Geodesic
Minimal Graphs with Disjoint Dominating and Paired-Dominating Sets
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 3, pp. 827-847

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A subset D ⊆ VG is a dominating set of G if every vertex in VG – D has a neighbor in D, while D is a paired-dominating set of G if D is a dominating set and the subgraph induced by D contains a perfect matching. A graph G is a DPDP -graph if it has a pair (D, P) of disjoint sets of vertices of G such that D is a dominating set and P is a paired-dominating set of G. The study of the DPDP -graphs was initiated by Southey and Henning [Cent. Eur. J. Math. 8 (2010) 459–467; J. Comb. Optim. 22 (2011) 217–234]. In this paper, we provide conditions which ensure that a graph is a DPDP -graph. In particular, we characterize the minimal DPDP -graphs.
Keywords: domination, paired-domination 
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     title = {Minimal {Graphs} with {Disjoint} {Dominating} and {Paired-Dominating} {Sets}},
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Henning, Michael A.; Topp, Jerzy. Minimal Graphs with Disjoint Dominating and Paired-Dominating Sets. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 3, pp. 827-847. http://geodesic.mathdoc.fr/item/DMGT_2021_41_3_a8/