Characterization of k-periodic graphs under the mincut operator

Characterize sufficient structural properties for a graph to be $k$-periodic under the mincut operator for every integer $k\geq 2$.

Background

A graph is defined in the paper to be periodic under an operator when some positive iterate is isomorphic to the original graph. Beyond fixed points and the exhibited 2-periodic Cartesian-product family, the structure of periodic orbits is not developed.

A general set of sufficient conditions for $k$-periodicity would extend the fixed-point characterization and clarify the possible limiting circuits of the mincut operator.