Optimal error‐detection system for identifying codes

Abstract

Assume that a graph models a detection system for a facility with a possible “intruder,” or a multiprocessor network with a possible malfunctioning processor. We consider the problem of placing detectors at a subset of vertices in to determine the location of an intruder if there is any. Many types of detection systems have been defined for different sensor capabilities; in particular, we focus on identifying codes, where each detector can determine whether there is an intruder within its closed neighborhood. In this research we explore a fault‐tolerant variant of identifying codes applicable to real‐world systems. Specifically, error‐detecting identifying codes permit a false‐negative transmission from any single detector. We investigate minimum‐sized error‐detecting identifying codes in several classes of graphs, including cubic graphs and infinite grids, and show that the problem of determining said minimum size in arbitrary graphs is NP‐complete.