Fault-tolerant Identifying Codes in Special Classes of Graphs

TL;DR

This paper introduces fault-tolerant identifying codes in graphs, allowing detection systems to remain functional despite a detector failure, and provides bounds for specific graph classes.

Contribution

It defines redundant identifying codes for fault tolerance and establishes NP-completeness and bounds for special graph classes.

Findings

NP-completeness of the minimum identifying code problem

Bounds on code size for trees, cylinders, and cubic graphs

Introduction of fault-tolerant (redundant) identifying codes

Abstract

A detection system, modeled in a graph, is composed of "detectors" positioned at a subset of vertices in order to uniquely locate an ``intruder" at any vertex. \emph{Identifying codes} use detectors that can sense the presence or absence of an intruder within distance one. We introduce a fault-tolerant identifying code called a \emph{redundant identifying code}, which allows at most one detector to go offline or be removed without disrupting the detection system. In real-world applications, this would be a necessary feature, as it would allow for maintenance on individual components without disabling the entire system. Specifically, we prove that the problem of determining the lowest cardinality of an identifying code for an arbitrary graph is NP-complete, we determine the bounds on the lowest cardinality for special classes of graphs, including trees, cylinders, and cubic graphs.