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](https://people.cs.clemson.edu/~goddard/papers/idomSurvey.pdf), it is also known that $\gamma(G) = i(G)$ for [Claw-free Graphs](https://en.wikipedia.org/wiki/Claw-free_graph) and $i(G) = \alpha(G)$ for [Block Graphs](https://en.wikipedia.org/wiki/Block_graph). 


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? 



Domination Number equals Independence Number?
Since $\gamma(G)\leq i(G)\leq \alpha(G)$ for all graphs $G$, the problem of characterizing all graphs $G$ for which $\gamma(G)=i(G)=\alpha(G)$ is equivalent to the characterizing all graphs $G$ for which $\gamma(G)=\alpha(G)$. The theorem "$\gamma(G)=i(G)$ holds for all claw-free graphs $G$" was proved by Bollobas and Cockayne in


"B. Bollobas and E.J. Cockayne, Graph-theoretic parameters concerning domination, independence, and irredundance, J. Graph Theory" 3 (1979), 241-249.


However, the problem of finding the family of all graphs $G$ for which $\gamma(G)=\alpha(G)$ is still open. 


