On L-shaped point set embeddings of trees: first non-embeddable examples

TL;DR

This paper presents the first known examples of trees requiring more points than vertices for L-shaped embeddings, establishing new lower bounds and exploring ordered tree embeddings.

Contribution

It provides the first non-trivial lower bounds for L-shaped embeddings of trees and analyzes ordered trees, including infinite families of non-embeddable cases.

Findings

Trees with 13-20 vertices can require more points than vertices for embedding.

All trees with up to 12 vertices can be embedded in any point set of the same size.

An infinite family of ordered trees does not always admit L-shaped embeddings.

Abstract

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 and every edge is drawn as a sequence of two axis-aligned line segments. There has been considerable work on establishing upper bounds on the minimum cardinality of a point set to guarantee that any tree of the same size with maximum degree 4 admits an L-shaped embedding on the point set. However, no non-trivial lower bound is known to this date, i.e., no known $n$-vertex tree requires more than $n$ points to be embedded. In this paper, we present the first examples of $n$-vertex trees for $n\in\{13,14,16,17,18,19,20\}$ that require strictly more points than vertices to admit an L-shaped embedding. Moreover, using computer help, we show that every tree on $n\leq 12$ vertices admits an L-shaped embedding in every set of $n$ points. We also consider embedding ordered trees, where the cyclic order of the neighbors of each vertex in the embedding is prescribed. For this setting, we determine the smallest non-embeddable ordered tree on $n=10$ vertices, and we show that every ordered tree on $n\leq 9$ or $n=11$ vertices admits an L-shaped embedding in every set of $n$ points. We also construct an infinite family of ordered trees which do not always admit an L-shaped embedding, answering a question raised by Biedl, Chan, Derka, Jain, and Lubiw.