On Sets of Lines Not-Supporting Trees

TL;DR

This paper proves that large sets of lines in the plane cannot universally support crossing-free straight-line embeddings of all trees, resolving an open problem in geometric graph theory.

Contribution

It demonstrates that sufficiently large line sets are not universal for trees, answering an open question posed by Dujmovic et al.

Findings

Large line sets are not universal for all trees

The result applies to sufficiently big line sets

It resolves an open problem in geometric graph theory

Abstract

We study the following problem introduced by Dujmovic et al. Given a tree $T = (V,E)$, on $n$ vertices, a set of $n$ lines $\mathcal{L}$ in the plane and a bijection $\iota: V \rightarrow \mathcal{L}$, we are asked to find a crossing-free straight-line embedding of $T$ so that $v\in \iota(v)$, for all $v\in V$. We say that a set of $n$ lines $\mathcal{L}$ is universal for trees if for any tree $T$ and any bijection $\iota$ there exists such an embedding. We prove that any sufficiently big set of lines is not universal for trees, which solves an open problem asked by Dujmovic et al.