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Independent Set Reconfiguration in H-Free Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14760))

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Abstract

Given a graph G and two independent sets of G, the independent set reconfiguration problem asks whether one independent set can be transformed into the other by moving a single vertex at a time, such that at each intermediate step we have an independent set of G. We study the complexity of this problem for H-free graphs under the token sliding and token jumping rule. Our contribution is twofold. First, we prove a reconfiguration analogue of Alekseev’s theorem for connected graphs H, showing that the problem is PSPACE-complete unless H is a path or a subdivision of the claw. We then show that under the token sliding rule the problem admits a polynomial-time algorithm if the input graph is fork-free, generalizing known results for \(P_4\)-free graphs and claw-free graphs. This implies a complete classification of the complexity of token sliding in H-free graphs, H being connected or not.

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Notes

  1. 1.

    If a token can jump to a non-neighbor then the independent set is not maximum.

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Acknowledgements

We would like to thank Martin Milanic for his insightful comments, in particular for pointing out that combining Theorems 2 and 3 with the result of [6] directly yields a complexity dichotomy for Token Sliding on H-free graphs for connected graphs H.

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Authors and Affiliations

  1. Univ. Lyon, Université Lyon 1, CNRS, LIRIS UMR CNRS 5205, 69621, Lyon, France

    Valentin Bartier & Nicolas Bousquet

  2. G-SCOP, Université Grenoble-Alpes, Saint-Martin-d’Hères, France

    Moritz Mühlenthaler

Authors
  1. Valentin Bartier
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Correspondence to Moritz Mühlenthaler .

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Editors and Affiliations

  1. Masaryk University, Brno, Czech Republic

    Daniel Kráľ

  2. University of Primorska, Koper, Slovenia

    Martin Milanič

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Bartier, V., Bousquet, N., Mühlenthaler, M. (2025). Independent Set Reconfiguration in H-Free Graphs. In: Kráľ, D., Milanič, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2024. Lecture Notes in Computer Science, vol 14760. Springer, Cham. https://doi.org/10.1007/978-3-031-75409-8_3

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