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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



3-connected planar spaces uniquely embed in the sphere

Authors: R. Bruce Richter and Carsten Thomassen
Journal: Trans. Amer. Math. Soc. 354 (2002), 4585-4595
MSC (2000): Primary 57M15; Secondary 05C10, 57M20
Published electronically: June 3, 2002
MathSciNet review: 1926890
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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 [Enhancements On Off] (What's this?)

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Additional Information

R. Bruce Richter
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

Carsten Thomassen
Affiliation: Mathematical Institute, Technical University of Denmark, Lyngby, Denmark

Received by editor(s): October 23, 2001
Published electronically: June 3, 2002
Additional Notes: The first author acknowledges the financial support of NSERC
Article copyright: © Copyright 2002 American Mathematical Society

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