On Roman balanced domination of graphs

Let G be a graph with vertex set V . A function f : V -> {-1, 0, 2} is called a Roman balanced dominating function (RBDF) of G if L E N G [ v ] f ( u ) = 0 for each vertex v E V . The maximum (resp. minimum) Roman balanced domination number gamma M Rb ( G ) (resp. gamma mRb ( G )) is the maximum (resp. minimum) value of v E V f (v) among all Roman balanced dominating functions f . A graph G is called Rd-balanced if gamma M Rb ( G ) = gamma m Rb ( G ) = 0. In this paper, we obtain several upper and lower bounds on gamma M Rb ( G ) and gamma m Rb ( G ) and further determine several classes of Rd-balanced graphs.