Monitoring arc-geodetic sets of oriented graphs

TL;DR

This work introduces monitoring arc-geodetic sets (MAG-sets) as the oriented-graph analogue of monitoring edge-geodetic sets and defines the associated MAG-number $mag(G)$ along with its lower and upper variants $mag^-(G)$ and $mag^+(G)$. It develops foundational results, computes MAG-numbers for fundamental classes (trees, paths, cycles, bipartite graphs), characterizes MAG-extremal graphs, and fully analyzes MAG-sets in tournaments, revealing that $mag$ is either $n-1$ or $n$ for a tournament of order $n$. The paper then establishes two core hardness results: determining whether a graph has $mag^+(G)=|V(G)|$ is NP-hard, and deciding whether an oriented graph has a MAG-set of size at most $k$ is NP-complete even for acyclic graphs of maximum degree 4. Collectively, the results offer structural, algorithmic, and complexity insights into MAG-sets and point to rich directions for future work on orientations, graph operations, and connections to other geodetic-type parameters.

Abstract

Monitoring edge-geodetic sets in a graph are subsets of vertices such that every edge of the graph must lie on all the shortest paths between two vertices of the monitoring set. These objects were introduced in a work by Foucaud, Krishna and Ramasubramony Sulochana with relation to several prior notions in the area of network monitoring like distance edge-monitoring. In this work, we explore the extension of those notions unto oriented graphs, modelling oriented networks, and call these objects monitoring arc-geodetic sets. We also define the lower and upper monitoring arc-geodetic number of an undirected graph as the minimum and maximum of the monitoring arc-geodetic number of all orientations of the graph. We determine the monitoring arc-geodetic number of fundamental graph classes such as bipartite graphs, trees, cycles, etc. Then, we characterize the graphs for which every monitoring arc-geodetic set is the entire set of vertices, and also characterize the solutions for tournaments. We also cover some complexity aspects by studying two algorithmic problems. We show that the problem of determining if an undirected graph has an orientation with the minimal monitoring arc-geodetic set being the entire set of vertices, is NP-hard. We also show that the problem of finding a monitoring arc-geodetic set of size at most $k$ is $NP$-complete when restricted to oriented graphs with maximum degree $4$.

Figures (7)

• Figure 1: The orientations $\overrightarrow{C_n^0}, \overrightarrow{C_n^1}$ and $\overrightarrow{C_n^2}$ of $C_n$.
• Figure 2: The graph $G_j$.
• Figure 3: Some of the different possibilities for the orientation of arcs in $\overrightarrow{G_j}$.
• Figure 4: When $u$ and $v$ are not part of an MAG-set, $\overrightarrow{uv}$ is not monitored.
• Figure 5: Construction of the graph $G(\Phi)$.

Theorems & Definitions (47)

• Definition 1.1
• Definition 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• proof