MSO-dependence and clique-width characterization

Prove or refute that, for every hereditary graph class $C$, bounded clique-width, MSO-dependence, monadic MSO-dependence, CMSO-dependence, and monadic CMSO-dependence are equivalent properties.

Background

Dependence, or NIP, is a model-theoretic dividing line generalizing stability. The paper notes that FO-dependent classes have recently received combinatorial characterizations and asks whether an analogous characterization exists for MSO-dependence. The proposed conjecture identifies bounded clique-width (equivalently, bounded rank-width) with several MSO- and CMSO-dependence notions. The equivalence between bounded clique-width and CMSO-dependence is already known through CMSO-transductions, but the remaining equivalences would have significant consequences, including implications for transducing all graphs and Seese’s conjecture on undecidability.

References

A second interesting question concerns the model theoretic notion of dependence (see \cref{sec:mstable} for a definition). Similar to FO-stable classes, also FO-dependent classes have recently been shown to admit nice combinatorial characterizations~. This raises the question whether also MSO-dependence can be combinatorially characterized. It is natural to conjecture the following.

— 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”

It is natural to conjecture the following.

For every hereditary graph class~$C$, the following are equivalent. \begin{enumerate} \item $C$ has bounded clique-width (or equivalently bounded rank-width). \item $C$ is MSO-dependent. \item $C$ is monadically MSO-dependent. \item $C$ is CMSO-dependent. \item $C$ is monadically CMSO-dependent. \end{enumerate}

— Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO  (2501.13903 - Mählmann, 23 Jan 2025) in Conjecture 7.1, Section 7, “Outlook: Stronger Obstructions and MSO-Dependence”