Theory of complexity: demonstrate that there is no polynomial algorithm for an independent set

I need some guidance in a task that I am doing.

I am completely lost, he says that the problem of MAXIMUM INDEPENDENT SET is NP-hard and then he asks me to prove that there is no polynomial time for the same algorithm and also guarantees that a certain inequality is maintained, which I also do not understand, since $ A (G) $ returns the size of a maximum independent set in $ G $ Y $ OBT (G) $ returns the size of a separate set of maximum size for $ G $ But is not it the case that a graph can have multiple maximum independent sets? Suppose that it returns the largest one that reduces the expression to $ 0 <= K $ that must always be maintained from $ K $ Is it a natural number?

As for the given track, I do not see how we can use $ H $ If I understand it correctly, it says split. $ G $ in disjoint sets. It also says for each case. $ G $ what confuses me

When I asked the teacher, she was very reluctant to help me, so please appreciate any guidance.

The complete question can be seen here.