Geodesic
On Total Domination in the Cartesian Product of Graphs
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 4, pp. 963-976

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Ho proved in [A note on the total domination number, Util. Math. 77 (2008) 97–100] that the total domination number of the Cartesian product of any two graphs without isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Lu and Hou from [Total domination in the Cartesian product of a graph and K_2 or C_n, Util. Math. 83 (2010) 313–322] by characterizing the pairs of graphs G and H for which γ_t (G □ H)=1/2 γ_t (G) γ_t (H), whenever γ_t (H) = 2. In addition, we present an infinite family of graphs G_n with γ_t (G_n) = 2n, which asymptotically approximate equality in γ_t (G_n □ H_n ) ≥ 1/2 γ_t (G_n)^2.
Keywords: total domination, Cartesian product, total domination quotient 
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Brešar, Boštjan; Hartinger, Tatiana Romina; Kos, Tim; Milanič, Martin. On Total Domination in the Cartesian Product of Graphs. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 4, pp. 963-976. http://geodesic.mathdoc.fr/item/DMGT_2018_38_4_a6/