[Paper Review] Moving Vertices to Make Drawings Plane

Xavier Goaoc, Jan Kratochvı́l+3 more|ArXiv.org|Jun 7, 2007

Computational Geometry and Mesh Generation12 references4 citations

TL;DR

This paper investigates the problem of making a non-plane straight-line drawing of a planar graph into a plane drawing by relocating the minimum number of vertices. It proves that computing and approximating this minimum number, called shift(G,δ), is NP-hard, and provides theoretical bounds for trees and general planar graphs, extending hardness results to the 1BendPointSetEmbeddability problem.

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.

Motivation & Objective

• To determine the minimum number of vertex moves required to transform a non-plane straight-line drawing of a planar graph into a plane drawing.
• To analyze the computational complexity of computing and approximating this minimum number, termed shift(G,δ).
• To establish explicit bounds on shift(G,δ) for specific graph classes, particularly trees and general planar graphs.
• To extend the hardness results to the 1BendPointSetEmbeddability problem, a well-known problem in graph drawing.

Proposed method

• Formalizing the problem as shift(G,δ), defined as the minimum number of vertices to move to achieve a plane straight-line drawing.
• Reducing known NP-hard problems to shift(G,δ) to prove its NP-hardness in both computation and approximation.
• Deriving theoretical upper and lower bounds on shift(G,δ) for trees and general planar graphs using structural graph properties.
• Extending the hardness results to the 1BendPointSetEmbeddability problem via reductions, showing similar intractability.

Experimental results

Research questions

• RQ1 What is the computational complexity of computing the minimum number of vertex moves required to make a given straight-line drawing of a planar graph plane?
• RQ2 Can the minimum number of vertex moves (shift(G,δ)) be efficiently approximated, and what is the approximation threshold?
• RQ3 What are tight bounds on shift(G,δ) when G is a tree or a general planar graph?
• RQ4 Does the hardness of computing shift(G,δ) extend to other related graph-drawing problems, such as 1BendPointSetEmbeddability?

Key findings

• Computing shift(G,δ) is NP-hard, meaning no polynomial-time algorithm is likely to solve it exactly unless P = NP.
• Approximating shift(G,δ) is also NP-hard, indicating that even finding a close-to-optimal solution is computationally infeasible.
• For trees, shift(G,δ) is bounded by a constant factor related to the number of edge crossings in the original drawing.
• For general planar graphs, shift(G,δ) is bounded in terms of the number of crossings and structural parameters of the graph.
• The hardness of shift(G,δ) extends to the 1BendPointSetEmbeddability problem, establishing its NP-hardness as well.