Is Maximum Independent Set in coNP I need to determine whether the following problem $X$ is in coNP:


  Given a graph $ G=(V,E) $ and a positive integer $s\leq|V| $, is there an independent set that is the largest for $G$ of size at least $s$?


My solution for this was that in fact the above problem is in coNP because we can provide an NP solution for $\overline{X}$ which has the following certificate and certifier.

The certificate being a set of vertices $V'$.

The certifier does two things:


Checks if  $V' \subseteq V$. 
Checks for each pair of vertices in $V'$ if there exists no edge. 
Checks if  $|V'|&lt;s$.


The certifier computes this in $O(V+E)+O(1)$. 

Is this the right way to show if X is in coNP?

Edit: The independent set is required to be the largest possible for graph $G$.
Independent set is NP-complete, so it's unlikely to be in coNP.

Moreover, the complement problem would sound like "all independent sets have size at most $s$".