On the Minimum Locating Number of Graphs with a Given Order

A locating set S in a connected graph is a set of vertices satisfying that \(N(u)\cap S\) is unique for each vertex u not in S. A locating set can be considered as a set of sensors which can determine the exact location of an intruder. The size of a smallest locating set of a graph G is called the locating number of the graph and denoted by \(ln(G)\) . We show that \(min\, \{ln(G): G \) is a connected graph with n vertices \(\} = s\) when \(2^{s-1}+(s-1) < n \leq 2^s+s\) .