Geodesic
Twin signed domination numbers in directed graphs
Algebra and discrete mathematics, Tome 24 (2017) no. 1, pp. 71-89

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Let $D=(V,A)$ be a finite simple directed graph (shortly digraph). A function $f\colon V\to \{-1,1\}$ is called a twin signed dominating function (TSDF) if $f(N^-[v])\ge 1$ and $f(N^+[v])\ge 1$ for each vertex $v\in V$. The twin signed domination number of $D$ is $\gamma_{s}^*(D)=\min\{\omega(f)\mid f \text{ is a TSDF of } D\}$. In this paper, we initiate the study of twin signed domination in digraphs and we present sharp lower bounds for $\gamma_{s}^*(D)$ in terms of the order, size and maximum and minimum indegrees and outdegrees. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs.
Keywords: twin signed dominating function, twin signed domination number, directed graph. 
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     title = {Twin signed domination numbers in directed graphs},
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     publisher = {mathdoc},
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     number = {1},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2017_24_1_a4/}
}
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M. Atapour; S. Norouzian; S. M. Sheikholeslami; L. Volkmann. Twin signed domination numbers in directed graphs. Algebra and discrete mathematics, Tome 24 (2017) no. 1, pp. 71-89. http://geodesic.mathdoc.fr/item/ADM_2017_24_1_a4/