Locating and identifying codes in circulant networks

• Authors: M. Ghebleh;L. Niepel
• Venue: Discrete Applied Mathematics
• Year: 2013

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 C"n(1,3). For an integer n=7, the graph C"n(1,3) has vertex set Z"n and edges xy where x,y@?Z"n and |x-y|@?{1,3}. We prove that a smallest locating code in C"n(1,3) has size @?n/3@?+c, where c@?{0,1}, and a smallest identifying code in C"n(1,3) has size @?4n/11@?+c^', where c^'@?{0,1}.