NP-hardness of language of graphs with $\alpha(G)=n/3$ How to prove the NP-hardness of the language $\{G \mid \alpha(G) =\frac{1}{3} |V(G)|\}$?
Here $G$ is a graph, $V(G)$ is its vertex set, and $\alpha(G)$ is the independence number of $G$, which is the maximum size of an independent set.
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.