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

-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

-complete to decide whether $d_t(G)\ge k$, where G is split 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.

中文翻译：

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NP-完全性结果，用于将图划分为总支配集

图的总半球形k分区是将其顶点集划分为k个子集，以便每个子集与每个顶点的开放邻域相交。存在总局部k分区的最大值k被称为图G的总局部数，表示为$d_{Ť}（G）$。通过显示，我们大大扩展了已知的硬度结果

-完成决定是否 $d_{Ť}（G）\ge 3$其中G是有界最大度的二部平面图。同样，对于每个$ķ\ge 3$， 它是

-完成决定是否 $d_{Ť}（G）\ge ķ$，其中G为分裂或k正则。特别是，这些结果补充了有关以下方面的最新组合结果：$d_{Ť}（G）$从某种意义上说，从某种意义上说，最好的结果就是对这些图类的使用。最后，对于一般的n-顶点图，我们表明问题可以解决$2^{ñ}{ñ}^{Ø（\mathrm{1个}）}$ 时间，并为特殊的图类推导甚至更快的算法。