Simultaneous Embeddings with Vertices Mapping to Pre-specified Points

  • Taylor Gordon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


We discuss the problem of embedding graphs in the plane with restrictions on the vertex mapping. In particular, we introduce a technique for drawing planar graphs with a fixed vertex mapping that bounds the number of times edges bend. An immediate consequence of this technique is that any planar graph can be drawn with a fixed vertex mapping so that edges map to piecewise linear curves with at most 3n + O(1) bends each. By considering uniformly random planar graphs, we show that 2n + O(1) bends per edge is sufficient on average.

To further utilize our technique, we consider simultaneous embeddings of k uniformly random planar graphs with vertices mapping to a fixed, common point set. We explain how to achieve such a drawing so that edges map to piecewise linear curves with \(O(n^{1-\frac{1}{k}})\) bends each, which holds with overwhelming probability. This result improves upon the previously best known result of O(n) bends per edge for the case where k ≥ 2. Moreover, we give a lower bound on the number of bends that matches our upper bound, proving our results are optimal.


Vertical Line Planar Graph Random Permutation Vertex Mapping Planar Embedding 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Taylor Gordon
    • 1
  1. 1.University of WaterlooCanada

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