Constructing embedded surfaces for cellular embeddings of leveled spatial graphs

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.