A subset X of V(G) is a dominating set of G if for every v is not an element of (V(G)\X), there exists x is an element of X such that xv is an element of E(G), that is, N[X] = V(G). It is a total dominating set if N(X) = V(G). A dominating set S of V(G) is a weakly connected dominating set of G if the subgraph < S >(w) = (V(G)E-w) weakly induced by S is connected. A total dominating set S of V(G) is a weakly connected total dominating set of G if < S >(w) = (V(G), E-w) is connected. The weakly connected domination number gamma(w)(G) (weakly connected total domination number gamma(wt)(G)) 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, gamma(wt)-critical if for each x, y is an element of V(G) with x not adjacent to y, gamma(wt)(G + xy) < gamma(wt)(G). Hence, G is k-gamma(wt)-critical if gamma(wt)(G) = k and for each xy is not an element of E(G), gamma(wt) (G + xy) < k. In this paper, we characterize weakly connected total domination critical graphs and give some classes of graphs which are weakly connected total domination critical.
