research-article

Inserting One Edge into a Simple Drawing is Hard

Authors:

Birgit Vogtenhuber,

Raimund Seidel,

Tilo WiederaAuthors Info & Claims

Discrete & Computational Geometry, Volume 69, Issue 3

Pages 745 - 770

https://doi.org/10.1007/s00454-022-00394-9

Published: 13 August 2022 Publication History

Abstract

A simple drawingD(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of

G + e

extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles

A

and a pseudosegment

σ

, it can be decided in polynomial time whether there exists a pseudocircle

Φ_{σ}

extending

σ

for which

A \cup {Φ_{σ}}

is again an arrangement of pseudocircles.

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Index Terms

Inserting One Edge into a Simple Drawing is Hard
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
2. Theory of computation
  1. Design and analysis of algorithms
  2. Randomness, geometry and discrete structures

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 69, Issue 3

Apr 2023

339 pages

ISSN:0179-5376

Issue’s Table of Contents

© The Author(s) 2022.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 13 August 2022

Accepted: 24 January 2022

Revision received: 14 January 2022

Received: 19 January 2021

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