Locating–dominating codes in paths

Bertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1], rr-locating–dominating codes in paths PnPn. They conjectured that if r≥2r≥2 is a fixed integer, then the smallest cardinality of an rr-locating–dominating code in PnPn, denoted by MrLD(Pn), satisfies MrLD(Pn)=⌈(n+1)/3⌉ for infinitely many values of nn. We prove that this conjecture holds. In fact, we show a stronger result saying that for any r≥3r≥3 we have MrLD(Pn)=⌈(n+1)/3⌉ for all n≥nrn≥nr when nrnr is large enough. In addition, we solve a conjecture on location–domination with segments of even length in the infinite path.