Convergence of the mincut operator to the null graph

Establish whether every simple connected graph iterated under the mincut operator eventually converges to the null graph, apart from a finite number of explicitly characterizable exceptional conditions.

Background

The paper proves that every graph converges under iteration of the mincut operator, meaning that sufficiently many iterations enter a finite periodic orbit. It also identifies several non-null limiting behaviors, including fixed graphs and 2-periodic graphs, while observing that a disconnected mincut graph immediately leads to the null graph on the next iteration.

The authors conjecture that convergence to the null graph is the generic behavior for simple connected graphs and leave open the task of determining the finite collection of conditions under which a non-null limit occurs.