In the Geodetic Set problem, an input is a digraph \(G\) and integer \(k\) , and the objective is to decide whether there exists a vertex subset \(S\) of size \(k\) such that any vertex in \(V(G)\setminus S\) lies on a shortest path between two vertices in \(S\) . The problem has been studied on undirected and directed graphs from both algorithmic and graph-theoretical perspectives. We focus on directed graphs and prove that Geodetic Set admits a polynomial-time algorithm on ditrees, that is, digraphs with possible \(2\) -cycles when the underlying undirected graph is a tree (after deleting possible parallel edges). This positive result naturally leads us to investigate cases where the underlying undirected graph is ‘close to a tree’. Towards this, we show that Geodetic Set on digraphs without \(2\) -cycles and whose underlying undirected graph has feedback edge set number \(\textsf {fen} \) , can be solved in time \(2^{\mathcal {O}(\textsf {fen})} \cdot n^{\mathcal {O}(1)}\) , where \(n\) is the number of vertices. To complement this, we prove that the problem remains NP-hard on DAGs (which do not contain \(2\) -cycles) even when the underlying undirected graph has constant feedback vertex set number. Our last result significantly strengthens the result of Araújo and Arraes [Discrete Applied Mathematics, 2022] that the problem is NP-hard on DAGs when the underlying undirected graph is either bipartite, cobipartite or split.