On the total restrained domination number of direct products of graphs
Wai Chee Shiu ; Hong-Yu Chen ; Xue-Gang Chen ; Pak Kiu Sun
Discussiones Mathematicae Graph Theory, Tome 32 (2012), p. 629-641 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V∖S is adjacent to a vertex in S as well as to another vertex in V∖S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γrt(G), is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds are sharp by presenting some infinite families of graphs that attain these bounds.

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:270917 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1632,
     author = {Wai Chee Shiu and Hong-Yu Chen and Xue-Gang Chen and Pak Kiu Sun},
     title = {On the total restrained domination number of direct products of graphs},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {32},
     year = {2012},
     pages = {629-641},
     zbl = {1293.05277},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1632}
}

Wai Chee Shiu; Hong-Yu Chen; Xue-Gang Chen; Pak Kiu Sun. On the total restrained domination number of direct products of graphs. Discussiones Mathematicae Graph Theory, Tome 32 (2012) pp. 629-641. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1632/

[000] [1] R. Chérifi, S. Gravier, X. Lagraula, C. Payan and I. Zigham, Domination number of cross products of paths, Discrete Appl. Math. 94 (1999) 101-139, doi: 10.1016/S0166-218X(99)00016-5. | Zbl 0923.05031

[001] [2] X.G. Chen, W.C. Shiu and H.Y. Chen, Trees with equal total domination and total restrained domination numbers, Discuss. Math. Graph. Theory 28 (2008) 59-66, doi: 10.7151/dmgt.1391. | Zbl 1169.05359

[002] [3] M. El-Zahar, S. Gravier and A. Klobucar, On the total domination number of cross products of graphs, Discrete Math. 308 (2008) 2025-2029, doi: 10.1016/j.disc.2007.04.034. | Zbl 1168.05344

[003] [4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs ( Marcel Dekker, New York, 1998). | Zbl 0890.05002

[004] [5] D.X. Ma, X.G. Chen and L. Sun, On total restrained domination in graphs, Czechoslovak Math. J. 55 (2005) 165-173, doi: 10.1007/s10587-005-0012-2. | Zbl 1081.05086

[005] [6] D.F. Rall, Total domination in categorical products of graphs, Discuss. Math. Graph Theory 25 (2005) 35-44, doi: 10.7151/dmgt.1257. | Zbl 1074.05068

[006] [7] M. Zwierzchowski, Total domination number of the conjunction of graphs, Discrete Math. 307 (2007) 1016-1020, doi: 10.1016/j.disc.2005.11.047. | Zbl 1112.05083