I would like to know how to answer the following problem:
Input: A family of sets $S$ over a universe $U$.
Question: Is there a directed flow network $N$ with edges $U$ such that the set of all $s$-$t$ min-cut edge sets is exactly $S$?
A min-cut edge set is the set of edges crossing a min-cut in $s$-$t$ direction.
I have come up with a partial solution: If there are three sets $A,B,C \in S$ such that the symmetric difference of $A$ and $B$ is a subset of $C$, then the answer to the problem is no. Unfortunately, this does not suffice.