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.