On Monitoring Edge-Geodetic Sets of Dynamic Graph

Abstract

The concept of a monitoring edge-geodetic set (MEG-set) in a graph $G$, denoted $MEG(G)$, refers to a subset of vertices $MEG(G)\subseteq V(G)$ such that every edge $e$ in $G$ is monitored by some pair of vertices $ u, v \in MEG(G)$, where $e$ lies on all shortest paths between $u$ and $v$. The minimum number of vertices required to form such a set is called the monitoring edge-geodetic number, denoted $meg(G)$. The primary motivation for studying $MEG$-sets in previous works arises from scenarios in which certain edges are removed from $G$. In these cases, the vertices of the $MEG$-set are responsible for detecting these deletions. Such detection is crucial for identifying which edges have been removed from $G$ and need to be repaired. In real life, repairing these edges may be costly, or sometimes it is impossible to repair edges. In this case, the original $MEG$-set may no longer be effective in monitoring the modified graph. This highlights the importance of reassessing and adapting the $MEG$-set after edge deletions. This work investigates the monitoring edge-geodetic properties of graphs, focusing on how the removal of $k$ edges affects the structure of a graph and influences its monitoring capabilities. Specifically, we explore how the monitoring edge-geodetic number $meg(G)$ changes when $k$ edges are removed. The study aims to compare the monitoring properties of the original graph with those of the modified graph and to understand the impact of edge deletions.