Monitoring arc-geodetic sets of oriented graphs

TL;DR

This paper extends the concept of monitoring edge-geodetic sets to oriented graphs, introduces the monitoring arc-geodetic number, and explores its properties, complexity, and specific graph classes.

Contribution

It defines monitoring arc-geodetic sets for oriented graphs, characterizes these sets for various graph classes, and analyzes the computational complexity of related problems.

Findings

Monitoring arc-geodetic number determined for bipartite graphs, trees, and cycles.

Characterization of graphs where every monitoring arc-geodetic set is the entire vertex set.

NP-hardness and NP-completeness results for related decision problems.

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 $4$.