Graphs with small or large Roman {3}-domination number

For an integer $k\geq1,$ a Roman $\{k\}$-dominating function (R$\{k\}$DF) on a graph $G=(V,E)$ is a function $f:V\rightarrow\{0,1,\dots,k\}$ such that for every vertex $v\in V$ with $f(v)=0$, $\sum_{u\in N(v)}f(u)\geq k$, where $N(v)$ is the set of vertices adjacent to $v.$ The weight of an R $\{k\}$DF is the sum of its function values over the whole set of vertices, and the Roman $\{k\}$-domination number $\gamma_{\{kR\}}(G)$ is the minimum weight of an R$\{k\}$DF on $G$. In this paper, we will be focusing on the case $k=3$, where trivially for every connected graphs of order $n\geq3,$ $3\leq$ $\gamma _{\{kR\}}(G)\leq n.$ We characterize all connected graphs $G$ of order $n\geq3$ such that $\gamma_{\{3R\}}(G)\in\{3,n-1,n\},$ and we improve the previous lower and upper bounds. Moreover, we show that for every tree $T$ of order $n\geq3$, $\gamma_{\{3R\}}(T)\geq\gamma(T)+2$, where $\gamma(T)$ is the domination number of $T,$ and we characterize the trees attaining this bound.