  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 perfect
matching. A graph $G$ is a $D\!P\!D\!P$-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 $D\!P\!D\!P$-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 $D\!P\!D\!P$-graph. In particular, we characterize the
minimal $D\!P\!D\!P$-graphs.

Combinatorics