Untangling Circular Drawings: Algorithms and Complexity

Sujoy Bhore, Guangping Li, Martin Nöllenburg, Ignaz Rutter, Hsiang Yun Wu

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


We consider the problem of untangling a given (non-planar) straight-line circular drawing δG of an outerplanar graph G = (V, E) into a planar straight-line circular drawing by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph G, it is clear that such a crossing-free circular drawing always exists and we define the circular shifting number shift(δG) as the minimum number of vertices that need to be shifted to resolve all crossings of δG. We show that the problem Circular Untangling, asking whether shift(δG) ≤ K for a given integer K, is NP-complete. Based on this result we study Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all the crossings. In this case we provide a tight upper bound shift(δG) ≤ ⌊n2 ⌋ - 1, where n is the number of vertices in G, and present a polynomial-time algorithm to compute the circular shifting number of almost-planar drawings.

Original languageEnglish
Title of host publication32nd International Symposium on Algorithms and Computation, ISAAC 2021
EditorsHee-Kap Ahn, Kunihiko Sadakane
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772143
StatePublished - 1 Dec 2021
Externally publishedYes
Event32nd International Symposium on Algorithms and Computation, ISAAC 2021 - Fukuoka, Japan
Duration: 6 Dec 20218 Dec 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference32nd International Symposium on Algorithms and Computation, ISAAC 2021


  • Graph drawing
  • NP-hardness
  • Outerplanarity
  • Straight-line drawing
  • Untangling

ASJC Scopus subject areas

  • Software


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