Vertex-minors and monadic dependence

Determine whether unrestricted vertex-minors can be integrated meaningfully into the theory of monadically dependent graph classes.

Background

Vertex-minors are discussed as a possible dense analogue of the minor order. However, forbidding vertex-minors does not imply monadic dependence, as illustrated by circle graphs. The survey therefore records as unresolved whether unrestricted vertex-minor theory has a meaningful relationship with monadic dependence.

References

However, so far it is unclear whether the notion of vertex-minors, without any constraints on the depth, can be combined with the theory of monadically dependent graph classes in any meaningful way.

— Graph classes through the lens of logic  (2501.04166 - Pilipczuk, 7 Jan 2025) in Section 6, paragraph preceding “Fine understanding of the transduction order”