Token Sliding on Split Graphs
R\'emy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, Yota, Otachi, Florian Sikora

TL;DR
This paper proves that the Token Sliding reconfiguration problem is PSPACE-complete on split graphs, introduces a polynomial algorithm for c-Colorable Reconfiguration on split graphs for fixed c>1, and establishes complexity bounds and hardness results.
Contribution
It resolves an open problem by showing PSPACE-completeness on split graphs and provides a polynomial algorithm for c-Colorable Reconfiguration for fixed c>1 on split graphs.
Findings
Token Sliding reconfiguration is PSPACE-complete on split graphs.
Polynomial-time algorithm for c-Colorable Reconfiguration on split graphs for fixed c>1.
Complexity bounds and hardness results for the problem parameterized by c and solution length.
Abstract
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 -Colorable Reconfiguration problem under the same rule, where the constraint is now to maintain the set -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 on chordal graphs. Somewhat surprisingly, we show that the same cannot be said for split…
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Taxonomy
TopicsAdvanced Graph Theory Research · DNA and Biological Computing · Optimization and Search Problems
