# Flow value over edges in Max flow/Min Cut Ford Fulkerson

Is it true that if a given edge e is in the min cut of a graph that there exists a max flow of the graph that has e with its full capacity?

• A cut is usually a collection of edges. Are you strictly interested in the restricted case where the cut is a single edge? – dkaeae Jan 1 at 21:57

Yes. More precisely, for every min cut $$(W,\bar{W})$$, there is some max flow $$f$$ such that $$f(x,y)=0$$ if edge $$(x,y)$$ goes from $$\bar{W}$$ to $$W$$, and $$f(x,y)$$ equals the capacity of $$(x,y)$$ if edge $$(x,y)$$ goes from $$W$$ to $$\bar{W}$$. This is usually proven using the linear programming formulations of min cut and max flow, and follows by the complementary slackness property.