Solving Exact2IS using IS


i encountered the following problem:

Exact2IS ={G has exactly 2 independent sets}

Assuming that given a graph G i can find an independent set how can i check if G has exactly 2 independent sets.

(i can check if a graph contains independent set in O(1) and also find some independent set in O(1))

I was thinking on find some independent set(if there is any) of size k and then remove one vertex from the set and check if the graph still has independent set of size k -after i check i return the vertex to the graph exactly as it was.

The problem my idea check only if a graph G contains at least 2 independent sets and not exactly.

Anyone have an idea how can i check if a graph has exactly 2 independent sets(in polynomial time and given the fact that find and check if there is independent set is O(1))?

Any clues or ideas will be appreciated 🙂 Thanks