Finite obstruction algorithms for unbounded shrub-depth

Determine whether every graph class of unbounded shrub-depth admits an algorithm that, given t, outputs a size-t obstruction consisting of either a flipped half-graph H_t or a k-flip of the path P_t, for some fixed k depending on the class.

Background

The paper observes that monadically FO-unstable classes admit algorithmically generable, periodic obstructions: for each requested size t, an algorithm can return either a flipped H_t or a bounded-flip path obstruction.

For arbitrary classes of unbounded shrub-depth, the proof only guarantees the presence of arbitrarily large flipped tP_t or flipped half-graphs and gives insufficient control over how the paths are flipped. A positive answer would provide finite descriptions of obstructions and could help transfer algorithmic hardness results from paths to hereditary classes of unbounded shrub-depth.