Let $\mathcal G_n$ be the set of (isomorphism classes of unlabelled) simple graphs on $n$ vertices.

I am interested in specific bijective maps $\mathcal G_n\to\mathcal G_n$, defined for all $n$. An example for what I have in mind would be the graph complement. Another example is to decompose the graph into connected components, and complement those components whose complement is connected.

Note that an easier variant of the same question is obtained by dropping the requirement that the map be bijective. Various graph powers then serve as examples.

This is a follow-up question on equidistributed parameters on graphs: for any map (bijective or not), we can check whether it affects various graph parameters in interesting ways.