Constructing embedded surfaces for cellular embeddings of leveled spatial graphs

Finding a closed orientable surface $\mathcal{S}$ embedded in $\mathbb{R}^3$ where a given spatial graph $\mathcal{G} \subset \mathbb{R}^3$ cellular embeds is in general not possible. We therefore restrict our interest to the special class of spatial graphs that are leveled. We 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 the idea of decomposing $\mathcal{G}$ into subgraphs that can be placed on a sphere and on handles that are attached to the sphere, together forming an embedding of $\mathcal{G}$ in $\mathcal{S}$. 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.