Clique in P when the input graph's connected components have at most $3\log|V|$ vertices?
Indeed, it is:

Consider the input $G,k$. First, it is easy to verify that the sizes of the connected components are as required.
Now, if $k&gt;3\log |V|$, then clearly there is no clique of size at least $k$.

Otherwise, we can find the maximal clique in each component by considering all the subsets of the component.

For a component of size $3\log |V|$, there are $2^{3\log |V|}=|V|^3$ possible subsets of vertices. Once we identified the maximal clique in each component, we can compare this size to $k$, and return the answer.