stability characterization conjecture for hereditary graph classes

A graph class is a collection of finite graphs, and a first-order transduction is a graph transformation definable in first-order logic with parameters. A class is monadically stable if every class obtained from it by adding arbitrary unary predicates is stable, and a class is nowhere dense if for every integer $r$ there is a bound excluding arbitrarily large complete graphs as depth-$r$ minors.

Monadic stability characterization conjecture. A class of graphs is monadically stable if and only if it is a first-order transduction of a nowhere dense class of graphs.

This conjecture proposes a structural characterization of monadic stability through sparse graph classes and is presented as a central question concerning first-order transductions of nowhere dense classes. Its resolution status is not specified in the source.