Abstract
Graphs are the figures which have some points (called vertices) and some curves (called edges) with two vertices as endpoints. Planar graphs are graphs which are able to be drawn on the plane without selfcrossings. The graphs drawn in this way are called plane graphs. The plane graphs whose edges are all straight line segments are said straight line plane graphs. Two plane graphs are ambient isotopic if one can be transformed into the other in the plane without selfcrossings. In 1948 I.Fary proved that every plane graph is ambient isotopic to a straight line plane graph. This theorem is called Fary's theorem. We consider the problem that if some edge-disjoint paths of a plane graph are given, then how conditions are satisfied that the plane graph is ambient isotopic to a straight line plane graph such that these paths are straight line segments. In this paper, we give the necessary conditions that the plane graph is ambient isotopic to a straight line plane graph like this.
