Disjoint dominating and total dominating sets in graphs

It has been shown [M.A. Henning, J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159–162] that every connected graph with minimum degree at least two that is not a cycle on five vertices has a dominating set DD and a total dominating set TT which are disjoint. We characterize such graphs for which D∪TD∪T necessarily contains all vertices of the graph and that have no induced cycle on five vertices.