Geodetic Set can be solved in polynomial time on ditrees and in FPT time parameterized by feedback edge set on 2-cycle-free digraphs, but is NP-hard on DAGs with constant feedback vertex set and pathwidth.
Discrete Applied Mathematics148(3), 256–262 (2005) 24
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Algorithms and Hardness for Geodetic Set on Tree-like Digraphs
Geodetic Set can be solved in polynomial time on ditrees and in FPT time parameterized by feedback edge set on 2-cycle-free digraphs, but is NP-hard on DAGs with constant feedback vertex set and pathwidth.