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