Suppose that someone gives you a black-box algorithm A that takes an undirected graph G =...

Suppose that someone gives you a black-box algorithm A that takes an undirected graph G = (V, E), and a number k, and behaves as follows.

• If G is not connected, it simply returns "G is not connected."

• If G is connected and has an independent set of size at least k,it returns "yes."

• If G is connected and does not have an independent set of size at least k, it returns "no."

Suppose that the algorithm A runs in time polynomial in the size of G and k.

Show how, using calls to A, you could then solve the Independent Set Problem in polynomial time: Given an arbitrary undirected graph G, and a number k, does G contain an independent set of size at least k?