An open-locating-dominating set of a graph models a detection system for a facility with a possible "intruder" or a multiprocessor network with a possible malfunctioning processor. A "sensor" or "detector" is assumed to be installed at a subset of vertices where it can detect an intruder or a malfunctioning processor in their neighborhood, but not at itself. We consider a fault-tolerant variant of an open-locating-dominating set called an error-correcting open-locating-dominating set, which can correct a false-positive or a false-negative signal from a detector. In particular, we prove the problem of finding a minimum error-correcting open-locating-dominating set in an arbitrary graph is NP-complete. Additionally, we characterize the existence criteria for an error-correcting open-locating-dominating sets for an arbitrary graph. We also consider extremal graphs that require every vertex to be a detector and minimum error-correcting open-locating-dominating sets in infinite grids.

翻译：图的开放定位控制集可建模为存在潜在“入侵者”的设施检测系统，或存在可能故障处理器的多处理器网络中的检测机制。假设在顶点子集上安装“传感器”或“检测器”，其能检测邻域内的入侵者或故障处理器，但无法检测自身所在顶点。本文研究一种容错的开放定位控制集变体——纠错开放定位控制集，其能够纠正检测器产生的误报或漏报信号。特别地，我们证明在任意图中寻找最小纠错开放定位控制集问题是NP完全的。此外，我们刻画了任意图存在纠错开放定位控制集的充要条件，并研究了要求每个顶点均作为检测器的极值图，以及无限网格中的最小纠错开放定位控制集。