So the question gives us some mysterious algorithm, that given a graph G and an integer k, it outputs true/false to whether there is an independent set of size k in G.
So we have to design an algorithm that can call this mysterious algorithm within a polynomial number of times, to return an independent set of size k from G if it exists, or outputs impossible. The hint is induction.
I'm having trouble figuring an algorithm to solve this. I know however, that the proof for this algorithm involves removing or adding a vertex in the inductive step.