On the distance-edge-monitoring numbers of graphs

Abstract

Foucaud et al. (2022) 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 dG(x,y)≠dG−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)∈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. In this paper, we continue the study of 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. In addition, we give some properties of the graph G with dem(G)=n−2.