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 gamma(1)-critical. If such a graph G has total domination number k, we call it k-gamma(1)-critical. We verify an open problem of k-gamma(1)-critical graphs and obtain some results on the characterization of total domination critical graphs of order n = Delta(G)(gamma(1)(G) - 1) + 1. (c) 2006 Elsevier GmbH. All rights reserved.