A subset X of V(G) is a dominating set of G if for every  there exists  such that  that is,  It is a total dominating set if  A dominating set S of V(G) is a weakly connected dominating set of G if the subgraph  weakly induced by S is connected. A total dominating set S of V(G) is a weakly connected total dominating set of G if is connected. The weakly connected domination number (weakly connected  total domination number   of G is the smallest cardinality of a weakly connected dominating (resp., weakly connected total dominating) set of G. A graph is said to be weakly connected total domination critical, -critical if for each  with x not adjacent to y, Hence, G is k--critical if and for each

In this paper, we characterize weakly connected total domination critical graphs and give some classes of graphs which are weakly connected total domination critical.

domination, weakly connected total domination, critical graphs, networks.