Given $ N = (G = (V , E),s, t, c)$ a flow network (assume that the capacity $c$ is always positive) and $e = (u,v) \in E$.
I would like to develop an algorithm that tell if there exist a min-cut (cut with a minimum capacity)that the edge $e$ cross him.
I was thinking about running the Ford–Fulkerson algorithm to find the max flow and it is actually the capacity of the minimum cut according to Max-flow min-cut theorem and then I go over the Residual Network that was created and find the cut base on the nodes that can be reached from $s$ and then find if $e$ cross the cut or not.But it is only one minimum cut. there can be more minimum cut with the same capacity