Given a directed graph $G = (V, E)$ and integer edge capacities $u(e)$, $e \in E$, you wish to find, among all minimum $s$-$t$ cuts, one that contains the smallest number of edges. Show how you can modify the instance $(G, u, s, t)$ to create a new max-flow instance such that solving the maximum flow problem on this new instance will find such a cut.
I understand the basics of finding a minimum $s$-$t$ cut in a directed graph using the max-flow min-cut theorem. However, I'm unsure how to adjust the graph or the capacities to prioritize cuts with fewer edges, while still ensuring that the cut remains minimum in terms of total capacity. I've considered adding a small epsilon value to the capacities to differentiate between edges, but I'm concerned this may not reliably select cuts with fewer edges. I've also thought about transforming each edge in some way or introducing auxiliary nodes/edges, but I'm stuck on the specifics of these transformations.