Taille minimum des codes identifiants dans les graphes différant par un sommet ou une arête Minimum Sizes of Identifying Codes in Graphs Differing by One Edge or One Vertex

Let G be a simple, undirected graph with vertex set V . For v ∈ V and r ≥ 1, we denote by BG,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 BG,r(v) ∩ C, v ∈ V , are all nonempty and distinct. A graph G admitting an ridentifying code is called r-twin-free, and in this case the size of a smallest r-identifying code in G is denoted by γr(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, or by adding or deleting an edge. If G∗ is still r-twin-free, we compare the behaviours of γr(G) and γr(G ∗), establishing results on their possible differences and ratios.