Geodesic
Some results on total domination in direct products of graphs
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 103-112

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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 
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Dorbec, Paul; Gravier, Sylvain; Klavžar, Sandi; Spacapan, Simon. Some results on total domination in direct products of graphs. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 103-112. http://geodesic.mathdoc.fr/item/DMGT_2006_26_1_a9/