A connected graph with no redundant edges that still admits a dominating set disjoint from a paired-dominating set is exactly a 2-subdivision graph of a connected graph free of isolated vertices and good subgraphs, except for cycles of length 3, 6, and 9.
Disjoint dominating and 2-dominating sets in graphs
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abstract
A graph $G$ is a $D\!D_2$-graph if it has a pair $(D,D_2)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $D_2$ is a 2-dominating set of $G$. We provide several characterizations and hardness results concerning $D\!D_2$-graphs.
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2019 1verdicts
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Minimal graphs with disjoint dominating and paired-dominating sets
A connected graph with no redundant edges that still admits a dominating set disjoint from a paired-dominating set is exactly a 2-subdivision graph of a connected graph free of isolated vertices and good subgraphs, except for cycles of length 3, 6, and 9.