Further results on the total Italian domination number of trees

Let $f:V(G)\to \{0,1,2\}$ 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 $\sum _{v\in N(x)}f(v)\ge 2$ for every vertex $x\in W_0$ and if $\sum _{v\in N(x)}f(v)\ge 1$ 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 $n(T)\ge 2$, then the following inequality chains are satisfied.

(ⅰ) $2\gamma (T)\le {\gamma }_{tI}(T)\le n(T)-\gamma (T)+s(T)$,

(ⅱ) $\frac{n(T)+\gamma (T)+s(T)-l(T)+1}{2}\le {\gamma }_{tI}(T)\le \frac{n(T)+\gamma (T)+l(T)}{2},$

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.