On the Global Roman Domination Number in Graphs

A Roman dominating function f on a graph G is a global Roman dominating function on G, if f is also a Roman dominating function on G¯. The weight of a global Roman dominating function f is the value w(f)=∑x∈V(G)f(x). The minimum weight of a global Roman dominating function on a graph G is called the global Roman domination number γgR(G) of G. In this paper, we present upper bounds for γgR(G) in terms of order, diameter, and girth. We give necessary and sufficient conditions for a graph G with property γgR(G)=γg(G)+i for all i=0,1,2,3, where γg(G) is the global domination number of G. We also describe all connected unicyclic graphs G for which γgR(G)−γR(G) is maximum.