Dense analogue characterization of bounded merge-width

Prove that, for every graph class, bounded merge-width, bounded flip-width, and membership in the dense analogue of bounded expansion are equivalent.

Background

The paper defines the dense analogue of bounded expansion through first-order transductions from weakly sparse classes. It proves the implication from bounded merge-width to bounded flip-width and cites the implication from bounded flip-width to the dense analogue, but proposes the stronger unresolved implication from the dense analogue back to bounded merge-width.

References

We propose the stronger conjecture, conj1:it3$\rightarrow$conj1:it1.

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

Moreover, is \cref{thm:dense-to-sparse} an instance of a more general dense-to-sparse collapse phenomenon?

— A Dense Weisfeiler-Leman Algorithm for Deciding Bounded-Cliquewidth Homomorphism Indistinguishability  (2608.13382 - Curticapean et al., 13 Aug 2026) in Section 9 (Conclusion)