Low Ply Drawings of Trees and 2-Trees

TL;DR

This paper investigates the ply number in graph drawings, showing that trees can have low ply with specific alpha values and bounded degree, while 2-trees can have significantly higher ply, highlighting differences in these graph classes.

Contribution

The paper introduces bounds on the ply number for trees and 2-trees, including constructions for low ply trees and a lower bound for 2-trees, advancing understanding of ply in graph drawing.

Findings

Trees with maximum degree Delta can be drawn with 1-ply when alpha = O(1/Delta)

Trees can have logarithmic ply with polynomial area for bounded degree when alpha=1/2

2-trees can have ply at least Omega(sqrt(n / log n)), showing higher complexity

Abstract

Ply number is a recently developed graph drawing metric inspired by studying road networks. Informally, for each vertex v, which is associated with a point in the plane, a disk is drawn centered on v with a radius that is alpha times the length of the longest edge incident to v, for some constant alpha in (0, 0.5]. The ply number is the maximum number of disks that overlap at a single point. We show that any tree with maximum degree Delta has a 1-ply drawing when alpha = O(1 / Delta). We also show that when alpha = 1/2, trees can be drawn with logarithmic ply number, with an area that is polynomial for bounded-degree trees. Lastly, we show that this logarithmic upper bound does not apply to 2-trees, by giving a lower bound of Omega(sqrt(n / log n)) ply for any value of alpha.