Drawing Graphs on the Torus

Let G be a 2-connected graph with a toroidal rotation system given. An algorithm for constructing a straight line drawing with no crossings on a rectangular representation of the torus is presented. It is based on Read’s algorithm for constructing a planar layout of a 2-connected graph with a planar rotation system. It is proved that the method always works. The complexity of the algorithm is linear in the number of vertices of G. 1. Toroidal Graphs Let G be a toroidal graph, that is, one which can be drawn on the torus with no edge crossings. We require G to be a 2-connected graph, and we work only with 2-cell embeddings on the torus. The vertex and edge sets of G are V (G) and E(G), respectively. If u, v ∈ V (G), then u → v means that u is adjacent to v (and so also v → u). The reader is referred to Bondy and Murty [1] for other graph-theoretic terminology. G is represented by a rotation system, that is, the edges incident on each vertex v ∈ V (G) are cyclically ordered. This is suffcient to determine the faces (2-cells) of the embedding. If G has n vertices, ε edges, and f faces, then Euler’s formula tells us that in a 2-cell embedding, n + f − ε = 0. Any rotation system which satisfies this formula is called a toroidal rotation system. We will find it useful to work with triangulations of the torus. In a triangulation, every face has degree 3, which gives us the further relations 2ε = 6n = 3f . 1.1 Loops and Multiple Edges We will allow G to have loops and multiple edges. This is necessary, since the duals of graphs we are interested in will often have loops or multiple edges. However, if vv is a loop, we require that the cycle vv be an essential cycle of the embedding, that is, if the torus is cut along the cycle vv, the * This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.