Let $G$ be a graph and define

$\mathscr{I}(G) = \{S \subset V(G)| S$ is a maximal indepedent set of $ G\}$

1. What is known about $\mathscr{I}(G)$?

2. What are some of the properties of $\mathscr{I}(G)$?

3. How does $\mathscr{I}(G)$ relates to other properties of $G$ for example chromatic number?

4. Is it possible to decide if a collection $\mathscr{A}$ is equal to $\mathscr{I}(H)$ for some graph $H$?