Move-reduced graphs on a torus

TL;DR

This paper classifies move-reduced bipartite graphs embedded on a torus, extending minimal graph classifications and providing a toric analog of known disk results, enhancing understanding of graph transformations in topological surfaces.

Contribution

It determines move-reduced bipartite graphs on a torus and classifies their equivalence classes under specific graph moves, extending prior minimal graph studies to a toric setting.

Findings

Classification of move-reduced bipartite graphs on a torus

Equivalence classes under square/spider moves established

Extension of minimal graph theory to toric surfaces

Abstract

We determine which bipartite graphs embedded in a torus are move-reduced. In addition, we classify equivalence classes of such move-reduced graphs under square/spider moves. This extends the class of minimal graphs on a torus studied by Goncharov-Kenyon, and gives a toric analog of Postnikov's results on a disk.