Geodesic
On the (2,2)-domination number of trees
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 2, pp. 185-199

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Let γ(G) and γ_2,2(G) denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that (2(γ(T)+1))/3 ≤ γ_2,2(T) ≤ 2γ(T). Moreover, we characterize all the trees achieving the equalities.
Keywords: domination number, total domination number, (2,2)-domination number 
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Lu, You; Hou, Xinmin; Xu, Jun-Ming. On the (2,2)-domination number of trees. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 2, pp. 185-199. http://geodesic.mathdoc.fr/item/DMGT_2010_30_2_a0/