Let $G = (V,E)$ be a flow network. Let $e = (u,v)$ be an edge in $E$ and let $f$ be a maximum flow in $G$. Prove or Disprove:
If the residual graph $G_f$ contains no directed path from $u$ to $v$ then $e$ crosses some minimum cut in $G$.
I might be mistaken, but I think that the statement is true. However, I've failed to prove that such a minimum cut actually exists.
I've also tried to assume by contradiction that $e$ doesn't cross any minimum cut in $G$. It follows that for every cut $(S,T)$ that $e$ crosses, the residual graph $G_f$ contains some edge $e'$ that crosses the cut (and this edge is obviously not saturated). But how does it help us to build a path from $u$ to $v$ in $G_f$ ?