Total domination critical graphs with respect to relative complements

A set S of vertices of a graph G is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number gamma(t)(G) is the minimum cardinality of a total dominating set of G. Let G be a spanning subgraph of K-s,K-s and let H be the complement of G relative to K-s,K-s; that is, K-s,K-s = G circle plus H is a factorization of K-s,K-s. The graph G is k(t)-critical relative to K-s,K-s if gamma(t)(G) = k and gamma(t)(G + e) < k for all e is an element of E(H). We study kt-critical graphs relative to K-s,K-s for small values of k. In particular, we characterize the 3(t)-critical and 4(t)-critical graphs.