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Reconfiguration of Labeled Matchings in Triangular Grid Graphs

Authors Naonori Kakimura , Yuta Mishima

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LIPIcs.ISAAC.2024.43.pdf
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Author Details

Naonori Kakimura
  • Department of Mathematics, Keio University, Yokohama, Japan
Yuta Mishima
  • The Japan Research Institute, Limited, Tokyo, Japan

Funding

  • Kakimura, Naonori: Supported by JSPS KAKENHI Grant Numbers JP22H05001, JP20H05795, and 23K21646, Japan and JST ERATO Grant Number JPMJER2301, Japan.

Acknowledgements

The authors are grateful to Yushi Uno for valuable discussions. We are also grateful for the anonymous referees of ISAAC 2024 for their helpful comments.

Cite As Get BibTex

Naonori Kakimura and Yuta Mishima. Reconfiguration of Labeled Matchings in Triangular Grid Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.43

Abstract

This paper introduces a new reconfiguration problem of matchings in a triangular grid graph. In this problem, we are given a nearly perfect matching in which each matching edge is labeled, and aim to transform it to a target matching by sliding edges one by one. This problem is motivated to investigate the solvability of a sliding-block puzzle called "Gourds" on a hexagonal grid board, introduced by Hamersma et al. [ISAAC 2020]. The main contribution of this paper is to prove that, if a triangular grid graph is factor-critical and has a vertex of degree 6, then any two matchings can be reconfigured to each other. Moreover, for a triangular grid graph (which may not have a degree-6 vertex), we present another sufficient condition using the local connectivity. Both of our results provide broad sufficient conditions for the solvability of the Gourds puzzle on a hexagonal grid board with holes, where Hamersma et al. left it as an open question.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • combinatorial reconfiguration
  • matching
  • factor-critical graphs
  • sliding-block puzzles

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