Geodesic
An upper bound on the double domination number of trees
Kragujevac Journal of Mathematics, Tome 39 (2015) no. 2, p. 133

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In a graph $G$, a vertex dominates itself and its neighbors. A set $S$ of vertices in a graph $G$ is a {\em double dominating set} if $S$ dominates every vertex of $G$ at least twice. The {\em double domination number} $\gamma_{\times2}(G)$ is the minimum cardinality of a double dominating set in $G$. The {\em annihilation number} $a(G)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $G$ is at most the number of edges in $G$. In this paper, we show that for any tree $T$ of order $n\ge 2$, different from $P_4$, $\gamma_{\times2}(T)\le \frac{3a(T)+1}{2}$.
Classification : 05C69
Keywords: domination number, double dominating set, double domination number, annihilation number, tree 
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J. Amjadi. An upper bound on the double domination number of trees. Kragujevac Journal of Mathematics, Tome 39 (2015) no. 2, p. 133 . http://geodesic.mathdoc.fr/item/KJM_2015_39_2_a1/