Algorithms and Hardness for Geodetic Set on Tree-like Digraphs

2026-03-24

AI Summary

This study addresses the geodetic set problem on directed graphs: determining whether there exists a vertex subset of size $k$ such that every other vertex lies on a shortest directed path between some pair of vertices in the subset. The authors present the first polynomial-time algorithm for ditrees (directed trees). For directed graphs without 2-cycles whose underlying undirected graphs have feedback edge number $\mathrm{fen}$, they devise a parameterized algorithm running in $2^{O(\mathrm{fen})} \cdot n^{O(1)}$ time. Furthermore, they establish that the problem remains NP-hard even on directed acyclic graphs (DAGs) with a constant feedback vertex number, significantly strengthening the known complexity landscape of the problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Geodetic Set
• Directed Graphs
• Tree-like Digraphs
• NP-hardness
• Shortest Paths

Innovation

• Geodetic Set
• ditree
• feedback edge number
• fixed-parameter tractability
• NP-hardness