Perturbation Results for Distance-edge-monitoring Numbers*

Abstract

Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Given a graph G = (V(G), E(G)), a set M ⊆ V(G) is a distance-edge-monitoring set if for every edge e ∈ E(G), there is a vertex x ∈ M and a vertex y ∈ V(G) such that the edge e belongs to all shortest paths between x and y. The smallest size of such a set in G is denoted by dem(G). Denoted by G – e (resp. G\u) the subgraph of G obtained by removing the edge e from G (resp. a vertex u together with all its incident edges from G). In this paper, we first show that dem(G – e) – dem(G) ≤ 2 for any graph G and edge e ∈ E(G). Moreover, the bound is sharp. Next, we construct two graphs G and H to show that dem(G) – dem(G\u) and dem(H \ v) – dem(H) can be arbitrarily large, where u ∈ V(G) and v ∈ V(H). We also study the relation between dem(H) and dem(G), where H is a subgraph of G. In the end, we give an algorithm to judge whether the distance-edge-monitoring set still remain in the resulting graph when any edge of a graph G is deleted.