On Straight Line Representation of Plane Graphs(I)

Abstract

A graph consists of a set of points, called vertices, and a set of line segments, called edges, such that each edge contains exactly two vertices as its endpoints. A plane graph is a graph embedded into the plane, and a geometric plane graph is a plane graph whose edges are straight-line segments. It is known that every plane graph is ambient isotopic to a geometric plane graph, and it is called Fary's theorem. From Fary's theorem we propose two problems: the first is how all plane graphs are lined up, and the second is how we generalize Fary's theorem. In this paper, on the first problem we focus on a grid geometoric plane graph, that is a geometric plane graph whose vertices are grid point. On the second problem we focus on a path decomposition of a plane graph.