Let \(G\) be a graph of order \(p\) such that both \(G\) and \(\overline{G}\) have no isolated vertices. Let \(\Upsilon_t\) and \(\overline{\Upsilon}_t\) denote respectively the total domination number of \(G\) and \(\overline{G}\). In this paper we obtain a characterization of all graphs \(G\) for which \\(i) \(\Upsilon_t +\overline{\Upsilon}_t= p+1\) and (ii) \(\Upsilon_t + \overline{\Upsilon}_t = p\).