Algorithmic generation of unbounded-shrub-depth obstructions

Determine whether graph classes of unbounded shrub-depth admit an algorithm that, given t, outputs a size-t obstruction—either a flipped half-graph H_t or a k-flip of P_t—contained in the class, for some class-dependent constant k.

Background

For every monadically FO-unstable class, the paper notes that periodic obstructions yield an algorithm producing a size-t obstruction for each input t. For general classes of unbounded shrub-depth, the authors lack control over how the tP_t graphs are flipped. An affirmative answer would provide a finite description of obstructions and could help transfer the non-elementary parameter dependence lower bound for MSO model checking on paths to hereditary classes of unbounded shrub-depth.