Finding a minimum equivalent graph of a digraph
This problem (or more precisely, its decision version) lies in NP, as a small enough equivalent digraph would give an NP-certificate.
What remains is to show that this problem is NP-hard.

This paper shows that finding a minimum equivalent digraph is NP-hard even when restricted to digraphs with cycles of length at most $5$, via a reduction from SAT. This means that the problem in general is NP-hard and therefore also NP-complete.