On the diameter of a total domination critical graph

A graph G with no isolated vertex is total domination vertex critical if for any vertex υ of G that is not adjacent to a vertex of degree one, the total domination number of G − υ is less than the total domination number of G. These graphs we call γ_t-critical. If such a graph G has total domination number k, we call it k-γ_t-critical. We study some open problems of k-γ_t-critical graphs and obtain a new upper bound for the diameter of k-γ_t-critical graphs.