On Redundant Locating-Dominating Sets

A locating-dominating set in a graph G is a subset of vertices representing "detectors" which can locate an "intruder" given that each detector covers its closed neighborhood and can distinguish its own location from its neighbors. We explore a fault-tolerant variant of locating-dominating sets called redundant locating-dominating sets, which can tolerate one detector malfunctioning (going offline or being removed). In particular, we characterize redundant locating-dominating sets and prove that the problem of determining the minimum cardinality of a redundant locating-dominating set is NP-complete. We also determine tight bounds for the minimum density of redundant locating-dominating sets in several classes of graphs including paths, cycles, ladders, k-ary trees, and the infinite hexagonal and triangular grids. We find tight lower and upper bounds on the size of minimum redundant locating-dominating sets for all trees of order $n$, and characterize the family of trees which achieve these two extremal values, along with polynomial time algorithms to classify a tree as minimum extremal or not.