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?