NP-completeness results for partitioning a graph into total dominating sets
Research output: Contribution to journal › Article › Scientific › peer-review
|Journal||Theoretical Computer Science|
|Publication status||E-pub ahead of print - 1 Jan 2018|
|Publication type||A1 Journal article-refereed|
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 dt(G). We extend considerably the known hardness results by showing it is[Formula presented]-complete to decide whether dt(G)≥3 where G is a bipartite planar graph of bounded maximum degree. Similarly, for every k≥3, it is[Formula presented]-complete to decide whether dt(G)≥k, where G is split or k-regular. In particular, these results complement recent combinatorial results regarding dt(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 2nnO(1) time, and derive even faster algorithms for special graph classes.