Discrete & Computational Geometry

, Volume 47, Issue 3, pp 461–491 | Cite as

The Shape of Orthogonal Cycles in Three Dimensions

  • Giuseppe Di BattistaEmail author
  • Ethan Kim
  • Giuseppe Liotta
  • Anna Lubiw
  • Sue Whitesides


Let σ be a directed cycle whose edges have each been assigned a desired direction in 3D (East, West, North, South, Up, or Down) but no length. We say that σ is a shape cycle. We consider the following problem. Does there exist an orthogonal representation Γ of σ in 3D space such that no two edges of Γ intersect except at common endpoints and such that each edge of Γ has the direction specified in σ? If the answer is positive, we say that σ is simple. This problem arises in the context of extending orthogonal graph drawing techniques from 2D to 3D. We give a combinatorial characterization of simple shape cycles that yields linear time recognition and drawing algorithms.


Graph drawing 3D orthogonal drawing Topology-shape-metrics Cycles 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Giuseppe Di Battista
    • 1
    Email author
  • Ethan Kim
    • 2
  • Giuseppe Liotta
    • 3
  • Anna Lubiw
    • 4
  • Sue Whitesides
    • 5
  1. 1.Dipartimento di Informatica ed AutomazioneUniversità di Roma TreRomaItaly
  2. 2.School of Computer ScienceMcGill UniversityMontrealCanada
  3. 3.Dipartimento di Ingegneria Elettronica e dell’InformazioneUniversità di PerugiaPerugiaItaly
  4. 4.Department of Computer ScienceUniversity of WaterlooWaterlooCanada
  5. 5.Department of Computer ScienceUniversity of VictoriaVictoriaCanada

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