The $k$-Total Bondage Number of a Graph

Let (G=(V,E)) be a connected, finite undirected graph. A set (S \subseteq V) is said to be a total dominating set of (G) if every vertex in (V) is adjacent to some vertex in (S). The total domination number, (\gamma_{t}(G)), is the minimum cardinality of a total dominating set in (G). We define the (k)-total bondage of $G$ to be the minimum number of edges to remove from (G) so that the resulting graph has a total domination number at least (k) more than (\gamma_{t}(G)). In