Abstract
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 \(\sqrt{(n\log n)/2}+\sqrt{n/2}\).
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I thank the referees for suggesting some improvements.
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Dubickas, A. Graphs with Diameter 2 and Large Total Domination Number. Graphs and Combinatorics 37, 271–279 (2021). https://doi.org/10.1007/s00373-020-02245-x
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DOI: https://doi.org/10.1007/s00373-020-02245-x