Vertex Cover of size at most $\log n$

Question

Consider the following language: $$ L = \{ G | G \text{ has a VC of size at most } \log n \} $$

Does $L\in P$ or $L\in NPC$?

Answer 1

If you solve Vertex Cover with the simple branching algorithm, you achieve an FPT running time of $2^k \text{poly}(n)$, where $k$ is an upper bound for the solution you are looking for.

Substituting $k$ for $\log n$ gives you the answer you are looking for.