A monitoring edge-geodetic set (or meg-set for short) of a graph is a set of vertices $M$ such that if any edge is removed, then the distance between some two vertices of $M$ increases. This notion was introduced by Foucaud et al. in 2023 as a way to monitor networks for communication failures. As computing a minimum meg-set is hard in general, recent works aimed to find polynomial-time algorithms to compute minimum meg-sets when the input belongs to a restricted class of graphs. Most of these results are based on the property of some classes of graphs to admit a unique minimal meg-set, which is then easy to compute. In this work, we prove that chordal graphs also admit a unique minimal meg-set, answering a standing open question of Foucaud et al.

翻译：图的监测边测地集（简称meg-集）是指顶点集$M$，使得任意删除一条边后，$M$中某两个顶点之间的距离必然增加。该概念由Foucaud等人于2023年提出，用于监测网络通信故障。由于计算最小meg-集在一般情况下较为困难，近期研究致力于寻找当输入图属于特定受限图类时计算最小meg-集的多项式时间算法。这些结果大多基于某些图类具有唯一最小meg-集的特性，从而易于计算。本文证明了弦图也具有唯一最小meg-集，解答了Foucaud等人提出的一个悬而未决的问题。