I know that the clique problem is NP-complete.
However, what if we change the problem a little bit?
Given a graph $G(V,E)$, an integer $k$ and a subset $S$ of $m$ vertices, we are given a decision problem to find a clique with size $k$ contained within $S$.
A real world example would be something like a social networking graph. Where each node is a person and an edge represents that they are friends with each other. Now what if we were to find clique with size $k$ amongst the set of teenagers.
Does that change the complexity? Since we are only looking into a subgraph, e.g. less vertices to look for?