Let $v$ be a vertex with in-degree 0 in an (acyclic) DAG $G$, and let $F$ be a subset of $G$'s vertices (the "forbidden") vertices. Now suppose $U$ is a set of vertices such that every path from $v$ to $F$ intersects $U$.
Is there a commonly-used term for such sets? I was thinking about using the term "Front", but a (cursory and lazy) search tells me that's not a popular term. I can't use "cut", because that's reserved for sets of edges with the similar property (cutting off $v$ from $F$; usually $F$ is the subgraph of all vertices reachable from $A$ but not in $A$).
Also, suppose that $U$ is minimal, in the sense that for every $u$ in $U$, there's a path from $v$ to $F$ which intersects no other vertex of $U$. Is there a commony-used term for this kind of sets?
If it helps, I'm willing to refer to $v$ as a "source" and to $F$ as the set of "sinks".