Identifying and locating-dominating codes on chains and cycles

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 Br (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 Br (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. © 2004 Elsevier Ltd. All rights reserved.