New bounds for (r,???2)-identifying codes in the infinite king grid
The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin. We call any subset \(C\subseteq V\) an (r,???k)-identifying code if the sets {c???C | d(c,v)???r, v???F} are distinct for all subset \(F\subseteq V\) with at most k vertices. Here, V denotes the vertex set of graph and d(c,v) the number of edges on any shortest path from c to v. We prove a new lower bound \(\frac{5}{12}\) on the density of (1,???2)-identifying codes and construct a (2,???2)-identifying code with density \(\frac{2}{7}\) in the infinite king grid.
identifying code king grid graph density