Monitoring the edges of product networks using distances

Foucaud 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 $V\left(G\right)$ and edge set $E\left(G\right)$. For any subset M in $V\left(G\right)$ and an edge e in $E\left(G\right)$, let $P(M,e)$ be the set of pairs $(x,y)$ such that $d_G(x,y)\ne d_{G-e}(x,y)$ where $x\in M$ and $y\in V\left(G\right)$. M is called a 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 distance-edge-monitoring number of G, denoted by $\mathrm{dem}\left(G\right)$, 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 $\max \left\{m\mathrm{dem}\right(H),n\mathrm{dem}(G\left)\right\}\le \mathrm{dem}(G\square H)\le m\mathrm{dem}\left(H\right)+n\mathrm{dem}\left(G\right)-\mathrm{dem}\left(G\right)\mathrm{dem}\left(H\right)$, where □ is the Cartesian product operation. Moreover, we characterize the networks 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.