Choice identification of a graph

Let GG be a graph, uu be a vertex of GG, and B(u)B(u)(or BG(u)BG(u)) be the set of uu with all its neighbors in GG. A set SS of vertices is called an identifying set   of GG if there exists a function ff from V(G)V(G) to the set of all nonempty subsets of SS such that (i) for each vertex uu in GG, f(u)⊆B(u)f(u)⊆B(u), and (ii) for every pair of distinct vertices uu and vv, f(u)f(u) and f(v)f(v) are distinct. ff is called a choice identification   of GG with respect to SS. The choice identification number  ιc(G)ιc(G) is the cardinality of a minimum identifying set of GG. In this paper, we study the identifying sets in graphs, give a polynomial-time algorithm to find a minimum identifying set of a tree, and determine the choice identification numbers of complete bipartite graphs.