Moving Vertices to Make Drawings Plane

Abstract

A straight-line drawing $δ$ of a planar graph $G$ need not be plane, but can be made so by moving some of the vertices. Let shift$(G,δ)$ denote the minimum number of vertices that need to be moved to turn $δ$ into a plane drawing of $G$. We show that shift$(G,δ)$ is NP-hard to compute and to approximate, and we give explicit bounds on shift$(G,δ)$ when $G$ is a tree or a general planar graph. Our hardness results extend to 1BendPointSetEmbeddability, a well-known graph-drawing problem.

This paper has been merged with http://arxiv.org/abs/0709.0170