This is exactly the minimum vertex cover problem which is known to be NP-complete. The key insight in seeing that computing the size of a minimum vertex cover is equivalent to computing the size of a maximum independent set is the following:
A set of vertices is a vertex cover, if and only if its complement is an independent set.
In particular, this means that the total number of vertices is equal to the size of a minimum vertex cover plus the size of a maximum independent set. This illustrates nicely how computing one number reduces to computing the other.