The Paradoxical Nature of Locating Sensors in Paths and Cycles: The Case of 2-Identifying Codes

For a graph G and a set D ⊆ V (G), define Nr[x] = {xi ∈ V (G) : d(x, xi) ≤ r} (where d(x, y) is graph theoretic distance) and Dr(x) = Nr[x] ∩D. D is known as an r-identifyingcode if for every vertex x, Dr(x) 6= ∅, and for every pair of vertices x and y, x 6= y ⇒ Dr(x) 6= Dr(y). The various applications of these codes include attack sensor placement in networks and fault detection/localization in multiprocessor or distributed systems. In [2] and [16], partial results about the minimum size of D for r-identifying codes are given for paths and cycles and complete closed form solutions are presented for the case r = 1, based in part on [14]. We provide complete solutions for the case r = 2 as well as present our own solutions (verifying earlier results) to the r = 1 case. We use these closed form solutions to illustrate some surprisingly counterintuitive behavior that arises when the length of the path or cycle or the value of r varies.