Are problems that are fractions of constraints of NP-complete problems also NP-complete? We know that Hamiltonian path, clique and independent sets are NP-complete but what about half or the square root of each problem or a fraction of $n$ ? That is, for a graph, $G$ of $n$ nodes,

is there a Hamiltonian path of size $\frac{n}{2}$ or of $\sqrt{n}$ or any number of nodes less than $n$?
is there an independent set of size $b$ in  $\frac{n}{2}$ or of $\sqrt{n}$ or any number of nodes less than $n$?
is there a clique of size $k$ in $\frac{n}{2}$ or of $\sqrt{n}$ or any number of nodes less than $n$?
There is no general rule that they necessarily need to be NP-complete in general.
For the specific examples you list, I expect the answer will be that each of those specific examples are NP-complete, but I haven't tried to prove or verify that; that's just speculation.