Identifying and locating-dominating codes on chains and cycles

Abstract

Consider a connected undirected graph G=( V, E), a subset of vertices C⊆ V, and an integer r≥1; for any vertex v∈ V, let B r ( v) denote the ball of radius r centered at v, i.e., the set of all vertices within distance r from v. If for all vertices v∈ V (respectively, v∈ V ⧹ C), the sets B r ( v)∩ C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the smallest cardinalities or densities of these codes in chains (finite or infinite) and cycles.