3-connected planar spaces uniquely embed in the sphere
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- by R. Bruce Richter and Carsten Thomassen PDF
- Trans. Amer. Math. Soc. 354 (2002), 4585-4595 Request permission
Abstract:
We characterize those locally connected subsets of the sphere that have a unique embedding in the sphere — i.e., those for which every homeomorphism of the subset to itself extends to a homeomorphism of the sphere. This implies that if $\bar G$ is the closure of an embedding of a 3-connected graph in the sphere such that every 1-way infinite path in $G$ has a unique accumulation point in $\bar G$, then $\bar G$ has a unique embedding in the sphere. In particular, the standard (or Freudenthal) compactification of a 3-connected planar graph embeds uniquely in the sphere.References
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Additional Information
- R. Bruce Richter
- Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada
- Email: brichter@math.uwaterloo.ca
- Carsten Thomassen
- Affiliation: Mathematical Institute, Technical University of Denmark, Lyngby, Denmark
- Email: c.thomassen@mat.dtu.dk
- Received by editor(s): October 23, 2001
- Published electronically: June 3, 2002
- Additional Notes: The first author acknowledges the financial support of NSERC
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 4585-4595
- MSC (2000): Primary 57M15; Secondary 05C10, 57M20
- DOI: https://doi.org/10.1090/S0002-9947-02-03052-0
- MathSciNet review: 1926890