Further results on the total Italian domination number of trees

Abstract

<abstract><p>Let $ f:V(G)\rightarrow \{0, 1, 2\} $ be a function defined from a connected graph $ G $. Let $ W_i = \{x\in V(G): f(x) = i\} $ for every $ i\in \{0, 1, 2\} $. The function $ f $ is called a total Italian dominating function on $ G $ if $ \sum_{v\in N(x)}f(v)\geq 2 $ for every vertex $ x\in W_0 $ and if $ \sum_{v\in N(x)}f(v)\geq 1 $ for every vertex $ x\in W_1\cup W_2 $. The total Italian domination number of $ G $, denoted by $ \gamma_{tI}(G) $, is the minimum weight $ \omega(f) = \sum_{x\in V(G)}f(x) $ among all total Italian dominating functions $ f $ on $ G $. In this paper, we provide new lower and upper bounds on the total Italian domination number of trees. In particular, we show that if $ T $ is a tree of order $ n(T)\geq 2 $, then the following inequality chains are satisfied.</p> <p>(ⅰ) $ 2\gamma(T)\leq \gamma_{tI}(T)\leq n(T)-\gamma(T)+s(T) $,</p> <p>(ⅱ) $ \frac{n(T)+\gamma(T)+s(T)-l(T)+1}{2}\leq \gamma_{tI}(T)\leq \frac{n(T)+\gamma(T)+l(T)}{2}, $</p> <p>where $ \gamma(T) $, $ s(T) $ and $ l(T) $ represent the classical domination number, the number of support vertices and the number of leaves of $ T $, respectively. The upper bounds are derived from results obtained for the double domination number of a tree.</p></abstract>