DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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

SOME RESULTS ON TOTAL DOMINATION IN DIRECT
PRODUCTS OF GRAPHS

Paul Dorbec

UJF, ERTé Maths à Modeler, GéoD research group, Leibniz laboratory
46 av. Félix Viallet, 38031 Grenoble CEDEX, France
e-mail: paul.dorbec@imag.fr

Sylvain Gravier

CNRS, ERTé Maths à Modeler, GéoD research group, Leibniz laboratory
46 av. Félix Viallet, 38031 Grenoble CEDEX, France
e-mail: sylvain.gravier@imag.fr

Sandi Klavžar

Department of Mathematics and Computer Science, PeF
University of Maribor
Koroska cesta 160, 2000 Maribor, Slovenia
e-mail: sandi.klavzar@uni-mb.si

Simon Spacapan

University of Maribor, FME
Smetanova 17, 2000 Maribor, Slovenia
e-mail: simon.spacapan@uni-mb.si

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.

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

2000 Mathematics Subject Classification: 05C69, 05C70.

References

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Received 9 February 2005
Revised 15 July 2005