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Untangling Circular Drawings: Algorithms and Complexity

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

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Author Details

Sujoy Bhore
  • Indian Institute of Science Education and Research, Bhopal, India
Guangping Li
  • Algorithms and Complexity Group, TU Wien, Austria
Martin Nöllenburg
  • Algorithms and Complexity Group, TU Wien, Austria
Ignaz Rutter
  • Universität Passau, Germany
Hsiang-Yun Wu
  • Research Unit of Computer Graphics, TU Wien, Austria

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Sujoy Bhore, Guangping Li, Martin Nöllenburg, Ignaz Rutter, and Hsiang-Yun Wu. Untangling Circular Drawings: Algorithms and Complexity. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 19:1-19:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


We consider the problem of untangling a given (non-planar) straight-line circular drawing δ_G 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°(δ_G) as the minimum number of vertices that need to be shifted to resolve all crossings of δ_G. We show that the problem Circular Untangling, asking whether shift°(δ_G) ≤ 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°(δ_G) ≤ ⌊n/2⌋-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.

Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Mathematics of computing → Permutations and combinations
  • Theory of computation → Problems, reductions and completeness
  • graph drawing
  • straight-line drawing
  • outerplanarity
  • NP-hardness
  • untangling


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