On the distance-edge-monitoring numbers of graphs

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) ≠ 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) ∈ 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.