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 vv of G   that is not adjacent to a vertex of degree one, the total domination number of G-vG-v is less than the total domination number of G  . We call these graphs γtγt-critical. If such a graph G has total domination number k, we call it k  -γtγt-critical. We verify an open problem of k  -γtγt-critical graphs and obtain some results on the characterization of total domination critical graphs of order n=Δ(G)(γt(G)-1)+1n=Δ(G)(γt(G)-1)+1.