On the Double Roman Domination in Spider Graphs

A double Roman dominating function (DRDF) $f$ on a given graph $G$ is a mapping from$V\left(G\right)$ to {0, 1, 2, 3} in such a way that a vertex $v$ for which $f\left(v\right)=0$ has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex $v$ for which $f\left(v\right)=1$ has at least a neighbor labeled 2 or 3. The weight of a DRDF $f$ is the value $w\left(f\right)={\displaystyle {\sum }_{v\in V\left(G\right)}f\left(v\right)}$ . The minimum weight of a DRDF on a graph $G$ is called the double Roman domination number of $G$ . In this paper, we determine the exact value of the double Roman domination number of the Spider graphs $S_{m,2}$ and $S_{m,3}$ , and obtain an upper bound of the Spider graphs $S_{}$