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

@article{RO_2017__51_4_1331_0,
     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://archive.numdam.org/articles/10.1051/ro/2017036/}
}

TY  - JOUR
AU  - Atapour, Maryam
AU  - Dehgardi, Nasrin
AU  - Volkmann, L.
TI  - An introduction to the twin signed total $k$-domination numbers in directed graphs
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2017
SP  - 1331
EP  - 1343
VL  - 51
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ro/2017036/
DO  - 10.1051/ro/2017036
LA  - en
ID  - RO_2017__51_4_1331_0
ER  -

%0 Journal Article
%A Atapour, Maryam
%A Dehgardi, Nasrin
%A Volkmann, L.
%T An introduction to the twin signed total $k$-domination numbers in directed graphs
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2017
%P 1331-1343
%V 51
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ro/2017036/
%R 10.1051/ro/2017036
%G en
%F RO_2017__51_4_1331_0

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://archive.numdam.org/articles/10.1051/ro/2017036/

Bibliographie
Cité par

M. Atapour, A. Bodaghli and S.M. Sheikholeslami, Twin signed total domination numbers in directed graphs, Ars Combin. (to appear). | MR

M. Atapour and A. Khodkar, Twin minus domination numbers in directed graphs, Commu. Combinat. Optimiz. 1 (2016) 149–164. | MR | Zbl

M. Atapour, S. Norouzian, S.M. Sheikholeslami and L. Volkmann, Twin signed domination numbers in directed graphs. Algebra and Discrete Math. (to appear). | MR

M. Atapour, S.M. Sheikholeslami, R. Hajypory and L. Volkmann, The signed k-domination number of directed graphs. Cent. Eur. J. Math. 8 (2010) 1048–1057. | DOI | MR | Zbl

A. Bodaghli, S.M. Sheikholeslami and L. Volkmann, Twin signed Roman domination number of a digraph. Tamkang J. Math. (to appear). | MR

G. Chartrand, P. Dankelmann, M. Schultz and H.C. Swart, Twin domination in digraphs. Ars Combin. 67 (2003) 105–114. | MR | Zbl

G. Chartrand, D. W. Vanderjagt and B.Q. Yue, Orientable domination in graphs. Congr. Numer. 119 (1996) 51–63. | MR | Zbl

M.A. Henning, Signed total domination in graphs. Discrete Math. 278 (2004) 109–125. | DOI | MR | Zbl

A. Khodkar and S.M. Sheikholeslami, Signed total $k$ -domatic numbers of graphs. J. Korean Math. Soc. 48 (2011) 551–563. | DOI | MR | Zbl

S.M. Sheikholeslami, Signed total domination numbers of directed graphs. Util. Math. 85 (2011) 273–279. | MR | Zbl

S.M. Sheikholeslami and L. Volkmann, Signed total k-domination numbers of directed graphs. An. St. Univ. Ovidius Constanta. 18 (2010) 241–252. | MR | Zbl

L. Volkmann, Lower bounds on the signed total k-domination number of graphs. Aequationes Math. (to appear). | MR

C.P. Wang, The signed k-domination numbers in graphs. Ars Combin. 106 (2012) 205–211. | MR | Zbl

D.B. West, Introduction to Graph Theory, Prentice-Hall, Inc (2000). | MR

B. Zelinka, Signed total domination number of a graph. Czechoslovak Math. J. 51 (2001) 225–229. | DOI | MR | Zbl

Cité par Sources :