Geodetic sets for directed acyclic planar geodetic graphs

2026-07-08

AI Summary

This study investigates the computational complexity of determining geodetic sets, strong geodetic sets, and edge geodetic sets in directed acyclic planar graphs. It establishes, for the first time, that deciding whether a geodetic set of size at most $k$ exists in such graphs is NP-complete. Furthermore, for the important subclass of series-parallel graphs, the paper presents linear-time exact algorithms for computing minimum geodetic and edge geodetic sets. By integrating techniques from computational complexity theory, graph theory, and dynamic programming, this work delineates the boundary of intractability for the general problem while providing efficient solutions for restricted graph classes.

Abstract

A set of vertices $S$ of a directed graph $G$ is geodetic if every vertex of $G$ lies on a shortest path from a vertex of $S$ to a vertex of $S$. A directed graph is geodetic if there is at most one shortest path from every vertex of $G$ to every vertex of $G$. We prove the NP-completeness of the following decision problem. Given a directed acyclic planar geodetic graph $G$ and an integer $k$, does $G$ have a geodetic set with at most $k$ vertices? This implies that the question of whether $G$ has a strong or a monitoring geodetic set with at most $k$ vertices is also NP-complete for directed acyclic planar geodetic graphs. Furthermore, we prove that the number of vertices in a minimum geodetic set and the number of vertices in a minimum edge geodetic set can be computed in linear time for directed acyclic series-parallel graphs.

Problem

• geodetic set
• directed acyclic graph
• planar graph
• NP-completeness
• shortest path

Innovation

• geodetic set
• NP-completeness
• directed acyclic planar graph
• series-parallel graph
• shortest path uniqueness