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Sequentially Swapping Tokens: Further on Graph Classes

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SOFSEM 2023: Theory and Practice of Computer Science (SOFSEM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13878))

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Abstract

We study the following variant of the 15 puzzle. Given a graph and two token placements on the vertices, we want to find a walk of the minimum length (if any exists) such that the sequence of token swappings along the walk obtains one of the given token placements from the other one. This problem was introduced as Sequential Token Swapping by Yamanaka et al. [JGAA 2019], who showed that the problem is intractable in general but polynomial-time solvable for trees, complete graphs, and cycles. In this paper, we present a polynomial-time algorithm for block-cactus graphs, which include all previously known cases. We also present general tools for showing the hardness of problem on restricted graph classes such as chordal graphs and chordal bipartite graphs. We also show that the problem is hard on grids and king’s graphs, which are the graphs corresponding to the 15 puzzle and its variant with relaxed moves.

Partially supported by JSPS KAKENHI Grant Numbers JP17H01698, JP17K19960, JP18H04091, JP20H05793, JP20H05967, JP21K11752, JP21K19765, JP21K21283, JP22H00513. The full version is available at https://arxiv.org/abs/2210.02835.

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Notes

  1. 1.

    By a coloring, we mean a mapping from the vertex set to a color set, which is not necessarily a proper coloring.

  2. 2.

    The proofs of the statements marked with \(\bigstar \) are omitted in this short version and can be found in the full version.

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Author information

Authors and Affiliations

  1. Kyushu University, Fukuoka, Japan

    Hironori Kiya

  2. Nagoya University, Nagoya, Japan

    Yuto Okada, Hirotaka Ono & Yota Otachi

Authors
  1. Hironori Kiya
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  2. Yuto Okada
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  3. Hirotaka Ono
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  4. Yota Otachi
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Corresponding author

Correspondence to Yota Otachi .

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Editors and Affiliations

  1. University of Liverpool, Liverpool, UK

    Leszek Gąsieniec

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Kiya, H., Okada, Y., Ono, H., Otachi, Y. (2023). Sequentially Swapping Tokens: Further on Graph Classes. In: Gąsieniec, L. (eds) SOFSEM 2023: Theory and Practice of Computer Science. SOFSEM 2023. Lecture Notes in Computer Science, vol 13878. Springer, Cham. https://doi.org/10.1007/978-3-031-23101-8_15

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  • DOI: https://doi.org/10.1007/978-3-031-23101-8_15

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  • Online ISBN: 978-3-031-23101-8

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