Locating–dominating codes in paths

AbstractBertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1], r-locating–dominating codes in paths Pn. They conjectured that if r≥2 is a fixed integer, then the smallest cardinality of an r-locating–dominating code in Pn, denoted by MrLD(Pn), satisfies MrLD(Pn)=⌈(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 location–domination with segments of even length in the infinite path