Abstract
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 D and a total dominating set T which are disjoint. We characterize such graphs for which D ∪ T necessarily contains all vertices of the graph and that have no induced cycle on five vertices.
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