Algorithmic construction of unbounded-shrub-depth obstructions

Construct an algorithm that, given an integer $t$ and a graph class $C$ of unbounded shrub-depth, outputs a size-$t$ obstruction contained in $C$, namely either a flipped half-graph $H_t$ or a $k$-flip of $P_t$ for a fixed $k$ associated with $C$.

Background

For monadically FO-unstable classes, the paper observes that periodic flipped paths or flipped half-graphs admit a finite description and can be generated algorithmically. For general classes of unbounded shrub-depth, the authors lack sufficient control over how the layers of the path obstructions are flipped. An algorithmic finite description would facilitate transferring algorithmic hardness from the class of all paths to hereditary classes of unbounded shrub-depth.