Identifying codes and locating-dominating sets on paths and cycles

Let G = (V,E) be a graph and let r ≥ 1 be an integer. For a set D ⊆ V , define Nr[x] = {y ∈ V : d(x, y) ≤ r} and Dr(x) = Nr[x] ∩ D, where d(x, y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x ∈ V (x ∈ V \D, respectively), Dr(x) are all nonempty and different. In this paper, we provide complete results for r-identifying codes in paths and odd cycles; we also give complete results for 2-locating-dominating sets in cycles.