Let $\gamma(G)$ and $\alpha(G)$ be the domination number and independence number for an Graph $G$. Further, let $i(G)$ be the minimum-size Independent Dominating Set. Then, it is known that $\forall G, \gamma(G) \leq i(G) \leq \alpha(G)$.

According to this paper, it is also known that $\gamma(G) = i(G)$ for Claw-free Graphs and $i(G) = \alpha(G)$ for Block Graphs.

What is the most general family of graphs for which $\gamma(G)=i(G)=\alpha(G)$? Further, what is the most general family of graphs for which $\frac{\alpha(G)}{c}\leq\gamma(G)\leq \alpha(G)$ for a constant c >1?