On total domination vertex critical graphs of high connectivity

Abstract: A graph is called -critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is -critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A -critical graph that has total domination number , is called --critical. In this paper, we introduce a class of --critical graphs of high connectivity for each integer . In particular, we provide a partial answer to the question “Which graphs are -critical and -critical or one but not the other?” posed in a recent work [W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255–261]. [Copyright &y& Elsevier]