  We consider the complexity of the Independent Set Reconfiguration problem
under the Token Sliding rule. In this problem we are given two independent sets
of a graph and are asked if we can transform one to the other by repeatedly
exchanging a vertex that is currently in the set with one of its neighbors,
while maintaining the set independent. Our main result is to show that this
problem is PSPACE-complete on split graphs (and hence also on chordal graphs),
thus resolving an open problem in this area. We then go on to consider the
$c$-Colorable Reconfiguration problem under the same rule, where the constraint
is now to maintain the set $c$-colorable at all times. As one may expect, a
simple modification of our reduction shows that this more general problem is
PSPACE-complete for all fixed $c\ge 1$ on chordal graphs. Somewhat
surprisingly, we show that the same cannot be said for split graphs: we give a
polynomial time ($n^{O(c)}$) algorithm for all fixed values of $c$, except
$c=1$, for which the problem is PSPACE-complete. We complement our algorithm
with a lower bound showing that $c$-Colorable Reconfiguration is W[2]-hard on
split graphs parameterized by $c$ and the length of the solution, as well as a
tight ETH-based lower bound for both parameters.
