The localization dominating set of a graph G is a subset of vertices representing “detectors”, each of which covers its closed neighborhood and locates an “intruder” if its own position can be distinguished from its neighborhood. can do. We investigate a fault-tolerant variant of the locating-dominating set, called the redundant locating-dominating set, that can tolerate one detector malfunctioning (going offline or being removed). In particular, we prove that the problem of characterizing redundant position domination sets and determining the minimum cardinality of redundant position domination sets is NP-complete. We also determine strict bounds on the minimal density of redundant position domination sets in several classes of graphs, including paths, cycles, ladders, k-ary trees, and infinite hexagonal and triangular grids. Finding hard lower and upper bounds on the size of the smallest redundant position domination set for all trees of order $n$, along with a polynomial-time algorithm to classify the tree as the smallest of the trees achieving these two extremes. characterize the family. extreme or not.
