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. Let G=(V, E) be a graph. A set of vertices M ⊆ V(G) is a distance-edge-monitoring set of G if any edges in G can be monitored by a vertex in M. The distance-edge-monitoring number dem(G) is the minimum cardinality of a distance-edge-monitoring set of 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 relationship between dem(H) and dem(G) for H⊂ 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.
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