Almost bounded merge-width and monadic dependence

Prove that, for every hereditary graph class, almost bounded merge-width, almost bounded flip-width, and monadic dependence are equivalent.

Background

Almost bounded merge-width allows the radius-fr width to grow subpolynomially with the number of vertices for each fixed radius. The paper proves that hereditary almost bounded merge-width classes are almost bounded flip-width and monadically dependent, while conjecturing the converse implications and the resulting full equivalence.

References

We conjecture that monadically dependent classes have almost bounded merge-width.

— Merge-width and First-Order Model Checking  (2502.18065 - Dreier et al., 25 Feb 2025) in Section “Discussion”, paragraph “Model checking and monadic dependence”, Conjecture \(\ref{conj:almostboundedmergewidth}\)