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Token Sliding on Split Graphs

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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 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 (nO(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. Finally, we study c-Colorable Reconfiguration under a relaxed rule called Token Jumping, where exchanged vertices are not required to be adjacent. We show that the problem on chordal graphs is PSPACE-complete even if c is some fixed constant. We then show that the problem is polynomial-time solvable for strongly chordal graphs even if c is a part of the input.

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Notes

  1. A path of five vertices is strongly chordal but not a split graph. The graph obtained from a triangle K3 by attaching to each edge a new vertex adjacent to both endpoints of the edge is a split graph but not strongly chordal.

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Acknowledgments

Supported by JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA) Project GRAPA 38593YJ, by PRC CNRS JSPS 2019-2020 program, project PARAGA (Parameterized Approximation Graph Algorithms), by FMJH program PGMO and EDF via project 2016-1760H/C16/1507 “Stability versus Optimality in Dynamic Environment Algorithmics” and project “ESIGMA” (ANR-17-CE23-0010), and by JSPS KAKENHI Grant Numbers JP18K11157, JP18K11168, JP18K11169, JP18H04091.

A preliminary version appeared in the proceedings of the 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019), vol. 126 of Leibniz International Proceedings in Informatics, pp. 13:1–13:17, 2019.

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Authors and Affiliations

  1. The University of Electro-Communications, Chofu, Tokyo, 182-8585, Japan

    Rémy Belmonte

  2. CNRS, LAMSADE, Université Paris-Dauphine, PSL University, 75016, Paris, France

    Eun Jung Kim, Michael Lampis & Florian Sikora

  3. IRIF, CNRS, Université Paris-Diderot, 75205, Paris, France

    Valia Mitsou

  4. Kumamoto University, Kumamoto, 860-8555, Japan

    Yota Otachi

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  1. Rémy Belmonte
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  2. Eun Jung Kim
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  3. Michael Lampis
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  4. Valia Mitsou
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  5. Yota Otachi
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  6. Florian Sikora
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Correspondence to Yota Otachi.

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This article belongs to the Topical Collection: Special Issue on Theoretical Aspects of Computer Science (2019)

Guest Editors: Rolf Niedermeier and Christophe Paul

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Belmonte, R., Kim, E.J., Lampis, M. et al. Token Sliding on Split Graphs. Theory Comput Syst 65, 662–686 (2021). https://doi.org/10.1007/s00224-020-09967-8

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  • DOI: https://doi.org/10.1007/s00224-020-09967-8

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