Independent set is polynomial in claw-free graphs, so I am wondering if this can approximate independent set.

By adding enough edges to $G$ and gets claw-free $G'$.

IS in $G'$ is IS in $G$, so this is a bound for the MIS.

If one adds very few edges, the bound is better.

Is it possible to efficiently find $G'$ with as few edges as possible?

One approach is to use integer programming, though in general this is NP hard. Limited experiments with this suggest the approximation of MIS is good.

Another approach is naive greedy algorithm, though this doesn't seem to approximate well.

The same question is for "claw-free" replaced by $X,Y$-free where IS for $X,Y$-free is polynomial.