Token Sliding on Split Graphs

Authors Rémy Belmonte , Eun Jung Kim, Michael Lampis , Valia Mitsou, Yota Otachi , Florian Sikora

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Rémy Belmonte
  • University of Electro-Communications, Chofu, Tokyo, 182-8585, Japan
Eun Jung Kim
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
Michael Lampis
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
Valia Mitsou
  • Université Paris-Diderot, IRIF, CNRS, 75205, Paris, France
Yota Otachi
  • Kumamoto University, Kumamoto, 860-8555, Japan
Florian Sikora
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France

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Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, Yota Otachi, and Florian Sikora. Token Sliding on Split Graphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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 >= 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
  • reconfiguration
  • independent set
  • split graph


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