On an open problem concerning 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. We call these graphs γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We verify an open problem of k-γt-critical graphs and obtain some results on the characterization of total domination critical graphs of order n=Δ(G)(γt(G)-1)+1.