3-connected planar spaces uniquely embed in the sphere

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 G ¯ \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 G has a unique accumulation point in G ¯ \bar G , then G ¯ \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.