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On the equality of domination number and $ 2 $-domination number

The 2-domination number $γ_2(G)$ of a graph $G$ is the minimum cardinality of a set $ D \subseteq V(G) $ for which every vertex outside $ D $ is adjacent to at least two vertices in $ D $. Clearly, $ γ_2(G) $ cannot be smaller than the domination number $ γ(G) $. We consider a large class of graphs and characterize those members which satisfy $γ_2=γ$. For the general case, we prove that it is NP-hard to decide whether $γ_2=γ$ holds. We also give a necessary and sufficient condition for a graph to satisfy the equality hereditarily.