Possible cardinalities for locating-dominating codes in graphs

Consider a connected undirected graph G =(V,E), a subset of vertices C ⊆ V, and an integer r ≥ 1; for any vertex v ∈ V, let B_r(v) denote the ball of radius r centered at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V \ C, the sets B_r(v) ∩ C are all nonempty and different, then we call C an r-locating-dominating code. It is known that the cardinality of a minimum r-locating-dominating code C in any connected undirected graph G having a given number, n, of vertices satisfies the inequalities |C| ≤ n - 1 and |C| + 2^|C| ≥ n + 1, and that these lower and upper bounds can be achieved. Here, we prove that any in-between value can also be reached by |C|.