Moving Vertices to Make Drawings Plane

TL;DR

This paper investigates the problem of transforming a straight-line drawing of a planar graph into a plane drawing by moving the fewest vertices, proving NP-hardness and providing bounds for trees and planar graphs.

Contribution

It establishes the NP-hardness of computing the minimum vertex moves needed to make a drawing planar and extends these results to related graph-drawing problems.

Findings

NP-hard to compute and approximate shift(G,δ)

Explicit bounds for trees and planar graphs

Hardness results extend to 1BendPointSetEmbeddability

Abstract

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