On total domination in the Cartesian product of graphs
  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 with no 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 $\gamma_t(G\Box H)=\frac{1}{2}\gamma_t(G) \gamma_t(H)\,$, whenever $\gamma_t(H)=2$. In addition, we present an infinite family of graphs $G_n$ with $\gamma_t(G_n)=2n$, which asymptotically approximate the equality in $\gamma_t(G_n\Box G_n)\ge \frac{1}{2}\gamma_t(G_n)^2$. 
Bo\v{s}tjan Bre\v{s}ar, Tatiana Romina Hartinger, Tim Kos, Martin   Milani\v{c}
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