Let $G=(V,E)$ be a graph and $\mathrm{Ind}(G)$ be the collection of its independent sets. 


We call a graph automorphism $f:V \to V$ of $G$ *good* if it is non-trivial and $f(\mathrm{Ind}(G))=\mathrm{Ind}(G)$, i.e. $f$ maps every independent set of $G$ into another independent set of $G$.


(i) Is there a (elegant/minimal) set of conditions under on the graph $G$ that guarantees the existence of a *good* automorphism? 


(ii) Is there any paper in the mathematical literature studying this class of *good* automorphisms?



Graph automorphisms that preserve independent sets
I thought that if $f$ is graph automorphism then it preserves both edges and non-edges. In that case, any automorphism would be good, and as @Turbo said, some graphs have trivial autormorphism groups only. See <https://en.wikipedia.org/wiki/Graph_automorphism>


