Symbols and Functions > Utility Functions >
AcyclicGraphTake
gives the intersection of the out-component of the first vertex
with the in-component of the second vertex:
In[] := graph = BlockRandom[
DirectedGraph[RandomGraph[{10, 10}], "Acyclic", VertexLabels -> Automatic],
RandomSeeding -> 2
]
In[] := AcyclicGraphTake[graph, {1, 9}]