# Some results on total domination in direct products of graphs

Paul Dorbec; Sylvain Gravier; Sandi Klavžar; Simon Spacapan

Discussiones Mathematicae Graph Theory (2006)

- Volume: 26, Issue: 1, page 103-112
- ISSN: 2083-5892

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topPaul Dorbec, et al. "Some results on total domination in direct products of graphs." Discussiones Mathematicae Graph Theory 26.1 (2006): 103-112. <http://eudml.org/doc/270561>.

@article{PaulDorbec2006,

abstract = {Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the \{2\}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direct products of graphs is also bounded from below.},

author = {Paul Dorbec, Sylvain Gravier, Sandi Klavžar, Simon Spacapan},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {direct product; total domination; k-tuple domination; open packing; domination; bound; domination number},

language = {eng},

number = {1},

pages = {103-112},

title = {Some results on total domination in direct products of graphs},

url = {http://eudml.org/doc/270561},

volume = {26},

year = {2006},

}

TY - JOUR

AU - Paul Dorbec

AU - Sylvain Gravier

AU - Sandi Klavžar

AU - Simon Spacapan

TI - Some results on total domination in direct products of graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2006

VL - 26

IS - 1

SP - 103

EP - 112

AB - Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direct products of graphs is also bounded from below.

LA - eng

KW - direct product; total domination; k-tuple domination; open packing; domination; bound; domination number

UR - http://eudml.org/doc/270561

ER -

## References

top- [1] B. Bresar, S. Klavžar and D.F. Rall, Dominating direct products of graphs, submitted, 2004. Zbl1116.05055
- [2] M. El-Zahar, S. Gravier and A. Klobucar, On the total domination of cross products of graphs, Les Cahiers du laboratoire Leibniz, No. 97, January 2004. Zbl1168.05344
- [3] F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000) 201-213. Zbl0993.05104
- [4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998).
- [5] W. Imrich, Factoring cardinal product graphs in polynomial time, Discrete Math. 192 (1998) 119-144, doi: 10.1016/S0012-365X(98)00069-7. Zbl0955.68089
- [6] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (J. Wiley & Sons, New York, 2000). Zbl0963.05002
- [7] P.K. Jha, S. Klavžar and B. Zmazek, Isomorphic components of Kronecker product of bipartite graphs, Discuss. Math. Graph Theory 17 (1997) 301-309, doi: 10.7151/dmgt.1057. Zbl0906.05050
- [8] R. Klasing and C. Laforest, Hardness results and approximation algorithms of k-tuple domination in graphs, Inform. Process. Lett. 89 (2004) 75-83, doi: 10.1016/j.ipl.2003.10.004. Zbl1178.68682
- [9] C.S. Liao and G.J. Chang, Algorithmic aspect of k-tuple domination in graphs, Taiwanese J. Math. 6 (2002) 415-420. Zbl1047.05032
- [10] R. Nowakowski and D. F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53-79, doi: 10.7151/dmgt.1023. Zbl0865.05071
- [11] D.F. Rall, Total domination in categorical products of graphs, Discuss. Math. Graph Theory 25 (2005) 35-44, doi: 10.7151/dmgt.1257. Zbl1074.05068
- [12] P.M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962) 47-52, doi: 10.1090/S0002-9939-1962-0133816-6. Zbl0102.38801

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