Reconfiguration of labeled matchings in triangular grid graphs
<|reference_start|>Reconfiguration of labeled matchings in triangular grid graphs: 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.<|reference_end|>
@article{kakimura2024reconfiguration,
    title={Reconfiguration of labeled matchings in triangular grid graphs},
    author={Naonori Kakimura and Yuta Mishima},
    journal={arXiv preprint arXiv:2409.11723},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.11723},
    primaryClass={cs.DS cs.DM}
}