Superpatterns and Universal Point Sets
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.
Cite
@article{arxiv.1308.0403,
title = {Superpatterns and Universal Point Sets},
author = {Michael J. Bannister and Zhanpeng Cheng and William E. Devanny and David Eppstein},
journal= {arXiv preprint arXiv:1308.0403},
year = {2015}
}
Comments
GD 2013 special issue of JGAA