3-connected planar spaces uniquely embed in the sphere |
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Authors: | R. Bruce Richter Carsten Thomassen |
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Affiliation: | Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada ; Mathematical Institute, Technical University of Denmark, Lyngby, Denmark |
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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 is the closure of an embedding of a 3-connected graph in the sphere such that every 1-way infinite path in has a unique accumulation point in , then 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. |
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Keywords: | |
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