We are given a flow network and two nodes $u$ and $v$. We want to create an algorithm that tells us whether or not there is a minimum s-t cut so that $u$ belongs to the same side of the cut as the source node $s$ and $v$ belongs to the same side of the cut as the sink node $t$.
My initial idea was to run Ford-Fulkerson to get the residual graph $G_f$ at the end. Then use that to find all nodes reachable from $s$ in $G_f$ (call this set $A$) then $B=V-A$ where $V$ is the vertex set. Then $(A,B)$ is a min s-t cut and if $u \in A, v \in B$ output true, else false.
But since there may be more than one min cut I'm a bit stuck.