Perturbation Results for Distance-edge-monitoring Numbers

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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.