Simpler Projective Plane Embedding

A projective plane is equivalent to a disk with antipodal points identiied. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar 20]. We provide a new approach that takes O(n 2) time is but much easier to implement. 1 Description of the problem A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V. Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. A graph can be used to model many things. Some examples with applications in computer science include modelling program structure, networks, or how documents on the web are linked together using hyperlinks. A graph visualization tool can help researchers to better understand the structure of such things. Usually, it is best to avoid having many crossing edges as this can complicate the picture of a graph. Algorithms for embedding graphs on surfaces are often used to help get a nice picture of a graph. A graph is planar if it can be embedded on the plane. A planar embedding of a graph is a description of how it can be embedded in the plane. A graph and one planar embedding of it are pictured in Figure 1. A planar embedding is commonly represented by giving the clockwise order of the neighbours of each vertex (the combinatorial embedding). This paper is concerned with combinatorial embeddings. The aesthetic issues of where to place the vertices and edges are discussed in a wide body of literature including the recent book 2]. It is well-known that a planar graph without loops or multiple edges has at most 3n ? 6 edges. Thus in discussing time complexities for algorithms, linear time should be interpreted as time in O(n). Often an algorithm which is the fastest in theory can be very complex. This makes it diicult to program. Further, when a program is developed, it is not that easy to determine whether the code is working correctly. Hence, algorithms which are simpler but possibly slower can be valuable from a programmer's perspective. There are several …