Dominating and total dominating partitions in cubic graphs

In this paper, we continue the study of domination and total domination in cubic graphs. It is known [Henning M. A., Southey J., A note on graphs with disjoint dominating and total dominating sets, Ars Combin., 2008, 89, 159-162] that every cubic graph has a dominating set and a total dominating set which are disjoint. In this paper we show that every connected cubic graph on n vertices has a total dominating set whose complement contains a dominating set such that the cardinality of the total dominating set is at most (n + 2)/2, and this bound is essentially best possible.