Double Roman Domination in Digraphs

Let D be a finite and simple digraph with vertex set V(D). A double Roman dominating function (DRDF) on a digraph D is a function $$f:V(D)\rightarrow \{0,1,2,3\}$$f:V(D)→{0,1,2,3} satisfying the condition that if $$f(v)=0$$f(v)=0, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3, while if $$f(v)=1$$f(v)=1, then the vertex v must have at least one in-neighbor assigned 2 or 3. The weight of a DRDF f is the sum $$\sum _{v\in V(D)}f(v)$$∑v∈V(D)f(v). The double Roman domination number of a digraph D is the minimum weight of a DRDF on D. In this paper, we initiate the study of the double Roman domination of digraphs, and we give several relations between the double Roman domination number of a digraph and other domination parameters such as Roman domination number, k-domination number and signed domination number. Moreover, various bounds on the double Roman domination number of a digraph are presented, and a Nordhaus–Gaddum type inequality for the parameter is also given.