Bounds on double Roman domination number of graphs

Bounds on double Roman domination number of graphs

Let G = (V,E) be a simple graph. A double Roman dominating function of a graph G is a function {0,1,2,3} :  V f having the property that if 0=) (v f , then the vertex v must have at least two neighbors 1w , 2w such that 2 = ) ( = ) ( 1 2 w f w f or one neighbor w such that 3 = ) (w f ; and if 1=) (v f , then the vertex v must have at least one neighbor w such that f (w)  2. The weight of a double Roman dominating function is the sum = ( ) ( ) w f v f v V G  , and the minimum weight of f w for every double Roman dominating function f on G is called double Roman domination number of G . We denote this number with ) (G dR  . In this paper; we obtain some new lower and upper bounds of double Roman domination number of graphs.

Let G = (V,E) be a simple graph. A double Roman dominating function of a graph G is a function {0,1,2,3} :  V f having the property that if 0=) (v f , then the vertex v must have at least two neighbors 1w , 2w such that 2 = ) ( = ) ( 1 2 w f w f or one neighbor w such that 3 = ) (w f ; and if 1=) (v f , then the vertex v must have at least one neighbor w such that f (w)  2. The weight of a double Roman dominating function is the sum = ( ) ( ) w f v f v V G  , and the minimum weight of f w for every double Roman dominating function f on G is called double Roman domination number of G . We denote this number with ) (G dR  . In this paper; we obtain some new lower and upper bounds of double Roman domination number of graphs.