Geodesic
Characterization of trees with equal 2-domination number and domination number plus two
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 687-697

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Let G = (V(G),E(G)) be a simple graph, and let k be a positive integer. A subset D of V(G) is a k-dominating set if every vertex of V(G) - D is dominated at least k times by D. The k-domination number γₖ(G) is the minimum cardinality of a k-dominating set of G. In [5] Volkmann showed that for every nontrivial tree T, γ₂(T) ≥ γ₁(T)+1 and characterized extremal trees attaining this bound. In this paper we characterize all trees T with γ₂(T) = γ₁(T)+2.
Keywords: 2-domination number, domination number, trees 
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     author = {Chellali, Mustapha and Volkmann, Lutz},
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Chellali, Mustapha; Volkmann, Lutz. Characterization of trees with equal 2-domination number and domination number plus two. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 687-697. http://geodesic.mathdoc.fr/item/DMGT_2011_31_4_a4/