Domination Number, Independent Domination Number and 2-Independence Number in Trees

For a graph G, let γ(G) be the domination number, i(G) be the independent domination number and β2(G) be the 2-independence number. In this paper, we prove that for any tree T of order n ≥ 2, 4β2(T) − 3γ(T) ≥ 3i(T), and we characterize all trees attaining equality. Also we prove that for every tree T of order n ≥ 2, i(T)≤3β2(T)4i(T) \le {{3{\beta _2}(T)} \over 4} , and we characterize all extreme trees

Domination Number, Independent Domination Number and 2-Independence Number in Trees

For a graph $G$, let $\gamma(G)$ be the domination number, $i(G)$ be the independent domination number and $\beta_2(G)$ be the 2-independence number. In this paper, we prove that for any tree $T$ of order $n ≥ 2, 4\beta_2(T) − 3\gamma(T) ≥ 3i(T)$, and we characterize all trees attaining equality. Also we prove that for every tree $T$ of order $n ≥ 2, i(T)≤\frac{3\beta_2(T)}{4}$, and we characterize all extreme trees