Modified Clique Problem
The problem is equivalent to deciding whether the complement graph of $G$ has a bipartition $(A,B)$ such that $|A|=|B|$. In order to check if such a bipartition exists, one can first compute an arbitrary bipartition $(A,B)$. If no such bipartition exists, we are finished. If it does exists, we must check whether we can flip single connected components of this bipartition (which mean all vertices in $A$ of a particular connected component are moved to $B$ and vice versa) such that we obtain a new partition where $A$ is equal to $B$. Using a dynamic programing approach similar to the partition problem, this can be decided in polynomial time. Thus the original problem is also in P and not NP-hard unless P equals NP.