GD 2013: Graph Drawing pp 208-219

# 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)

## Abstract

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.

## Keywords

Planar Graph Canonical Representation Outer Face Clockwise Order Planar Embedding
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 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