Q:

Poly-time reduction: D and D Comp

Looking at the Independent Set problem and its complement, I want to show that IS is poly-time reducible to its complement, however I am struggling on coming up with the reduction function.
I will define its complement for further clarity, does every subset such that its size is at least $k$ in $V$ contain at least one edge between its vertices? My intuition is below.
$$f(G, k) = (G, i-k),$$
where $i = |V|$? 

A:

Independent set and its complement are NP-complete and co-NP-complete, respectively. If you found a polynomial-time reduction between those two problems, you'd have proven that NP$\,=\,$co-NP, resolving one of the biggest open problems in theoretical CS.

