Constructing embedded surfaces for cellular embeddings of leveled spatial graphs

TL;DR

This paper introduces a method to construct embedded surfaces for leveled spatial graphs, enabling cellular embeddings in three-dimensional space, with an algorithmic approach and a conjecture for broader applicability.

Contribution

It defines leveled spatial graphs and provides an algorithm to construct embedded surfaces for their cellular embedding, extending previous methods.

Findings

The algorithm successfully constructs surfaces for leveled graphs with few levels.

A decomposition method into subgraphs on spheres and cylinders is developed.

Conjecture that all connected leveled embeddings can be embedded using this approach.

Abstract

For a given spatial graph $\mathcal{G} \subset \mathbb{R}^3$, we would like to find a closed orientable surface $\mathcal{S}$ embedded in $\mathbb{R}^3$ in which $\mathcal{G}$ is cellular embedded. However, for general $\mathcal{G}$ this is not possible. We therefore define a property of spatial graphs, called leveled, to show that for leveled spatial graphs with a small number of levels, a surface $\mathcal{S}$ can always be found. The argument is based on decomposing $\mathcal{G}$ into spatial subgraphs that can be placed on a sphere and on cylinders attached as handles, in such a way that the resulting surface contains a cellular embedding of $\mathcal{G}$. We generalize the procedure to an algorithm that, if successful, constructs $\mathcal{S}$ for leveled spatial graphs with any number of levels. We conjecture that all connected leveled embeddings can be cellular embedded with the presented algorithm.