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Counting unlabelled toroidal graphs with no k3,3subdivisions
 Advances in Applied Mathematics
"... We provide a description of unlabelled enumeration techniques, with complete proofs, for graphs that can be canonically obtained by substituting 2pole networks for the edges of core graphs. Using structure theorems for toroidal and projectiveplanar graphs containing no K3,3subdivisions, we apply t ..."
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We provide a description of unlabelled enumeration techniques, with complete proofs, for graphs that can be canonically obtained by substituting 2pole networks for the edges of core graphs. Using structure theorems for toroidal and projectiveplanar graphs containing no K3,3subdivisions, we apply these techniques to obtain their unlabelled enumeration. 1
Structure and labelled enumeration of K3,3subdivisionfree projectiveplanar graphs
, 2008
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The obstructions for toroidal graphs with no K3,3’s
, 2005
"... Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3subdivisions that coincide with the toroidal graphs with no K3,3minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide ..."
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Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3subdivisions that coincide with the toroidal graphs with no K3,3minors. 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.
Characterization and enumeration of toroidal K3,3subdivisionfree graphs
, 2004
"... We describe the structure of 2connected nonplanar toroidal graphs with no K3,3subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing ..."
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We describe the structure of 2connected nonplanar toroidal graphs with no K3,3subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K5subdivision in [A. Gagarin and W. Kocay, “Embedding graphs containing K5subdivisions”, Ars Combin. 64 (2002), 3349]. It allows to recognize these graphs in lineartime and makes possible to enumerate labelled 2connected toroidal graphs containing no K3,3subdivisions and having minimum vertex degree two or three by using an approach similar to [A. Gagarin, G. Labelle, and P. Leroux, ”Counting labelled projectiveplanar graphs without a K3,3subdivision”, submitted, arXiv:math.CO/ 0406140, (2004)].
A Faster Algorithm for Torus Embedding
, 2004
"... 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 ..."
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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.