English

Abstract

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 $$f(V) = \sum\nolimits_{v \in V} {f(v)} $$. The minimum weight of a double Roman dominating function on G is called the double Roman domination number of G and is denoted by γdR(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 ≠ C5 satisfies the inequality $${\gamma _{{\rm{dR}}}}(G)\leqslant \left\lfloor {{\textstyle{{13} \over {11}}}n} \right\rfloor $$. One open question posed by R. A. Beeler et al. has been settled.