A note on extremal trees for a bound on the double domination number

Abstract

In a graph $G,$ a vertex is said to dominate itself and its neighbors. A subset $D$ of $V\left(G\right)$ is a double dominating set if every vertex of $V\left(G\right)$ is dominated at least twice by the vertices of $D$. The double domination number ${\gamma }_{\times 2}\left(G\right)$ is the minimum cardinality among all double dominating sets of $G$. Cabrera-Martínez proved that for every nontrivial tree $T$ of order $n\left(T\right)$ with $\ell \left(T\right)$ leaves and $s\left(T\right)$ support vertices, ${\gamma }_{\times 2}\left(T\right)\ge \frac{1}{2}(n\left(T\right)-\gamma \left(T\right)+\ell \left(T\right)+s\left(T\right)+1),$ where $\gamma \left(T\right)$ stands for the domination number of $T.$ In this note, we provide a constructive characterization of trees attaining this bound in response to the problem raised by Cabrera-Martínez.