Global Roman domination in graphs

Abstract

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