Minimum sizes of identifying codes in graphs differing by one edge

Abstract

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 ${\mathcal C} \subseteq V$ is said to be an r-identifying code in G if the sets B G , r ( v ) ? 𝒞 $B_{G,r}(v)\cap {\mathcal C}$ , 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 ? 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 an edge. If G ? is still r-twin-free, we compare the behaviours of ? r (G) and ? r (G ?), establishing results on their possible differences and ratios.