We study the following independent set reconfiguration problem, called
TAR-Reachability: given two independent sets I and J of a graph G, both
of size at least k, is it possible to transform I into J by adding and
removing vertices one-by-one, while maintaining an independent set of size at
least k throughout? This problem is known to be PSPACE-hard in general. For
the case that G is a cograph (i.e. P4-free graph) on n vertices, we show
that it can be solved in time O(n2), and that the length of a shortest
reconfiguration sequence from I to J is bounded by 4n−2k, if such a
sequence exists.
More generally, we show that if X is a graph class for which (i)
TAR-Reachability can be solved efficiently, (ii) maximum independent sets can
be computed efficiently, and which satisfies a certain additional property,
then the problem can be solved efficiently for any graph that can be obtained
from a collection of graphs in X using disjoint union and complete join
operations. Chordal graphs are given as an example of such a class X