Identifying codes and locating-dominating sets on paths and cycles

Let $G=(V,E)$ be a graph and let $r\ge 1$ be an integer. For a set $D \subseteq V$,define$N_r[x] = {y \in V: d(x, y) \leq r}$and$D_r(x) = N_r[x] \cap D$,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 $x\in V$ ($x \in V\backslash D$,respectively),$D_r(x)$ 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.