ABSTRACT
The plane and the sphere are the simplest topological surfaces. The structure of planar graphs, and algorithms for embedding graphs on the plane are well understood. Much less is known about graph embeddings on other surfaces, and the structure of these graphs. We begin with the torus, the doughnut-shaped surface shown in Figure 13.1. We imagine this surface made out of rubber, and using scissors, cut it along the two circumferences shown in the diagram. The surface of the torus then unfolds into a rectangle, which is indicated on the right. The opposite sides of the rectangle labeled a must be glued together with the arrows aligned, as must the sides labeled b, in order to reconstruct the torus. We could glue the edges in the order a, then b; or else 6, then a. Both represent the same torus.
