# On L-shaped Point Set Embeddings of Trees: First Non-embeddable Examples

An L-shaped embedding of a tree in a point set is a planar drawing of the tree where the vertices are mapped to distinct points of the set and every edge is drawn as a sequence of two axis-aligned line segments. Let f_d(n) denote the minimum number N of points such that every n-vertex tree with maximum degree d∈{3,4} admits an L-shaped embedding in every point set of size N, where no two points have the same abscissa or ordinate. The best known upper bounds for this problem are f_3(n)=O(n^1.22) and f_4(n)=O(n^1.55), respectively. However, no lower bound besides the trivial bound f_d(n) ≥ n is known to this date. In this paper, we present the first examples of n-vertex trees for n∈{13,14,16,17,18,19,20} that require strictly more points than vertices to admit an L-shaped embedding, proving that f_4(n)≥ n+1 for those n. Moreover, using computer assistance, we show that every tree on n≤ 11 vertices admits an L-shaped embedding in every set of n points, proving that f_d(n)=n, d∈{3,4}, for those n.

• 19 publications
• 22 publications
research
09/05/2017

### Improved Bounds for Drawing Trees on Fixed Points with L-shaped Edges

Let T be an n-node tree of maximum degree 4, and let P be a set of n poi...
research
02/28/2023

### Linear Size Universal Point Sets for Classes of Planar Graphs

A finite set P of points in the plane is n-universal with respect to a c...
research
12/18/2020

### Upward Point Set Embeddings of Paths and Trees

We study upward planar straight-line embeddings (UPSE) of directed trees...
research
03/11/2023

### A Geometric Statistic for Quantifying Correlation Between Tree-Shaped Datasets

The magnitude of Pearson correlation between two scalar random variables...
research
03/08/2023

### Improved Bounds for Covering Paths and Trees in the Plane

A covering path for a planar point set is a path drawn in the plane with...
research
01/07/2022

### InRS: implementing the indicator function of NURBS-shaped planar domains

We provide an algorithm that implements the indicator function of NURBS-...
research
08/09/2018

### Low Ply Drawings of Trees and 2-Trees

Ply number is a recently developed graph drawing metric inspired by stud...