Token Sliding on Split Graphs [article]

Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, Yota Otachi, Florian Sikora
2019 arXiv   pre-print
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
more » ... 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> 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.
arXiv:1807.05322v2 fatcat:45ooim6lvbbijddgfbt4ih5czq