Classification of higher-periodic graphs under the mincut operator

Determine whether graphs other than the Cartesian products $K_n\Box K_2$ are 2-periodic under the mincut operator and whether graphs with period greater than 2 exist.

Background

The paper characterizes 1-periodic graphs, namely fixed graphs, as precisely the super-edge-connected regular graphs, and identifies the Cartesian products Kn□K2K_n\Box K_2 as a class of 2-periodic graphs.

The authors leave unresolved both the existence of additional 2-periodic graph families and the existence of graph orbits with period greater than 2. These questions are prerequisites for a complete classification of periodic behavior under mincut iteration.

References

But are there other graphs that are $2$-periodic? Are there graphs that are have a higher periodicity than $2$?

— Iteration of the mincut graph operator  (2501.15883 - Kriel et al., 27 Jan 2025) in Section 'Periodicity'