On the distance-edge-monitoring numbers of graphs

TL;DR

This paper studies the new concept of distance-edge-monitoring numbers in graphs, providing bounds, characterizations of extremal graphs, and analyzing graphs with specific monitoring numbers, to enhance network failure detection.

Contribution

It introduces bounds and characterizations for the distance-edge-monitoring number and extremal graphs, advancing understanding of network monitoring capabilities.

Findings

Established bounds for P(M, e), EM(x), and dem(G).

Characterized extremal graphs that attain these bounds.

Identified and characterized graphs with dem(G)=3.

Abstract

Foucaud et al. [Discrete Appl. Math. 319 (2022), 424-438] recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. For a set $M$ of vertices and an edge $e$ of a graph $G$, let $P(M, e)$ be the set of pairs $(x, y)$ with a vertex $x$ of $M$ and a vertex $y$ of $V(G)$ such that $d_G(x, y)\neq d_{G-e}(x, y)$. For a vertex $x$, let $EM(x)$ be the set of edges $e$ such that there exists a vertex $v$ in $G$ with $(x, v) \in P(\{x\}, e)$. A set $M$ of vertices of a graph $G$ is 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 a graph $G$, denoted by $dem(G)$, is defined as the smallest size of distance-edge-monitoring sets of $G$. The vertices of $M$ represent distance probes in a network modeled by $G$; when the edge $e$ fails, the distance from $x$ to $y$ increases, and thus we are able to detect the failure. It turns out that not only we can detect it, but we can even correctly locate the failing edge. In this paper, we continue the study of \emph{distance-edge-monitoring sets}. In particular, we give upper and lower bounds of $P(M,e)$, $EM(x)$, $dem(G)$, respectively, and extremal graphs attaining the bounds are characterized. We also characterize the graphs with $dem(G)=3$.