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

The 2-domination number $\gamma_2(G)$ 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, $\gamma_2(G)$ cannot be smaller than the domination number $\gamma(G)$. We consider a large class of graphs and characterize those members which satisfy $\gamma_2=\gamma$. For the general case, we prove that it is NP-hard to decide whether $\gamma_2=\gamma$ holds. We also give a necessary and sufficient condition for a graph to satisfy the equality hereditarily.