  We present a polynomial-time algorithm that, given two independent sets in a
claw-free graph $G$, decides whether one can be transformed into the other by a
sequence of elementary steps. Each elementary step is to remove a vertex $v$
from the current independent set $S$ and to add a new vertex $w$ (not in $S$)
such that the result is again an independent set. We also consider the more
restricted model where $v$ and $w$ have to be adjacent.
