NP-hardness of language of graphs with $\alpha(G)=n/3$
1-IN-3SAT is the following problem: given a 3SAT instance, determine whether there is a truth assignment in which each clause contains exactly one true literal. It is known that 1-IN-3SAT is NP-complete.
Given an instance of 1-IN-3SAT, construct an instance of your problem in which each clause corresponds to a triangle, and contradicting literals are connected by edges. Details left to you.