Fault-tolerant locating-dominating sets on the infinite king grid

Let \(G\) be a graph of a network system with vertices, \(V(G)\), representing physical locations and edges, \(E(G)\), representing informational connectivity. A locating-dominating (LD) set \(S \subseteq V(G)\) is a subset of vertices representing detectors capable of sensing an “intruder” at precisely their location or at some unknown point in their open-neighborhood. An LD set must be capable of locating an intruder anywhere in the graph using this collection of detectors. We explore three types of fault-tolerant LD sets: redundant LD sets, which allow at most one detector to be removed or disabled, error-detecting LD sets, which allow at most one false negative, and error-correcting LD sets, which allow at most one error (false positive or false negative). In particular, we determine lower and upper bounds for the minimum density of these three fault-tolerant locating-dominating sets in the infinite king grid.