# Check if a pair of vertices belongs to the min-cut

Given a digraph $$G = (V,A)$$ with a source $$s \in V$$ and a sink $$t \in V$$, I need to adapt the graph to known if a pair of vertices $$u \in V$$ and $$v \in V$$ belongs to the min-cut $$S$$ between $$s$$ and $$t$$. That is, a new vertex $$a$$ belongs to the min-cut $$S$$ if and only if $$u \in S$$ and $$v \in S$$. This new vertex must preserve the original max-flow from $$s$$ to $$t$$. I tried creating new arcs $$(u,a)$$ and $$(a,v)$$ with infinity capacities, but while $$a$$ belongs to $$S$$ when $$u \in S$$, this immediately forces $$v$$ to belong to $$S$$ also, which is not necessarily true if the max-flow was computed in the original graph $$G$$. So, is there a way to force a new node to be in the min-cut set $$S$$ if a pair of vertices belongs to $$S$$?