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

The metric dimension dim(G) of a graph G is the minimum cardinality of a set of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. Let v and e respectively denote a vertex and an edge of a graph G. We show that, for any integer k, there exists a graph G such that dim(G − v) − dim(G) = k. For an arbitrary edge e of any graph G, we prove that dim(G − e) ≤ dim(G) + 2. We also prove that dim(G − e) ≥ dim(G) − 1 for G belonging to a rather general class of graphs. Moreover, we give an example showing that dim(G)− dim(G− e) can be arbitrarily large.