Embedding Plane 3-Trees in ℝ2 and ℝ3

  • Stephane Durocher
  • Debajyoti Mondal
  • Rahnuma Islam Nishat
  • Md. Saidur Rahman
  • Sue Whitesides
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7034)


A point-set embedding of a planar graph G with n vertices on a set P of n points in ℝ d , d ≥ 1, is a straight-line drawing of G, where the vertices of G are mapped to distinct points of P. The problem of computing a point-set embedding of G on P is NP-complete in ℝ2, even when G is 2-outerplanar and the points are in general position. On the other hand, if the points of P are in general position in ℝ3, then any bijective mapping of the vertices of G to the points of P determines a point-set embedding of G on P. In this paper, we give an O(n 4/3 + ε )-expected time algorithm to decide whether a plane 3-tree with n vertices admits a point-set embedding on a given set of n points in general position in ℝ2 and compute such an embedding if it exists, for any fixed ε>0. We extend our algorithm to embed a subclass of 4-trees on a point set in ℝ3 in the form of nested tetrahedra. We also prove that given a plane 3-tree G with n vertices, a set P of n points in ℝ3 that are not necessarily in general position and a mapping of the three outer vertices of G to three different points of P, it is NP-complete to decide if G admits a point-set embedding on P respecting the given mapping.


Plane Graph General Position Outer Face Outerplanar Graph Triangular Face 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Debajyoti Mondal
    • 1
  • Rahnuma Islam Nishat
    • 2
  • Md. Saidur Rahman
    • 3
  • Sue Whitesides
    • 2
  1. 1.Department of Computer ScienceUniversity of ManitobaCanada
  2. 2.Department of Computer ScienceUniversity of VictoriaCanada
  3. 3.Graph Drawing and Information Visualization Laboratory, Department of Computer Science and EngineeringBangladesh University of Engineering and TechnologyBangladesh

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