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arXiv · 1308.0403

Superpatterns and Universal Point Sets

Michael J. Bannister·Zhanpeng Cheng·William E. Devanny·David Eppstein
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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 + Theta(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 - Theta(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(n log^O(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.

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Michael J. Bannister, Zhanpeng Cheng, William E. Devanny, David Eppstein. 2014-04-03. Superpatterns and Universal Point Sets. https://doi.org/10.7155/jgaa.00318

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