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arXiv · 2609.10916

Almost Linear Universal Point Sets for Planar Graphs

Taylor Gordon
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Abstract

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

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Taylor Gordon. 2026-09-10. Almost Linear Universal Point Sets for Planar Graphs. https://arxiv.org/abs/2609.10916

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