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 BG,r(v) the ball of radius r and centre v. A set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal C} \subseteq V$\end{document} is said to be an r-identifying code in G if the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B_{G,r}(v)\cap {\cal C}$\end{document}, 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 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. If G* is still r-twin-free, we compare the behaviours of γr(G) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma_r(G^*)$\end{document}, establishing results on their possible differences and ratios.