# Cut In a Flow Network

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

Add the edges $$su$$ and $$vt$$ to the network with infinite capacity to obtain the modified network $$N'$$. It is not difficult to see that the finite ($$st$$-)cuts in $$N'$$ are precisely those finite cuts in $$N$$ that contain the edge $$e$$. This means that a min-cut in $$N'$$ is a min-cut in $$N$$ among those cuts that contain $$e$$. So you just have to check whether the min-cut in $$N'$$ has the same capacity as the min-cut in $$N$$, which can be done using the Ford-Fulkerson (or, to be more precise, the Edmonds-Karp) algorithm.