Restrained Italian domination in trees

Abstract

Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is a \textit{restrained dominating set} if every vertex in $V\setminus D$ is adjacent to a vertex in $D$ and to a vertex in $V\setminus D$. The \textit{restrained domination number}, denoted by ${\gamma }_r(G)$, is the smallest cardinality of a restrained dominating set of $G$. A function $f:V\to \{0,1,2\}$ is a \textit{restrained Italian dominating function} on $G$ if (i) for each vertex $v\in V$ for which $f(v)=0$, it holds that ${\sum }_{u\in N_G(v)}f(u)\ge 2$, (ii) the subgraph induced by $\{v\in V\mid f(v)=0\}$ has no isolated vertices. The \textit{restrained Italian domination number}, denoted by ${\gamma }_{rI}(G)$, is the minimum weight taken over all restrained Italian dominating functions of $G$. It is known that ${\gamma }_r(G)\le {\gamma }_{rI}(G)\le 2{\gamma }_r(G)$ for any graph $G$. In this paper, we characterize the trees $T$ for which ${\gamma }_r(T)={\gamma }_{rI}(T)$, and we also characterize the trees $T$ for which ${\gamma }_{rI}(T)=2{\gamma }_r(T)$.