On total domination vertex critical graphs of high connectivity
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 ?t ? t -critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A ?t ? t -critical graph that has total domination numberk k , is called k k -?t ? t -critical. In this paper, we introduce a class of k k -?t ? t -critical graphs of high connectivity for each integer k?3 k ? 3 . In particular, we provide a partial answer to the question Which graphs are ? ? -critical and ?t ? 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) 255261].
total domination vertex critical connectivity diameter