A Basic Structure for Grids in Surfaces

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.