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