Let $A$ be a polynomial time algorithm which receives a graph $G$ and returns a stable set $SA(G)$ of $G$ with the property: $\alpha(G) - |SA(G)| \leq k$, for some constant $k$.
Prove that $A$ could be used to determine, in polynomial time, a stable set of maximum cardinality in a certain given graph.
I have tried by taking a graph $G_1$ which is formed by disjoin union of $k+1$ copies of $G$. $\alpha(G_1) = (K+1) \alpha(G)$. I need to prove that $|SA| = |SA(G_1)| / (k+1)$ in order to complete the proof.
But I got stuck at this point. Any hints or help would be appreciated.