Relating total double Roman domination to 2-independence in trees Relating total double Roman domination to 2-independence in trees

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A {\em double Roman dominating function} (DRDF) on a graph $G=(V, E)$ is a function $f:V\rightarrow \{0,1,2,3\}$ having the property that if $f(u)=0$, then vertex $u$ has at least two neighbors assigned $2$ under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(u)=1$, then vertex $u$ must have at least one neighbor $w$ with $f(w)\ge 2$. The 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 $\{v\in V: f(v)\neq 0\}$ has no isolated vertices. The weight of a total double Roman dominating function $f$ is the value, $f(V)=\Sigma_{u\in V(G)}f(u)$. The {\em total double Roman domination number} $\gamma_{tdR}(G)$ is the minimum weight of a TDRDF on $G$.A subset $S$ of $V$ is a 2-independent set of $G$ if every vertex of $S$ has at most one neighbor in $S$. The maximum cardinality of a 2-independent set of $G$ is the 2-independence number $\beta_2(G)$. In this paper, we show that if $T$ is a tree, then $\gamma_{tdR}(T)\le 2\beta_2(T)$ and we characterize all trees attaining the equality.