A Basic Structure for Grids in Surfaces

TL;DR

This paper characterizes $S$-grids, graphs embedded in surfaces with all facial boundaries of length four, by relating their structure to degree sequences and immersions, providing a unique correspondence and classification.

Contribution

It offers a succinct characterization of $S$-grids with nonempty curvature sequences based on degree sequences and immersion properties, establishing a unique correspondence.

Findings

Characterization of $S$-grids via degree sequences and immersions

Unique association between $S$-grids and their immersion representations

Partitioning of all $S$-grids based on the characterization

Abstract

A graph $G$ embedded in a surface $S$ is called an $S$-grid when every facial boundary walk has length four, that is, the topological dual graph of $G$ in $S$ is 4-regular. Aside from the case where $S$ is the torus or Klein bottle, an $S$-grid must have vertices of degrees other than four. Let the sequence of degrees other than four in $G$ be called the curvature sequence of $G$. We give a succinct characterization of $S$-grids with nonempty curvature sequence $L$ in terms of graphs that have degree sequence $L$ and are immersed in a certain way in $S$; furthermore, the immersion associated with the $S$-grid $G$ is unique and so our characterization of $S$-grids also partitions the collection of all $S$-grids.