Maximum difference about the size of optimal identifying codes in graphs differing by one vertex

LetG = (V,E) be a simple undirected graph. We call any subset C ⊆ V an identifying code if the sets I(v) = {c ∈ C | {v, c} ∈ E or v = c} are distinct and non-empty for all vertices v ∈ V . A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is denoted by γ(G). Let GS denote the induced subgraph of G where the vertex set S ⊂ V is deleted. We provide a tight upper bound for γ(GS) − γ(G) when both graphs are twin-free and |V | is large enough with respect to |S|. Moreover, we prove tight upper bound when G is a bipartite graph and |S| = 1.