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

Let $$G=(V(G),E(G))$$ G = ( V ( G ) , E ( G ) ) be a connected graph. A set of vertices $$S\subseteq V(G)$$ 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 $$edim(G-e)-edim(G)$$ e d i m ( G - e ) - e d i m ( G ) for a connected graph G and any edge $$e\in E(G)$$ e ∈ E ( G ) . On the other hand, we show that the value of $$edim(G)-edim(G-e)$$ e d i m ( G ) - e d i m ( G - e ) is unbounded for some graph G and some edge $$e\in E(G)$$ e ∈ E ( G ) . However, for a unicyclic graph H, we obtain that $$edim(H)-edim(H-e)\le 1$$ e d i m ( H ) - e d i m ( 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 $$edim(G)-edim(G-u)$$ e d i m ( G ) - e d i m ( G - u ) and $$edim(H-v)-edim(H)$$ e d i m ( H - v ) - e d i m ( H ) can be arbitrarily large, where $$u\in V(G)$$ u ∈ V ( G ) and $$v\in V(H)$$ v ∈ V ( H ) .