Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T07:18:40.650Z Has data issue: false hasContentIssue false
  • English
  • Français

Obstructions of Connectivity Two for Embedding Graphs into the Torus

Published online by Cambridge University Press:  20 November 2018

Bojan Mohar  and
Petr Škoda
Bojan Mohar
Affiliation:
Department of Mathematics, Simon Fraser University, BurnabyBC. e-mail: mohar@sfu.ca
Petr Škoda
Affiliation:
Department of Mathematics, Simon Fraser University, BurnabyBC Current address: Google Inc., Mountain View, CA. e-mail: pskoda@sfu.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The complete set of minimal obstructions for embedding graphs into the torus is still not determined. In this paper, we present all obstructions for the torus of connectivity 2. Furthermore, we describe the building blocks of obstructions of connectivity 2 for any orientable surface.

Keywords

05C1005C83torusobstructionminorconnectivity 2
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 6 , 01 December 2014 , pp. 1327 - 1357
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Battle, J., Harary, F., Kodama, Y., and Youngs, J.W. T., Additivity of the genus of a graph. Bull. Amer. Math. Soc. 68(1962), 565–568. http://dx.doi.org/10.1090/S0002-9904-1962-10847-7 Google Scholar
[2] Chambers, J., Hunting for torus obstructions. Master's thesis, Deptartment of Computer Science, University of Victoria, 2002.Google Scholar
[3] Decker, R.W., Glover, H. H., and Huneke, J. P., The genus of the 2-amalgamations of graphs. J.Graph Theory 5(1981), no. 1, 95–102.http://dx.doi.org/10.1002/jgt.3190050107 Google Scholar
[4] Decker, R.W., Glover, H. H., and Huneke, J. P., Computing the genus of the 2-amalgamations of graphs. Combinatorica 5(1985), no. 4, 271–282.http://dx.doi.org/10.1007/BF02579241 Google Scholar
[5] Gagarin, A., Myrvold, W., and Chambers, J., The obstructions for toroidal graphs with no K3,3’s. Discrete Math. 309(2009), no. 11, 3625–3631.http://dx.doi.org/10.1016/j.disc.2007.12.075 Google Scholar
[6] Mohar, B. and Thomassen, C., Graphs on surfaces. Johns Hopkins Studies in Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, 2001.Google Scholar
[7] Mohar, B. and Škoda, P., Obstructions for two-vertex alternating embeddings of graphs in surfaces. . arxiv:1112.0800 Google Scholar
[8] Neufeld, E. and Myrvold, W., Practical toroidality testing. In: Proceedings of the eigth annual ACM-SIAM symposium on discrete algorithms (New Orleans, LA, 1997), ACM, New York, 1997, pp. 574–580.Google Scholar
[9] Robertson, N. and Seymour, P. D., Graph minors. VIII. A Kuratowski theorem for general surfaces. J. Combin. Theory Ser B. 48(1990), no. 2, 255–288.http://dx.doi.org/10.1016/0095-8956(90)90121-F Google Scholar
[10] Stahl, S. and Beineke, L.W., Blocks and the nonorientable genus of graphs. J. Graph Theory 1(1977), no. 1, 75–78.http://dx.doi.org/10.1002/jgt.3190010114 Google Scholar
[11] Woodcock, J., A faster algorithm for torus embedding. . Master's thesis, Department of Computer Science, University of Victoria, 2007.Google Scholar