ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2019: 0.755

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 26(1) (2006) 103-112
DOI: 10.7151/dmgt.1305


Paul Dorbec

UJF, ERTé Maths à Modeler, GéoD research group, Leibniz laboratory
46 av. Félix Viallet, 38031 Grenoble CEDEX, France

Sylvain Gravier

CNRS, ERTé Maths à Modeler, GéoD research group, Leibniz laboratory
46 av. Félix Viallet, 38031 Grenoble CEDEX, France

Sandi Klavžar

Department of Mathematics and Computer Science, PeF
University of Maribor
Koroska cesta 160, 2000 Maribor, Slovenia

Simon Spacapan

University of Maribor, FME
Smetanova 17, 2000 Maribor, Slovenia


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.

Keywords: direct product, total domination, k-tuple domination, open packing, domination.

2000 Mathematics Subject Classification: 05C69, 05C70.


[1] B. Bresar, S. Klavžar and D.F. Rall, Dominating direct products of graphs, submitted, 2004.
[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.
[3] F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000) 201-213.
[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.
[6] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (J. Wiley & Sons, New York, 2000).
[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.
[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.
[9] C.S. Liao and G.J. Chang, Algorithmic aspect of k-tuple domination in graphs, Taiwanese J. Math. 6 (2002) 415-420.
[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.
[11] D.F. Rall, Total domination in categorical products of graphs, Discuss. Math. Graph Theory 25 (2005) 35-44, doi: 10.7151/dmgt.1257.
[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.

Received 9 February 2005
Revised 15 July 2005