Geodesic
Fault-tolerant identifying codes in special classes of graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 2, pp. 591-611

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A detection system, modeled in a graph, is composed of “detectors quot; positioned at a subset of vertices in order to uniquely locate an “intruder quot; at any vertex. 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 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 a redundant identifying code for an arbitrary graph is NP-hard, and we determine the bounds on the lowest cardinality for special classes of graphs, including trees, ladders, cylinders, and cubic graphs.
Keywords: domination, detection system, identifying-code, fault-tolerant, redundant-identifying-code, density 
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     title = {Fault-tolerant identifying codes in special classes of graphs},
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Jean, Devin C.; Seo, Suk J. Fault-tolerant identifying codes in special classes of graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 2, pp. 591-611. http://geodesic.mathdoc.fr/item/DMGT_2024_44_2_a9/