Foucaud, Krishna and Lekshmi recently introduced the concept of monitoring edge-geodetic sets in graphs, and a related graph invariant. These are sets of vertices such that the removal of any edge changes the distance between some pair of vertices in the set. They studied the minimum possible size of such a set in a given graph, which we call the monitoring edge-geodetic number. We show that the decision problem for the monitoring edge-geodetic number is NP-complete. We also give best-possible upper and lower bounds for the Cartesian and strong products of two graphs. These bounds establish the exact value in many cases, including many new examples of graphs whose only monitoring edge-geodetic set is the whole vertex set.

翻译：Foucaud、Krishna和Lekshmi近期引入了图中监测边测地集的概念及其相关图不变量。这类顶点集具有如下性质：移除任意一条边都会改变集合中某对顶点之间的距离。他们研究了给定图中此类集合的最小可能基数，我们称之为监测边测地数。本文证明监测边测地数的判定问题是NP完全的。同时，我们给出了两个图笛卡尔积与强积的最优上下界。这些界在多种情况下确定了精确值，并提供了许多反例——这些图唯一的监测边测地集即为全部顶点集。