Graphs where every k-subset of vertices is an identifying set

Let G = (V, E) be an undirected graph without loops and multiple edges. A subset C ⊆ V is called identifying if for every vertex x ∈ V the intersection of C and the closed neighbourhood of x is nonempty, and these intersections are different for different vertices x. Let k be a positive integer. We will consider graphs where every ksubset is identifying. We prove that for every k > 1 the maximal order of such a graph is at most 2k − 2. Constructions attaining the maximal order are given for infinitely many values of k. The corresponding problem of k-subsets identifying any at most ` vertices is considered as well.