Hassler Whitney Collected Papers pp 135-146 | Cite as

# Planar Graphs

## Abstract

Kuratowski^{3}) 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## Preview

Unable to display preview. Download preview PDF.