The effect of vertex and edge deletion on the edge metric dimension of graphs

Let G=(V(G),E(G)) be a connected graph. A set of vertices S⊆ V(G) is an edge metric generator of G if any pair of edges in G can be distinguished by their distance to a vertex in S . The edge metric dimension edim ( G ) is the minimum cardinality of an edge metric generator of G . In this paper, we first give a sharp bound on edim(G-e)-edim(G) for a connected graph G and any edge e∈ E(G) . On the other hand, we show that the value of edim(G)-edim(G-e) is unbounded for some graph G and some edge e∈ E(G) . However, for a unicyclic graph H , we obtain that edim(H)-edim(H-e)≤ 1 , where e is an edge of the unique cycle in H . And this conclusion generalizes the result on the edge metric dimension of unicyclic graphs given by Knor et al. Finally, we construct graphs G and H such that both edim(G)-edim(G-u) and edim(H-v)-edim(H) can be arbitrarily large, where u∈ V(G) and v∈ V(H) .