Minimum sizes of identifying codes in graphs differing by one vertex

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 ${\cal C} \subseteq V$ is said to be an r-identifying code in G if the sets $B_{G,r}(v)\cap {\cal C}$, 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 $\gamma_r(G^*)$, establishing results on their possible differences and ratios.