global restrained roman domination in graphs

A global restrained roman dominating function on a graph $g=(v,e)$ to be a function $f:vrightarrow{0,1,2}$ 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.