On locating-dominating codes in the infinite king grid. (Q2828920)

In this paper, better lower bounds of densities of locating-dominating codes in infinite king grids are presented. Vertices of the king grid correspond to pairs of integers. Two vertices \(u=[u_x,u_y]\), \(v=[v_x,v_y]\) are adjacent if \(|u_x-v_x|\leq 1\) and \(|u_y-v_y|\leq 1\). A code is any set of vertices in the grid. Vertices in the code are called codewords. The ball \(B_r(v)\) with center \(v\) is the set of all vertices \(u\) with distance \(d(u,v)\leq r\). \(C\) is an \(r\)-locating-dominating code if the sets \(I_r(v)=B_r(v)\cap C\) are non-empty and distinct for all non-codewords \(v\in V\setminus C\). The author finds better bounds for the minimum density of locating \(r\)-locating-dominating codes for values \(r>1\).