NP-completeness results for partitioning a graph into total dominating sets

A total domatic k-partition of a graph is a partition of its vertex set into k subsets such that each intersects the open neighborhood of each vertex. The maximum k for which a total domatic k-partition exists is known as the total domatic number of a graph G, denoted by d:t(G). We extend considerably the known hardness results by showing it istextsc {NP} -complete to decide whether d:t(G)ge 3 where G is a bipartite planar graph of bounded maximum degree. Similarly, for every kge 3, it istextsc {NP} -complete to decide whether d:t(G)ge k, where G is a split graph or k-regular. In particular, these results complement recent combinatorial results regarding d:t(G) on some of these graph classes by showing that the known results are, in a sense, best possible. Finally, for general n-vertex graphs, we show the problem is solvable in 2^n n^{O(1)} time, and derive even faster algorithms for special graph classes.