Monitoring the edges of product networks using distances

TL;DR

Addresses the problem of determining the distance-edge-monitoring number $\text{dem}(G)$ and its behavior under standard graph operations, with emphasis on the Cartesian product. The authors derive lower and upper bounds, provide decomposition lemmas, and obtain exact values for join, corona, cluster, and Cartesian products, including sharpness and equality conditions. Key contributions include formulas such as $\text{dem}(G\vee H)=\min\{c(G)+|V(H)|, c(H)+|V(G)|\}$, and tight bounds for $\text{dem}(G\Box H)$ along with explicit results for paths, cycles, trees and complete graphs, plus comparisons to other distance-based parameters. This work offers a practical framework for monitoring in networks and connects the new parameter to classical invariants, aiding network design and analysis.

Abstract

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.

Figures (2)

• Figure 1: The comparison of different parameter on $P_m\Box P_n$
• Figure 2: The comparison of different parameter on $K_m \Box K_n(m\leq n\leq 2m-1)$

Theorems & Definitions (55)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4