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

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.

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.