Give an example of a directed graph in which a depth-ﬁrst search backs up from a vertex $v$ before all the vertices that can be reached from $v$ via one or more edges are discovered.
My professor recently asked this question as a warm up to lecture, but never answered it. I still have not figure how that is possible. Why would it return if it's not complete?
I just can't see a scenario where this would happen.
It would never return, since DFS is (essentially) recursive and it can't return without having hit all base cases.