Foucaud {\it et al.} recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Let $G$ be a graph with vertex set $V(G)$, $M$ a subset of $V(G)$, and $e$ be an edge in $E(G)$, and let $P(M, e)$ be the set of pairs $(x,y)$ such that $d_G(x, y)\neq d_{G-e}(x, y)$ where $x\in M$ and $y\in V(G)$. $M$ is called a \emph{distance-edge-monitoring set} if every edge $e$ of $G$ is monitored by some vertex of $M$, that is, the set $P(M, e)$ is nonempty. The {\em distance-edge-monitoring number} of $G$, denoted by $\operatorname{dem}(G)$, is defined as the smallest size of distance-edge-monitoring sets of $G$. For two graphs $G,H$ of order $m,n$, respectively, in this paper we prove that $\max\{m\operatorname{dem}(H),n\operatorname{dem}(G)\} \leq\operatorname{dem}(G\,\Box \,H) \leq m\operatorname{dem}(H)+n\operatorname{dem}(G) -\operatorname{dem}(G)\operatorname{dem}(H)$, where $\Box$ is the Cartesian product operation. Moreover, we characterize the graphs attaining the upper and lower bounds and show their applications on some known networks. We also obtain the distance-edge-monitoring numbers of join, corona, cluster, and some specific networks.

翻译：Foucaud等人近期提出并开创性地研究了一个新的图论概念在网络监控领域的应用。设$G$为顶点集$V(G)$的图，$M$为$V(G)$的子集，$e$为$E(G)$中的一条边，并令$P(M, e)$表示满足$d_G(x, y)\neq d_{G-e}(x, y)$的配对$(x,y)$集合，其中$x\in M$，$y\in V(G)$。若图$G$的每条边$e$都被$M$中的某个顶点监控，即集合$P(M, e)$非空，则称$M$为\textit{距离边监控集}。图$G$的\textit{距离边监控数}记作$\operatorname{dem}(G)$，定义为$G$的最小距离边监控集的大小。针对阶数分别为$m,n$的两个图$G,H$，本文证明了$\max\{m\operatorname{dem}(H),n\operatorname{dem}(G)\} \leq\operatorname{dem}(G\,\Box \,H) \leq m\operatorname{dem}(H)+n\operatorname{dem}(G) -\operatorname{dem}(G)\operatorname{dem}(H)$，其中$\Box$为笛卡尔乘积运算。此外，我们刻画了达到上下界的图，并展示了其在若干已知网络中的应用。同时，本文还得到了联图、冠状图、簇图及特定网络的距离边监控数。