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-vG-v is less than the total domination number of G. These graphs we call γtγt-critical. If such a graph G has total domination number k, we call it k-γtγt-critical. We characterize the connected graphs with minimum degree one that are γtγt-critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k-γtγt-critical graph for k⩽8k⩽8 and provide an example which shows that the maximum diameter is in general at least 5k/3-O(1)5k/3-O(1).