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A characterization of (γₜ,γ₂)-trees

You Lu;  Xinmin Hou;  Jun-Ming Xu;  Ning Li — 2010

Discussiones Mathematicae Graph Theory

Let γₜ(G) and γ₂(G) be the total domination number and the 2-domination number of a graph G, respectively. It has been shown that: γₜ(T) ≤ γ₂(T) for any tree T. In this paper, we provide a constructive characterization of those trees with equal total domination number and 2-domination number.

On the (2,2)-domination number of trees

You Lu;  Xinmin Hou;  Jun-Ming Xu — 2010

Discussiones Mathematicae Graph Theory

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 ) ≤ 2 γ ( T ) . Moreover, we characterize all the trees achieving the equalities.

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