Strong obstruction conjecture for unbounded shrub-depth

Prove or refute that for every graph class C of unbounded shrub-depth, there exists an integer k such that C contains, for every t, either a flipped half-graph H_t or a k-flip of the path P_t as an induced subgraph.

Background

The paper establishes a weaker obstruction theorem using either flipped half-graphs or flipped tP_t graphs, and shows that monadically FO-unstable classes already contain the stronger types of obstructions described here. The conjecture asks whether the same conclusion holds for every class of unbounded shrub-depth, independently of monadic FO-instability. Establishing it would provide finitely describable obstructions and could support the transfer of algorithmic hardness results from paths to hereditary classes of unbounded shrub-depth.

References

While the obstructions we found were sufficient to achieve our goal of characterizing MSO-stability, we initially had stronger obstructions in mind, which we could neither prove nor refute: For every graph class~$C$ of unbounded shrub-depth there exists~$k\in$ such that~$C$ contains as induced subgraphs either a flipped~$H_t$ for every~$t\in$, or a~$k$-flip of~$P_t$ for every~$t \in $.

— Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO  (2501.13903 - Mählmann, 23 Jan 2025) in Section 6, “Outlook: Stronger Obstructions and MSO-Dependence”