A projective plane is equivalent to a disk with antipodal points identified. 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. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background 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. Archdeacon's survey [2] provides an excellent introduction to topologica...

The proof of Wagner's Conjecture by Robertson and Seymour gives a finite description of any family of graphs which is closed under the minor ordering. This description is a finite set of graphs called the obstructions of the family. Since the intersection and the union of two minor closed graph families is again a minor closed graph family, an interesting question is that of computing the obstructions of the new family given the obstructions for the original two families. It is easy to compute the obstructions of the intersection but, until very recently, it was an open problem to compute the obstructions of the union. We show that if the original families are planar then the obstructions of the union are no larger than n O(n 2 ) where n is the size of the largest obstruction of the original family. 1 Introduction Robertson and Seymour's proof of Wagner's conjecture [RSb] raises somes interesting computational questions. An immediate corollary of the theorem is that for every fami...

this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1

Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3-subdivisions that coincide with the toroidal graphs with no K3,3-minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide the complete lists of four forbidden minors and eleven forbidden subdivisions for the toroidal graphs with no K3,3’s and prove that the lists are sufficient.

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Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and discuss how it was inspired by the quadratic time planar embedding algorithm of Demoucron, Malgrange and Pertuiset. We show that it is faster in practice than the only fully implemented alternative (also exponential) and explain how both the algorithm itself and the knowledge gained during its development might be used to solve the well-studied problem of finding the complete set of torus obstructions.