A global restrained Roman dominating function on a graph G = (V; E) to be a function f : V ! f0; 1; 2g such that f is a restrained Roman dominating function of both G and its complement G. The weight of a global restrained Roman dominating function is the value w(f) = Σu2V 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 γgrR(G). In this paper we initiate the study of global restrained Roman domination number of graphs. We then prove that the problem of computing γgrR 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 γrR and global domination number γg by bounding γgrR from below and above involving γrR and γg for general graphs, respectively. We characterize graphs G for which γgrR(G) 2 f1; 2; 3; 4; 5g. It is shown that: for trees T of order n, γgrR(T ) = n if and only if diameter of T is at most 5. Finally, the triangle free graphs G for which γgrR(G) = jV j are characterized.
