Reconfiguring Independent Sets in Claw-Free Graphs
  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. 
Paul Bonsma, Marcin Kami\'nski, Marcin Wrochna
cs.DM math.CO