I am referring to the definition of the minimum vertex cover problem from the book Approximation Algorithms by Vijay V. Vazirani (page 23):
Is the size of the minimum vertex cover in $G$ at most $k$?
and right after this definition, the author states that this problem is in NP.
My question: What would be a yes certificate?
Indeed, our non-deterministic algorithm could guess a subset of vertices, denoted by $V'$, and we can verify if $V'$ is a vertex cover of some cardinality in polynomial time, but how could we possibly show that $V'$ is minimum in polynomial time?