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 minimal meg-set is hard in general, recent works aimed to find polynomial-time algorithms to compute minimal 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 minimum 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-set）是一个顶点集合$M$，使得若移除任意一条边，则$M$中某两个顶点之间的距离会增大。这一概念由Foucaud等人于2023年提出，作为监测网络通信故障的一种方法。由于计算最小meg-set在一般情况下是困难的，近期研究致力于在输入图属于受限图类时，寻找计算最小meg-set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-set的性质，该集合因此易于计算。本文证明了弦图同样具有唯一的最小meg-set，从而回答了Foucaud等人提出的一个悬而未决的开放性问题。