论文标题
使用距离监视产品网络的边缘
Monitoring the edges of product networks using distances
论文作者
论文摘要
Foucaud {\ it等人}最近在网络监视领域引入并启动了对新的图理论概念的研究。让$ g $是带有顶点套装$ v（g）$，$ m $的图形，$ v（g）$，$ e $是$ e（g）$中的优势，让$ e（m，e）$是一组$（x，y）$的集合，$ d_g（x，y）$ d_g（x，y） v（g）$。 $ m $称为\ emph {距离 - 边缘监测集}，如果每个边缘$ e $ e $ of $ g $都由$ m $的某些顶点监视，即，套装$ p（m，e）$是非空的。 $ g $的{\ em距离 - 边缘图}，由$ \ operatatorName {dem}（g）$表示为$ g $的距离 - 边缘监测集的最小尺寸。对于两张图，分别为$ m，n $的$ g，h $，在本文中，我们证明了$ \ max \ {m \ operatorname {dem}（h），n \ opperatorname {dem}（g）\} \} \ leq \ leq \ leq \ leq \ leq \ pereratatOrname {dem}（dem}（g \ \，\ box \ box \ box \ box \ box \ box \ box \ box， m \ operatorName {dem}（h）+n \ operatotorName {dem}（g） - \ operatatorName {dem}（g）\ propatatorName {dem}（h）$，其中$ \ box $是笛卡尔产品操作。此外，我们表征了达到上限和下限的图形，并在某些已知网络上显示了它们的应用。我们还获得了连接，电晕，群集和某些特定网络的距离边缘监测数。
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.
