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 and constant pathwidth. 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$k$，目标是判断是否存在大小为$k$的顶点子集$S$，使得$V(G)\setminus S$中的任意顶点都位于$S$中两个顶点之间的某条最短（有向）路径上。该问题从算法和图论两个角度在无向图和有向图上均已有研究。我们专注于有向图，并证明测地集问题在有向树（即底层无向图为树且可能包含2-环的有向图，在删除可能的平行边后）上存在多项式时间算法。这一正向结果自然引导我们研究底层无向图“接近树”的情形。为此，我们证明：对于不含2-环、且底层无向图的反馈边集数为$\textsf{fen}$的有向图，测地集问题可在$2^{\mathcal{O}(\textsf{fen})} \cdot n^{\mathcal{O}(1)}$时间内求解，其中$n$为顶点数。作为补充，我们证明：即使底层无向图具有常数反馈顶点集数和常数路径宽度，该问题在有向无环图（不含2-环）上仍是NP-难的。我们的最后一个结果显著加强了Araújo和Arraes [Discrete Applied Mathematics, 2022]的结论，即当底层无向图为二分图、共二分图或分裂图时，该问题在有向无环图上也是NP-难的。