Superpatterns and Universal Point Sets

  • Michael J. Bannister
  • Zhanpeng Cheng
  • William E. Devanny
  • David Eppstein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)


An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n 2/4 + Θ(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n 2/4 − Θ(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size O(nlog O(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.


Planar Graph Canonical Representation Outer Face Clockwise Order Planar Embedding 
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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Michael J. Bannister
    • 1
  • Zhanpeng Cheng
    • 1
  • William E. Devanny
    • 1
  • David Eppstein
    • 1
  1. 1.Department of Computer ScienceUniversity of CaliforniaIrvineUSA

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