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Minimal doubly resolving sets of necklace graph

Abstract:

Consider a simple connected undirected graph G = (V;E), where V represents the vertex set and E represents the edge set, respectively. A subset D of V is called doubly resolving set if for every two vertices x; y of G, there are two vertices u; v ∈ D such that d(u; x) - d(u; y) ≠= d(v; x) - d(v; y). A doubly resolving set with minimum cardinality is called minimal doubly resolving set. This minimum cardinality is denoted by (G). In this paper, we find the minimal doubly resolving set for necklace graph Nen, n ≥ 2. Also, we prove that (Nen) = 3 for n ≥ 2.