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Abstract

Kuratowski3) has shown that a topologicagraph is planar, i. e. it can be mapped in a 1 — i continuous manner on the surface of a sphere, if and only if it contains neither of two certain graphs within it. The author has shown 4) (I, Theorem 29) that a graph is planar if and only if it has a dual as defined in I. It is the main purpose of the present paper to give a proof of the purely combinatorial theorem (Theorem 12 that a graph has a dual if and only if it contains neither of Kuratowski’s graphs as a subgraph 5). This together with the above mentioned theorem of the author gives a proof (for graphs) of Kuratowski’s theorem which involves little of a point set nature.

Keywords

Planar Graph Common Vertex Dual Graph Simple Closed Curve Combinatorial Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 1992

Authors and Affiliations

  • Hassler Whitney
    • 1
  1. 1.CambridgeUSA

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