Straight-line representations of maps on the torus and other flat surfaces

It is shown that every map on the torus satisfying the obvious necessary conditions has a straight-line representation on the flat torus R2/Z 2. The same holds for the Klein bottle, and the two-bordered flat surfaces the cylinder and the M6bius band. I . Stra ightl ine m a p s By a well-known result of Wagner [13] (usually attributed to Ffiry [4]), every simple plane graph admits a straight-line representation, i.e. there is a homeomorphism of the plane such that the edges of the graph become straight-line segments after performing this homeomorphism. A companion result is the theorem of Steinitz [11] that every 3-connected planar graph can be represented as the graph of a convex 3-polytope. There were attempts to generalize Steinitz's theorem to maps on surfaces of positive genus, e.g. [5,9]. But it seems that no one tried to extend the Wagner-F~iry's Theorem to non-simply connected surfaces. In this paper we fill in this gap by proving a corresponding result for the toms, the Klein bottle, the cylinder, and the M6bius band. These are the only flat surfaces with the boundary components being straight. The existence of straight-line representations of maps on the toms and the Klein bottle might have some applications in the theory of filings of the plane since their universal covers give rise to plane filings. A short discussion about this can be found in [7, p. 202]. Our proof of the existence of straight-line representations is fairly elementary. It can be extended to surfaces of higher genera, applied to their models with constant 1 Supported in part by the Ministry for Science and Technology of Slovenia, Research Project P1-0210-10192. 0012-365X/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSD! 0012-365X(94)00381-5 174 B. Mohar/Discrete Mathematics 155 (1996) 173 181 curvature 1 (the hyperbolic metric), and with the geodesics playing the role of straight-line segments. We will not give details for these cases since it recently came to our attention that the existence of such geodesic representations (for surfaces without boundary) follows from the Circle Packing Theorem of Koebe [8], Andreev [1,2], and Thurston [12]. The advantage of our proof compared to the circle packing results is that it is elementary and that it also yields a polynomial-time algorithm to produce straight-line drawings of given maps. Let S be a compact surface. A map on S is a pair M = (G,S), where G is a connected graph embedded in S. To get more freedom, we do not require the embedding to be cellular, so a map can have non-simply connected faces. If S has non-empty boundary, as ~ ~, then we require for each edge of G either to be disjoint from 0S, having only one or both endvertices on aS, or entirely lying on as. A map on the torus is also said to be a toroidal map. Two maps M = (G,S) and M I = (GI, S ~) are equivalent if there is a homeomorphism h : S ~ S t mapping the graph G of the first map isomorphically to the graph G ~ of the second map. It is well-known (cf., e.g., [6,10]) that two maps on an orientable surface without boundary and with all faces simply connected are equivalent if and only if they determine the same rotation system on the graph. A compact Riemannian surface S (possibly with boundary) is flat if every point p E S has a neighbourhood which is affinely diffeomorphic to an open set in the closed upper half-plane of the Euclidean plane. This is equivalent to the condition that the curvature and torsion are identically zero, including the curvature of the boundary aS. The special case of a flat surface is the flat torus, the quotient space R2/Z 2 (R2/~ where (x,y) ~ (x~,y ') means ( x xt, y y~) E Z2). Another fiat surface is the flat Klein bottle. This surface is the quotient of the Euclidean plane R 2 corresponding to the relation ~ given by (x,y) ~ (x + n , ( -1)ny + m), n,m E Z. The fiat torus and the fiat Klein bottle are usually represented as the identification space of the unit square by identifying the top and the bottom side and then identifying the left and the right, with a previous turn by 180 ° in case of the Klein bottle. There are two additional fiat surfaces with boundary the flat cylinder and the flat Mfbius band. They are obtained from the unit square as well, by identifying only one pair of sides, the left and the right. To get the cylinder they are identified without a turn, and to get the M6bius band we perform a twist of 180 ° of one side before the identification. It can be shown by using the Gauss-Bonnet formula (cf. [3]) that every compact flat surface is homeomorphic to one of these four surfaces. A straight-line segment on a fiat surface S is a segment of a geodesic on S. A map M on S is said to be a straight-line map if each edge of the graph of M is a straight-line segment. Every straight-line map M on S is simple, i.e. M has the following properties (see Fig. 1): 1. Each pair of parallel edges (edges with the same endvertices) gives rise to a noncontractible cycle on S. 2. No loop of M is contractible. 0 B. Mohar/Discrete Mathematics 155 (1996) 173-181 175 Ca) (b) (c) Fig. 1. Forbidden submaps of simple maps. 3. I f e is a loop at the vertex v then no other loop at v is homotopic to e (i.e., does not bound a disk together with e). This is easily seen by using the Gauss-Bonnet formula (cf. [3]). It is clear that the map is simple if and only if its universal cover has no loops and no parallel edges. For the flat torus and other flat surfaces we will also prove the converse: A map on a flat surface is equivalent to a straight-line map if and only if it is simple (Theorems 3.1, 4.1, 5.2). 2. Triangular maps and contractible edges A map M = (G,S) is triangular if d S C G and every face is o f size three and homeomorphic to an open disk. The smallest triangular simple map on the torus is shown in Fig. 2. Its graph consists of a single vertex with three loops. We will show that every triangular simple map on the torus can be reduced to the map of Fig. 2 by means of edge contractions. Let M be a triangular simple map. An interior edge e of M is contractible if the operation shown in Fig. 3 gives rise to another simple map on the same surface. Notice the edges ~, fl, and, respectively, 7, 6, of the two triangles containing e, are pairwise identified after the contraction. An edge e on the boundary of the surface is contractible i f the similar operation as shown on Fig. 3, adopted to the fact