Is the NO-VERTEX-COVER in NP or not?

Question

This is the reverse version of vertex cover problem. Consider a decision problem that asks whether, given a graph G = (V, E) and a nonnegative integer k, there does not exist a vertex cover of size no larger than k. Answer whether this problem is NP or not ? Please explain to me.

Answer 1

A short answer would be no (unless co-NP=NP).

Your decision problem, NO-VERTEX-COVER, is exactly the complement of the well-known VERTEX-COVER problem; the latter problem is NP-complete (and is, of course, in NP). Your problem NO-VERTEX-COVER is thus in co-NP. (Recall that a problem is in co-NP if and only if its complement is in NP.)

Because VERTEX-COVER, the complement of you problem, is NP-complete, it follows that unless co-NP=NP, the NO-VERTEX-COVER problem is not in NP. (This follows from a theorem that says that if co-NP is not equal to NP then no NP-complete problem is in co-NP.)