Double Roman domination

For a graph G = ( V , E ) , a double Roman dominating function is a function f : V ź { 0 , 1 , 2 , 3 } having the property that if f ( v ) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor with f ( w ) = 3 , and if f ( v ) = 1 , then vertex v must have at least one neighbor with f ( w ) ź 2 . The weight of a double Roman dominating function f is the sum f ( V ) = ź v ź V f ( v ) , and the minimum weight of a double Roman dominating function on G is the double Roman domination number of G . We initiate the study of double Roman domination and show its relationship to both domination and Roman domination. Finally, we present an upper bound on the double Roman domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound.