In the GEODETIC SET problem, an input is a (di)graph $G$ 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 (directed) 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.

翻译：在测地集问题中，输入为一个（有向）图$G$和整数$k$，目标是判定是否存在大小为$k$的顶点子集$S$，使得$V(G)\setminus S$中任意顶点都位于$S$中两个顶点之间的某条最短（有向）路径上。该问题已在无向图和有向图上从算法与图论两个角度得到研究。我们聚焦于有向图，并证明测地集问题在有向树（即：其底层无向图为树（删除可能存在的平行边后）且可能包含2-圈的有向图）上存在多项式时间算法。这一肯定性结果自然引导我们考察底层无向图"接近树"的情形。为此，我们证明：对于不含2-圈且底层无向图的反馈边集数为$\textsf{fen}$的有向图，可在时间$2^{\mathcal{O}(\textsf{fen})} \cdot n^{\mathcal{O}(1)}$内求解测地集问题，其中$n$为顶点数。作为补充，我们证明该问题在有向无环图（不含2-圈）上仍为NP难问题，即便底层无向图具有常数大小的反馈顶点集。最后这一结果显著强化了Araújo和Arraes [Discrete Applied Mathematics, 2022]的结论——当底层无向图为二部图、共二部图或分裂图时，该问题在有向无环图上为NP难问题。