The diameter of total domination vertex 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. These graphs we call gamma(t)-critical. If such a graph G has total domination number k, we call it k-gamma(t)-critical. We characterize the connected graphs with minimum degree one that are gamma(t)-critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k-gamma(t)-critical graph for kless than or equal to8 and provide an example which shows that the maximum diameter is in general at least 5k/3 - O(1). (C) 2004 Elsevier B.V. All rights reserved.