Restrained double Roman domination of a graph

For a graph G=(V,E), a restrained double Roman dominating function is a function f:V\rightarrow\{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 w with f(w)\geq2, and at the same time, the subgraph G[V_0] which includes vertices with zero labels has no isolated vertex. The weight of a restrained double Roman dominating function f is the sum f(V)=\sum_{v\in V}f(v), and the minimum weight of a restrained double Roman dominating function on G is the restrained double Roman domination number of G. We initiate the study of restrained double Roman domination with proving that the problem of computing this parameter is NP-hard. Then we present an upper bound on the restrained double Roman domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. We study the restrained double Roman domination versus the restrained Roman domination. Finally, we characterized all trees T attaining the exhibited bound.