# Reducing Vertex Cover (or Independent Set) to Vertex Cover and Independent Set at the same time

In order to show that the next problem is NP-hard:

Problem: Vertex Cover and Independent Set

Input: Graph G and integer k

Output: Does G have a vertex cover of k and an independent set of k?

The sets does not have to be complementary (just about size k). I have tried to reduce Vertex Cover but I cannot find how to ensure that when there is a vertex cover in G, the problem above outputs true and deal with the independent set of the same size k. I have thought of doing something with the fact that vertex cover is the complement of independent set but since there is the same value for k I have no idea for that.

If you want a reduction from Vertex Cover: given $$(G, k)$$ an input of VC, create a graph $$G'$$ with a copy of $$G$$ and an additionnal $$k$$ vertices with no additionnal edge.
Then $$G'$$ has a vertex cover and an independant set of size $$k$$ if and only if $$G$$ has a vertex cover of size $$k$$.