A Note on the Double Roman Domination Number of Graphs

for a graph G = (V, E), a double Roman dominating function is a function f: V -> {0, 1, 2, 3} having the property that if f (v) = 0, then the 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 the vertex v must have at least one neighbor with f (w) > 2. The weight of a double Roman dominating function f is the sum f(V)= n-ary sumation v is an element of Vf(v). The minimum weight of a double Roman dominating function on G is called the double Roman domination number of G and is denoted by gamma(dR)(G). In this paper, we establish a new upper bound on the double Roman domination number of graphs. We prove that every connected graph G with minimum degree at least two and G not equal C-5 satisfies the inequality gamma dR(G)<= L1311n. One open question posed by R. A. Beeler et al. has been settled.