دوستعلی مژده

A double Roman dominating function on a graph $G$ is a function $ f:V\rightarrow \{0,1,2,3\}$ such that the following conditions hold. If $f(v)=0$, then vertex $v$ must have at least two neighbors in $V_2$ or one neighbor in $V_3$ and if $f(v)=1$, then vertex $v$ must have at least one neighbor in $V_2\cup V_3$. The weight of a double Roman dominating function is the sum $w_f=\sum_{v\in V(G)}{f(v)}$. A total double Roman dominating function $(TDRDF)$ on a graph $G$ with no isolated vertex is a $DRDF$ $f$ on $G$ with the additional property that the subgraph of $G$ induced by the set $\{v\in V :f(v)\neq0\}$ has no isolated vertices. The total double Roman domination number $\gamma_{tdR} (G)$ is the minimum weight of a $TDRDF$ on $G$. We initiate the improvement of the upper bounds of $\gamma_{dR}(G)$ and we show that $\gamma_{tdR}(G)\leq\dfrac{4n}{3}$, for any graph with $\delta(G) \ge 2$.