a b s t r a c tBertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 , r-locatingdominating 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 LD r (P n ), satisfies M LD r (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 LD r (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.
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