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 is \(\textsc {NP}\)-complete to decide whether \(d_t(G) \ge 3\) where G is a bipartite planar graph of bounded maximum degree. Similarly, for every \(k \ge 3\), it is \(\textsc {NP}\)-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.