On total domination vertex critical graphs of high connectivity

Abstract

A graph is called γ -critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is γ t -critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γ t -critical graph that has total domination number k , is called k - γ t -critical. In this paper, we introduce a class of k - γ t -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 γ t -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].