Identifying codes and locating-dominating sets on paths and cycles

Abstract

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