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.
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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.