Independent Set Reconfiguration in Cographs

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(n), 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 G 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 G using disjoint union and complete join operations. Chordal graphs are given as an example of such a class G.