Geodesic
Twin Minus Total Domination Numbers In Directed Graphs
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 989-1004.

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Let D = (V,A) be a finite simple directed graph (shortly, digraph). A function f : V →−1, 0, 1 is called a twin minus total dominating function (TMTDF) if f(N^−(v)) ≥ 1 and f(N^+(v)) ≥ 1 for each vertex v ∈ V. The twin minus total domination number of D is γ_mt^∗ (D) = min{ w(f) | f is a TMTDF of D }. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for γ_mt^∗ (D) in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs.
Keywords: twin minus total dominating function, twin minus total domination number, directed graph 
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Dehgardi, Nasrin; Atapour, Maryam. Twin Minus Total Domination Numbers In Directed Graphs. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 989-1004. http://geodesic.mathdoc.fr/item/DMGT_2017_37_4_a9/

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