Superpatterns and Universal Point Sets

TL;DR

This paper links the problem of universal point sets in graph drawing to permutation pattern theory, constructing smaller superpatterns for specific permutation classes and resulting in smaller universal point sets for certain planar graphs.

Contribution

It introduces generalized superpatterns for permutation classes and uses them to create smaller universal point sets for planar graphs with bounded pathwidth.

Findings

Superpatterns of size n^2/4 + Theta(n) for 213-avoiding permutations

Universal point sets of size n^2/4 - Theta(n) for certain planar graphs

Proper subclasses of 213-avoiding permutations have superpatterns of size O(n log^O(1) n)

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.