Forcing an edge to be in S-T min-cut

Question

Given a flow-network $N=(G,c,s,t)$ and an edge $e=(u,v)$, I am trying to build an algorithm that finds a minimum $(S,T)$ cut in the given network, that includes e.

So, I tried couple of steps, first, I know I must saturate $e$ so it could be in any min-cut. I tried to add a supersource that connects to $u$ (with infinite capacity) and a supersink that connects to $v$. But that will only saturate $e$ if it is not yet been saturated, not force it to be in a min cut $s,t$. I also thought maybe to reduce an epsilon from the capacity of $e$, but I am not sure if it is enough.

I also tried to connect the supersource to all of u neighbors and the supersink to all of v neighbors and then reduce an epsilon from e. Again I don’t know if that’s enough.

I’m not sure how to continue (or if i can) from here or if I’m missing something.

Answer 1

Remove $u$ and $v$ (as well as all edges connected to them), and for any removed edge $(u,x)$, add an edge from $s$ to $x$ with the same capacity; for any removed edge $(y,v)$, add an edge from $y$ to $t$ with the same capacity. Now find a min cut in this new graph. The partition of nodes in this cut suggests a min cut among those including $e$ in the original graph.