Reconfiguration of labeled matchings in triangular grid graphs

TL;DR

This paper studies a reconfiguration problem of matchings in triangular grid graphs, providing conditions under which any two matchings can be transformed into each other, thereby addressing an open question related to a sliding-block puzzle called "Gourds."

Contribution

It proves that factor-critical triangular grids with a degree-6 vertex allow universal reconfiguration of matchings, and introduces a local connectivity condition for broader cases.

Findings

Any two matchings can be reconfigured if the graph is factor-critical with a degree-6 vertex.

A sufficient local connectivity condition guarantees reconfigurability in general triangular grid graphs.

Addresses the open problem of the Gourds puzzle solvability on hexagonal grids with holes.

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.