Controlled-flip obstructions for all unbounded-shrub-depth classes

Establish whether every graph class $C$ of unbounded shrub-depth contains, for some fixed $k$, either a flipped half-graph of every order or a $k$-flip of a path $P_t$ for every $t$ as induced subgraphs.

Background

The main theorem obtains a forbidden-induced-subgraph characterization using arbitrarily large flipped half-graphs and flipped disjoint unions of paths. The conjecture seeks substantially more controlled obstructions: a single flipped half-graph family or bounded-complexity flips of individual paths. The paper notes that this stronger assertion is known for monadically FO-unstable classes, including periodic flips, but is unresolved for arbitrary classes of unbounded shrub-depth. A positive answer could support algorithmic reductions and help transfer 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 Conjecture, Section 8 “Outlook: Stronger Obstructions and MSO-Dependence”