A monitoring edge-geodetic set, or simply an MEG-set, of a graph $G$ is a vertex subset $M \subseteq V(G)$ such that given any edge $e$ of $G$, $e$ lies on every shortest $u$-$v$ path of $G$, for some $u,v \in M$. The monitoring edge-geodetic number of $G$, denoted by $meg(G)$, is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem. In this article, we compare $meg(G)$ with some other graph theoretic parameters stemming from the network monitoring problem and provide examples of graphs having prescribed values for each of these parameters. We also characterize graphs $G$ that have $V(G)$ as their minimum MEG-set, which settles an open problem due to Foucaud \textit{et al.} (CALDAM 2023), and prove that some classes of graphs fall within this characterization. We also provide a general upper bound for $meg(G)$ for sparse graphs in terms of their girth, and later refine the upper bound using the chromatic number of $G$. We examine the change in $meg(G)$ with respect to two fundamental graph operations: clique-sum and subdivisions. In both cases, we provide a lower and an upper bound of the possible amount of changes and provide (almost) tight examples.

翻译：监控边测地集（简称MEG集）是图$G$的一个顶点子集$M \subseteq V(G)$，满足对$G$中任意边$e$，存在$u,v \in M$使得$e$位于$G$的每条最短$u$-$v$路径上。图$G$的监控边测地数记为$meg(G)$，即此类MEG集的最小基数。该概念为网络监控问题提供了图论模型。本文比较$meg(G)$与源于网络监控问题的其他图论参数，并给出各参数取预设值的图例。我们刻画了以$V(G)$为最小MEG集的图$G$，解决了Foucaud等人（CALDAM 2023）提出的开放问题，并证明若干图类属于此刻画范围。针对稀疏图，我们基于其围长给出了$meg(G)$的通用上界，随后利用色数进一步优化该上界。我们考察了$meg(G)$在图团和与细分两种基本运算下的变化情况，在这两种情况下均给出了可能变化量的上下界，并提供了（近乎）紧的实例。