Straight Line PlanarEmbeddings of Graphs I

Abstract

A Graph is a figure consisted of some points, called vertices, and some line segments, called edges, which join two vertices. An embedding f : G → R2 of into the Euclidean plane R2 is called a planar embedding of G. Then G is said to be a planar graph, and the image f(G) is said to be a plane graph of G. An planar embedding of G is called a straight line planar embedding if the image of each edge of G is a straight line segment of R2. Szele posed the question what are the necessary and sufficient conditions that a given graph could be have a straight line planar embedding. He conjectured that every planar graph has a straight line planar embedding. In 1936 Wagner proved that this conjecture is true. In 1948 Fary also solved this conjecture, and furthermore proved that for every planar embedding f : G → R2 of every planar graph G, there exists a straight line planar embedding f : G → R2 such that f is ambient isotopic to f.This theorem is called Fary's theorem. In this paper we explain Fary's proof of Fary's theorem in detail.