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 this work we establish general properties of \(k\)-total bondage and find exact values for certain graph classes including paths, cycles, wheels, complete and complete bipartite graphs.