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Embedding graphs containing K5subdivisions
 Ars Combinatoria
"... Given a nonplanar graph G with a subdivision of K5 as a subgraph, we can either transform the K5subdivision into a K3,3subdivision if it is possible, or else we obtain a partition of the vertices of G\K5 into equivalence classes. As a result, we can reduce a projective planarity or toroidality al ..."
Abstract

Cited by 8 (2 self)
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Given a nonplanar graph G with a subdivision of K5 as a subgraph, we can either transform the K5subdivision into a K3,3subdivision if it is possible, or else we obtain a partition of the vertices of G\K5 into equivalence classes. As a result, we can reduce a projective planarity or toroidality algorithm to a small constant number of simple planarity checks [6] or to a K3,3subdivision in the graph G. It significantly simplifies algorithms presented in [7], [10] and [12]. We then need to consider only the embeddings on the given surface of a K3,3subdivision, which are much less numerous than those of K5. 1.
A Faster Algorithm for Torus Embedding
, 2006
"... 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 di ..."
Abstract
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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 wellstudied problem of finding the complete set of torus obstructions.