Abstract :
A double Roman dominating function on a graph G with vertex set V(G) is defined in cite{bhh} as a function f:V(G)→{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 w with f(w)=3, and if f(v)=1, then the vertex v must have at least one neighbor u with f(u)≥2. The weight of a double Roman dominating function f is the sum ∑v∈V(G)f(v), and the minimum weight of a double Roman dominating function on G is the double Roman domination number γdR(G) of G. A set {f1,f2,…,fd} of distinct double Roman dominating functions on G with the property that ∑di=1fi(v)≤3 for each v∈V(G) is called a double Roman dominating family (of functions) on G. The maximum number of functions in a double Roman dominating family on G is the double Roman domatic number of G. In this note we continue the study the double Roman domination and domatic numbers. In particular, we present a sharp lower bound on γdR(G), and we determine the double Roman domination and domatic numbers of some classes of graphs.
