Geodesic
Minimal graphs with disjoint dominating and total dominating sets
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 47-74

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A graph G is a DTDP-graph if it has a pair (D,T) of disjoint sets of vertices of G such that D is a dominating set and T is a total dominating set of G. Such graphs were studied in a number of research papers. In this paper we study further properties of DTDP-graphs and, in particular, we characterize minimal DTDP-graphs without loops.
Keywords: domination, total domination 
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Henning, Michael A.; Topp, Jerzy. Minimal graphs with disjoint dominating and total dominating sets. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 47-74. http://geodesic.mathdoc.fr/item/DMGT_2024_44_1_a3/