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چکیده
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A subset $S$ of vertices of a digraph $D$ is a total $2$-dominating set if every vertex not in $S$ is adjacent from at least two vertices in $S$, and the subdigraph induced by $S$ has no isolated vertices. Let $D^{-1}$ be a digraph obtained by reversing the direction of every arc of $D$. In this work, we investigate this concept which can be considered as an extension of double domination in graphs $G$ to digraphs $D$, along with the concept total $2$-limited packing ($L^{t}_{2}(D)$) of digraphs $D$ which has close relationships with the above-mentioned concept. We prove that the problems of computing these parameters are NP-hard, even when the digraph is bipartite. We give several lower and upper bounds on these parameters. In dealing with these two parameters our main emphasis is on directed trees, by which we prove that $L^{t}_{2}(D)+L^{t}_{2}(D^{-1})$ can be bounded from above by $16n/9$ for any digraph $D$ of order $n$. Also, we bound the total $2$-domination number of a directed tree from below and characterize the directed trees attaining the bound.
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