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.
KeywordsPlanar Graph Common Vertex Dual Graph Simple Closed Curve Combinatorial Theorem
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