دوستعلی مژده

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 investigate the bounds for the restrained double Roman domination of trees and determine trees $T$ attaining the exhibited bounds.