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A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in$V_G-D$ 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 perfectmatching. A graph $G$ is a $D\!P\!D\!P$-graph if it has a pair $(D,P)$ ofdisjoint sets of vertices of $G$ such that $D$ is a dominating set and $P$ is apaired-dominating set of $G$. The study of the $D\!P\!D\!P$-graphs wasinitiated 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 whichensure that a graph is a $D\!P\!D\!P$-graph. In particular, we characterize theminimal $D\!P\!D\!P$-graphs.

Minimal graphs with disjoint dominating and paired-dominating sets