Progress on fault-tolerant locating-dominating sets

A locating-dominating set is a subset of vertices representing “detectors” in a graph G 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 error-detecting locating-dominating (DET:LD) sets, which can tolerate one false negative. The concept of DET:LD sets was originally introduced in 2002, along with several properties in various classes of graphs; we extend this work by fully characterizing DET:LD sets for general graphs and establishing existence criteria. We prove that the problem of determining the minimum density of a DET:LD set in arbitrary graphs is NP-complete. Additionally, we determine that the minimum density in cubic graphs is at least [Formula: see text], and we show an infinite family of cubic graphs achieving this density.