Global restrained Roman domination in graphs

A global restrained Roman dominating function on a graph $G=(V,E)$ to be a function $f:V\rightarrow\{0,1,2\}$ such that $f$ is a restrained Roman dominating function of both $G$ and its complement $\overline G$. The weight of a global restrained Roman dominating function is the value $w(f)=\Sigma_{u \in V} f(u)$. The minimum weight of a global restrained Roman dominating function of $G$ is called the global restrained Roman domination number of $G$ and denoted by $\gamma_{grR}(G)$. In this paper we initiate the study of global restrained Roman domination number of graphs. We then prove that the problem of computing $\gamma_{grR}$ is NP-hard even for bipartite and chordal graphs. The global restrained Roman domination of a given graph is studied versus to the other well known domination parameters such as restrained Roman domination number $\gamma_{rR}$ and global domination number $\gamma_g$ by bounding $\gamma_{grR}$ from below and above involving $\gamma_{rR}$ and $\gamma_g$ for general graphs, respectively. We characterize graphs $G$ for which $\gamma_{grR}(G)\in \{1,2,3,4,5\}$. It is shown that: for trees $T$ of order $n$, $\gamma_{grR}(T)=n$ if and only if diameter of $T$ is at most $5$. Finally, the triangle free graphs $G$ for which $\gamma_{grR}(G)=|V|$ are characterized.