Global Roman domination in graphs

A Roman dominating function   (RDF) on a graph G=(V,E)G=(V,E) is defined to be a function f:V→{0,1,2}f:V→{0,1,2} satisfying the condition that every vertex uu for which f(u)=0f(u)=0 is adjacent to at least one vertex vv for which f(v)=2f(v)=2. A set S⊆VS⊆V is a global dominating set   if SS dominates both GG and its complement G¯. The global domination number γg(G)γg(G) of a graph GG is the minimum cardinality of SS. We define a global Roman dominating function   on a graph G=(V,E)G=(V,E) to be a function f:V→{0,1,2}f:V→{0,1,2} such that ff is an RDF for both GG and its complement G¯. The weight   of a global Roman dominating function is the value f(V)=∑u∈Vf(u)f(V)=∑u∈Vf(u). The minimum weight of a global Roman dominating function on a graph GG is called the global Roman domination number   of GG and denoted by γgR(G)γgR(G). In this paper, we initiate a study of this parameter.