On the total domination critical graphs

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G ∖ { v } 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 verify an open problem of k- γ t -critical graphs and obtain some results on the characterization of total domination critical graphs of order Δ ( G ) + γ t ( G ) .