Inserting One Edge into a Simple Drawing Is Hard

Alan Arroyo, Fabian Klute, Irene Parada, Raimund Seidel, Birgit Vogtenhuber*, Tilo Wiedera

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

Abstract

A simple drawing D(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 Φσ extending σ for which A∪{Φσ} is again an arrangement of pseudocircles.
Original languageEnglish
Title of host publication46th International Workshop on Graph-Theoretic Concepts in Computer Science (WG'20)
PublisherSpringer
Pages325-338
Number of pages14
ISBN (Electronic)978-3-030-60440-0
ISBN (Print)978-3-030-60439-4
DOIs
Publication statusPublished - 2020

Publication series

NameLNCS
PublisherSpringer
Volume12301

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