Dual objects for bounded-depth treewidth

Construct or characterize a dual object analogous to brambles for bounded-depth treewidth, so that the resulting duality can support structural or game-theoretic characterizations of graphs admitting tree-decompositions with simultaneously bounded width and depth.

Background

The paper proves monotonicity for a Cops-and-Robber game characterizing graphs with tree-decompositions whose width and depth are simultaneously bounded. For ordinary treewidth, monotonicity proofs can use brambles as a dual object.

The authors note that an analogous dual theory for bounded-depth treewidth is not known and leave open whether such an object exists. Such a duality could provide an alternative structural explanation of the game characterization and clarify the relationship between width and depth constraints.

References

We also leave open whether a dual object similar to brambles can be defined for bounded depth treewidth.

— Going deep and going wide: Counting logic and homomorphism indistinguishability over graphs of bounded treedepth and treewidth  (2505.01193 - Adler et al., 2 May 2025) in Section 7, Section “Conclusion”