How to verify the unique min-cut

How to verify that there is only one min-cut in a graph G(V, E), which have one sink, t, and a source, s. The first idea is as followed:

• run the max-flow algorithm, such as Ford-Fulkerson or Edmonds-Karp, and keep the residual graph as G`.
• run DFS, from s and t, on G` and find out a reachable set R₁ and R₂.
• if and only if R₁ + R₂ = E, then return true, otherwise, return false.

This method uses a constraint of min-cut that all the edges through a min-cut in a max-flow graph have 0 capacity. So, if there are two min-cut, there must be another “bottleneck” that does not allow DFS to reach the node.
For a directed graph G(V, E). We could do the following to verify the unique or not.

• run the max-flow algorithm;
• for each edge e though the min cut, increase the capacity;
• if the max flow does not change, the min cut is not unique; otherwise, it is unique.