Graphs with Diameter 2 and Large Total Domination Number

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 $$\sqrt{(n\log n)/2}+\sqrt{n/2}$$ .