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

Atapour, Maryam¹ ; Dehgardi, Nasrin² ; 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), $N^{-}\left(v\right)$ and $N^{+}\left(v\right)$ 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 $\sum u\in \left(N^{-}\left(v\right)\right)f\left(u\right)\ge k$ and $\sum u\in \left(N^{+}\left(v\right)\right)f\left(u\right)\ge k$ for each vertex $v\in V$. The twin signed total $k$-domination number of $D$ is $\gamma {}_{stk}^{*}\left(D\right)=min\left\{\omega \left(f\right)\right|f$ is a TST$k$DF of $D\}$, where $\omega \left(f\right)={\sum }_{v\in V}f\left(v\right)$ 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 $\gamma {}_{stk}^{*}\left(D\right)$. 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.

Détail

Reçu le : 2016-11-12
Accepté le : 2017-03-06
DOI : 10.1051/ro/2017036

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