Universal Point Sets for Planar Three-Trees

Abstract

For every n ∈ N, we present a set S n of O(n 3/2 log n) points in the plane such that every planar 3-tree with n vertices has a straight-line embedding in the plane in which the vertices are mapped to a subset of S n . This is the first subquadratic upper bound on the size of universal point sets for planar 3-trees, as well as for the class of 2-trees and serial parallel graphs.

Superpatterns and Universal Point Sets

“…It is natural to represent a permutation σ by the points with Cartesian coordinates (i, σ i ), but for our purposes we need to stretch this representation in the vertical direction; we use a transformation closely related to one used by Bukh, Matoušek, and Nivasch [21] for weak epsilon-nets, and by Fulek and Tóth [11] for universal point sets for plane 3-trees. Letting q = |σ|, we define Proof.…”

“…For instance, every set of n points in general position (no three collinear) is universal for the n-vertex outerplanar graphs [9]. Universal point sets of size O n(log n/ loglog n) 2 exist for simply-nested planar graphs (graphs that can be decomposed into nested induced cycles) [10], and planar 3-trees have universal point sets of size O(n 5/3 ) [11]. Based in part on the results in this paper, the graphs of line and pseudoline arrangements have been shown to have universal point sets of size O(n log n) [12].…”

Drawing Arrangement Graphs in Small Grids, or How to Play Planarity

“…Subquadratic bounds are known on the size of universal point sets for subclasses of the planar graphs including the outerplanar graphs [22], simply-nested planar graphs [21,23], planar 3-trees [24], and graphs of bounded pathwidth [21]; however, these results do not apply to arrangement graphs. The grid drawing technique of Theorem 1 immediately provides a universal point set for arrangement graphs of size O(n 7/6 ); in this section we significantly improve this bound, while only increasing the area of our drawings by a constant factor.…”

A Universal Point Set for 2-Outerplanar Graphs

“…A subclass of k-outerplanar graphs, in which the value of k is unbounded, but every level is restricted to be a chordless simple cycle, was known to have a universal point arXiv:1508.05784v1 [cs.CG] 24 Aug 2015 set of size O(n( log n log log n ) 2 ) [1], which was subsequently reduced to O(n log n) [2]. It is also known that planar 3-trees -graphs not defined in terms of k-outerplanarity -have a universal point set of size O(n 5/3 ) [8]. Note that planar 3-trees have treewidth equal to 3, while 2-outerplanar graphs have treewidth at most 5.…”