Locating and identifying codes in circulant networks

A set SS of vertices of a graph GG is a dominating set of GG if every vertex uu of GG is either in SS or it has a neighbour in  SS. In other words, SS is dominating if the sets S∩N[u]S∩N[u] where u∈V(G)u∈V(G) and N[u]N[u] denotes the closed neighbourhood of uu in GG, are all nonempty. A set S⊆V(G)S⊆V(G) is called a locating code   in GG, if the sets S∩N[u]S∩N[u] where u∈V(G)∖Su∈V(G)∖S are all nonempty and distinct. A set S⊆V(G)S⊆V(G) is called an identifying code   in GG, if the sets S∩N[u]S∩N[u] where u∈V(G)u∈V(G) are all nonempty and distinct. We study locating and identifying codes in the circulant networks  Cn(1,3)Cn(1,3). For an integer n⩾7n⩾7, the graph Cn(1,3)Cn(1,3) has vertex set ZnZn and edges xyxy where x,y∈Znx,y∈Zn and |x−y|∈{1,3}|x−y|∈{1,3}. We prove that a smallest locating code in Cn(1,3)Cn(1,3) has size ⌈n/3⌉+c⌈n/3⌉+c, where c∈{0,1}c∈{0,1}, and a smallest identifying code in Cn(1,3)Cn(1,3) has size ⌈4n/11⌉+c′⌈4n/11⌉+c′, where c′∈{0,1}c′∈{0,1}.