In this paper we show that for each sufficiently large n there exist graphs G of order n and diameter 2 whose total domination number \(\gamma _t(G)\) is greater than \(\sqrt{(3n\log n)/8}-\sqrt{n}\). On the other hand, it is shown that the total domination number of a graph of order \(n \geqslant 3\) and diameter 2 is always less than \(\sqrt{(n\log n)/2}+\sqrt{n/2}\).

中文翻译：

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直径2和大控制数的图形

在本文中，我们表明，对于每个足够大的n，都存在阶数为n且直径为2的图G，其总支配数\（\ gamma _t（G）\）大于\（\ sqrt {（3n \ log n）/ 8}-\ sqrt {n} \）。另一方面，表明阶为\（n \ geqslant 3 \）和直径2的图的总支配数始终小于\（\ sqrt {（n \ log n）/ 2} + \ sqrt {n / 2} \）。