Restrained roman domination in graphs

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 restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in v − s. we define a restrained roman dominating function on a graph g = (v, e) 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 and at least one vertex w for which f (w) = 0. the weight of a restrained roman dominating function is the value f (v ) = ∑u v f (u). the minimum weight of a restrained roman dominating function on a graph g is called the restrained roman domination number of g and denoted by γrr (g). in this paper, we initiate a study of this parameter.