On rr-locating–dominating sets in paths

Assume that G=(V,E)G=(V,E) is a simple undirected graph, and CC is a nonempty subset of VV. For every v∈Vv∈V, we define Ir(v)={u∈C∣dG(u,v)≤r}Ir(v)={u∈C∣dG(u,v)≤r}, where dG(u,v)dG(u,v) denotes the number of edges on any shortest path between uu and vv. If the sets Ir(v)Ir(v) for v∉Cv∉C are pairwise different, and none of them is the empty set, we say that CC is an rr-locating–dominating set in GG. It is shown that the smallest 2-locating–dominating set in a path with nn vertices has cardinality ⌈(n+1)/3⌉⌈(n+1)/3⌉, which coincides with the lower bound proved earlier by Bertrand, Charon, Hudry and Lobstein. Moreover, we give a general upper bound which improves a result of Bertrand, Charon, Hudry and Lobstein.