Reconfiguration of labeled matchings in triangular grid graphs
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 , 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.
Cite
@article{arxiv.2409.11723,
title = {Reconfiguration of labeled matchings in triangular grid graphs},
author = {Naonori Kakimura and Yuta Mishima},
journal= {arXiv preprint arXiv:2409.11723},
year = {2024}
}