# NP-complete problem

Is the following line true?

Consider three problems A, B and C. If $$A$$ $$ $$B$$ and $$B$$ $$ $$C$$ and $$B$$ is NP-complete problem, then $$C$$ is also NP-complete.

If B is NP-complete, then C would have also been NP-complete if it was in NP but it is not given here. Can we consider it false? Or is there any other reason?

• You are correct, all we know about $C$ is that it is $\mathsf{NP}$-hard. By the way, the fact that $A \leq^m_p B$ is irrelevant. Jun 13, 2021 at 9:08