Bertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 , r-locatingdominating codes in paths Pn. They conjectured that if r≥2 is a fixed integer, then the smallest cardinality of an r-locatingdominating code in Pn, denoted by MrLD(Pn), satisfies MrLD(P n)=(n+1)/3⌉ for infinitely many values of n. We prove that this conjecture holds. In fact, we show a stronger result saying that for any r≥3 we have MrLD(Pn)=(n+1)3⌉ for all n≥nr when nr is large enough. In addition, we solve a conjecture on locationdomination with segments of even length in the infinite path. © 2011 Elsevier B.V. All rights reserved.
Exoo, G., Junnila, V., & Laihonen, T. (2011). Locatingdominating codes in paths. Discrete Mathematics, 311(17), 1863–1873. https://doi.org/10.1016/j.disc.2011.05.004