Locating and Identifying Codes in Circulant Networks

A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words S is dominating if the sets S ∩N [u] where u ∈ V (G) and N [u] denotes the closed neighbourhood of u in G, are all nonempty. A set S ⊆ V (G) is called a locating code in G, if the sets S ∩ N [u] where u ∈ V (G) \ S are all nonempty and distinct. A set S ⊆ V (G) is called an identifying code in G, if the sets S ∩ N [u] where u ∈ V (G) are all nonempty and distinct. We study locating and identifying codes in the circulant networks Cn(1, 3). For an integer n > 7, the graph Cn(1, 3) has vertex set Zn and edges xy where x, y ∈ Zn and |x − y| ∈ {1, 3}. We prove that a smallest locating code in Cn(1, 3) has size dn/3e + c, where c ∈ {0, 1}, and a smallest identifying code in Cn(1, 3) has size d4n/11e+ c′, where c′ ∈ {0, 1}.