Locating–dominating codes in paths

Bertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1], r -locating–dominating codes in paths P n . They conjectured that if r ≥ 2 is a fixed integer, then the smallest cardinality of an r -locating–dominating code in P n , denoted by M r L D ( P n ) , satisfies M r L D ( 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 M r L D ( P n ) = ⌈ ( n + 1 ) / 3 ⌉ for all n ≥ n r when n r is large enough. In addition, we solve a conjecture on location–domination with segments of even length in the infinite path.