An introduction to the twin signed total $k$-domination numbers in directed graphs

Atapour, Maryam; Dehgardi, Nasrin; Volkmann, L.

doi:10.1051/ro/2017036

An introduction to the twin signed total

k

-domination numbers in directed graphs

Atapour, Maryam ¹ ; Dehgardi, Nasrin ² ; Volkmann, L.³

¹ Department of Mathematics, Faculty of basic sciences, University of Bonab, Bonab, I.R. Iran.
² Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, I.R. Iran.
³ Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 4, pp. 1331-1343.

Résumé

Let $D = (V, A)$ be a finite simple directed graph (shortly digraph), $N^{-} (v)$ and $N^{+} (v)$ denote the set of in-neighbors and out-neighbors of a vertex $v \in V$ , respectively. A function $f : V - \to {- 1, 1}$ is called a twin signed total $k$ -dominating function (TST $k$ DF) if $\sum u \in (N^{-} (v)) f (u) \geq k$ and $\sum u \in (N^{+} (v)) f (u) \geq k$ for each vertex $v \in V$ . The twin signed total $k$ -domination number of $D$ is $γ_{s t k}^{*} (D) = m i n {ω (f) | f$ is a TST $k$ DF of $D}$ , where $ω (f) = \sum_{v \in V} f (v)$ is the weight of $f$ . In this paper, we initiate the study of twin signed total $k$ -domination in digraphs and present different bounds on $γ_{s t k}^{*} (D)$ . In addition, we determine the twin signed total $k$ -domination number of some classes of digraphs. Our results are mostly extensions of well-known bounds of the twin signed total domination numbers of directed graphs.

Reçu le : 2016-11-12
Accepté le : 2017-03-06

MR Zbl

DOI : 10.1051/ro/2017036

Classification : 05C69
Mots clés : Twin signed total k-dominating function, twin signed total k-domination number, directed graph

Affiliations des auteurs :

Atapour, Maryam ¹ ; Dehgardi, Nasrin ² ; Volkmann, L. ³

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     author = {Atapour, Maryam and Dehgardi, Nasrin and Volkmann, L.},
     title = {An introduction to the twin signed total $k$-domination numbers in directed graphs},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {1331--1343},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {4},
     year = {2017},
     doi = {10.1051/ro/2017036},
     mrnumber = {3783948},
     zbl = {1393.05195},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2017036/}
}

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Atapour, Maryam; Dehgardi, Nasrin; Volkmann, L. An introduction to the twin signed total $k$-domination numbers in directed graphs. RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 4, pp. 1331-1343. doi : 10.1051/ro/2017036. http://www.numdam.org/articles/10.1051/ro/2017036/

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