Reconfiguration of labeled matchings in triangular grid graphs

TL;DR

This work studies reconfiguration of labeled matchings on triangular grid graphs, motivated by the Gourds sliding-puzzle on hexagonal boards. It develops two broad sufficient conditions for reconfigurability: (i) if the graph is $2$-connected and factor-critical with at least one vertex of degree $6$, then any two labeled matchings are mutually reconfigurable via ear-decomposition-based sequences; and (ii) if the graph is locally-connected (excluding the Star of David), then reconfiguration is possible in polynomial time using a Hamilton-cycle framework. The methods combine ear-decomposition techniques with rotation along odd cycles and a Hamilton-cycle-based reconfiguration strategy, yielding bounds such as $O(kn)$ for aligning placements and $O(n^3)$ for Hamilton-cycle reconfigurations. These results broaden solvability conditions for the Gourds puzzle on hexagonal grids with holes and highlight both graph-theoretic and puzzle-theoretic implications of reconfiguration on 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 $6$, 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.

Figures (17)

• Figure 1: The Star of David graph.
• Figure 2: A triangular grid graph with $2$ holes.
• Figure 3: Slide operations. The colored, thick edges correspond to pieces.
• Figure 4: Rotation operations for a placement aligned with an odd cycle when $k=3$.
• Figure 5: A factor-critical graph that is not reconfigurable. We cannot change the ordering of the pieces by slide.

Theorems & Definitions (27)

• Theorem 3
• Proposition 4: Theorem 5.5.2 in Lovász--Plummer ref:ear_decomposition
• Lemma 4
• Lemma 4
• Theorem 5
• Corollary 6
• Lemma 7
• Lemma 7
• Lemma 10
• Lemma 11