On total domination vertex critical graphs of high connectivity

A graph is called γ -critical if the removal of any vertex from the graph decreases the dominationnumber,while a graphwith no isolated vertex isγt -critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γt -critical graph that has total domination number k, is called k-γt -critical. In this paper, we introduce a class of k-γt -critical graphs of high connectivity for each integer k ≥ 3. In particular, we provide a partial answer to the question ‘‘Which graphs are γ -critical and γt -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]. © 2008 Elsevier B.V. All rights reserved.