Minimum sizes of identifying codes in graphs differing by one vertex

Let G be a simple, undirected graph with vertex set V. For v ∈ V and r ≥ 1, we denote by B_G,r(v) the ball of radius r and centre v. A set C ⊆ V is said to be an r-identifying code in G if the sets B_G,r(v) ∩ C, v ∈ V, are all nonempty and distinct. A graph G admitting an r-identifying code is called r-twin-free, and in this case the size of a smallest r-identifying code in G is denoted by γ_r(G). We study the following structural problem: let G be an r-twin-free graph, and G^* be a graph obtained from G by adding or deleting a vertex. If G^* is still r-twin-free, we compare the behaviours of γ_r(G) and γ_r(G^*), establishing results on their possible differences and ratios.