Theoretical Computer Science ( IF 0.9 ) Pub Date : 2018-08-14 , DOI: 10.1016/j.tcs.2018.04.006 Mikko Koivisto , Petteri Laakkonen , Juho Lauri
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 . We extend considerably the known hardness results by showing it is -complete to decide whether where G is a bipartite planar graph of bounded maximum degree. Similarly, for every , it is -complete to decide whether , where G is split or k-regular. In particular, these results complement recent combinatorial results regarding 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 time, and derive even faster algorithms for special graph classes.
中文翻译:
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。通过显示,我们大大扩展了已知的硬度结果
-完成决定是否 其中G是有界最大度的二部平面图。同样,对于每个, 它是 -完成决定是否 ,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: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 时间,并为特殊的图类推导甚至更快的算法。