A Flow Approach to Upward Drawings of Toroidal Maps

1 D e f i n i t i o n s Given a graph G, let V(G) be the set of vertices of G, E(G) the set of edges of G. For A, B subsets of V(G), let E(A) be the set of edges of G with both ends in A, and let E(A, B) be the set of edges of G with one end in A, the other in B. A map M on a surface S is a connected graph G toge~..her with a 2-cell embedding of G in S. Two maps are equivalent if there is a homeomorphism of S mapping the graph of the first map onto the graph of the second. It is wellknown that the equivalence classes of maps on orientable surfaces correspond natural ly to rotat ion systems on the underlying graphs [5], where a rotation system on a graph G is a set of cyclic rotations of edges emanat ing from each vertex of G, the local rotations corresponding to the cyclic order of these edges on the surface. The rotat ion system around the faces of M defines the dual map M*. A map M and its dual M* can be simultaneously drawn in S such that each vertex of M* corresponds to an interior point of the corresponding face of M (and vice versa), and such that precisely dual pairs of edges cross each other. From now on we shall assume that the surface S is the torus. An angle of a map M (respectively, M*) on S is a pair of consecutive arcs at a vertex v of M (respectively, a face f of M). The angle map of M is a map .4 on S whose vertices are the vertices plus the faces of M, and whose edges are the angles of/l~r, each angle being incident with the corresponding vertex and face of M [9]. The set * Supported in part by the Ministry of Science and Technology of Slovenia, Research Project P1-0210-101-94. ** This work was partially supported by the ESPRIT Basic Research Action No. 7141 (ALCOM II).