Total domination supercritical 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 γt(G) is the minimum cardinality of a total dominating set of G. Let G be a connected spanning subgraph of Ks,s, and let H be the complement of G relative to Ks,s; that is, Ks,s=G⊕H is a factorization of Ks,s. The graph G is k-supercritical relative to Ks,s if γt(G)=k and γt(G+e)=k−2 for all e∈E(H). Properties of k-supercritical graphs are presented, and k-supercritical graphs are characterized for small k.