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 N r [ x ] = { y ∈ V : d ( x , y ) ≤ r } and D r ( x ) = N r [ 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), D r ( x ) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72–82] provided complete results for the paths and cycles when r = 2 . In this paper, we provide results for a remaining open case in cycles and complete results in paths for r -identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating–dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969–987]. ► We provide results for a remaining open case in cycles for r -identifying codes. ► We provide complete results in paths for r -identifying codes. ► We give complete results for 2-locating-dominating sets in cycles.