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Bertrand, 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...
The inequality ρ(G)≤γ(G) between the packing number ρ(G) and the domination number γ(G) of a graph G is well known. For general graphs G, there exists no upper bound on γ(G) of the form γ(G)≤f(ρ(G)) where f is a function, as is remarked in [Discrete Math. 309 (2009), 2473–2478]. In this paper, we observe that γ(G)≤Δ(G)ρ(G), where Δ(G) denotes the maximum degree of G. We characterize connected graph...
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