Trees in simple Polygons

TL;DR

This paper proves that every simple polygon contains a degree 3 tree covering specified vertices, establishes bounds on such trees, and applies these findings to disjoint line segments, enhancing understanding of geometric graph structures.

Contribution

Introduces a new result on degree 3 trees in simple polygons and applies it to planar segment sets, providing tight bounds and alternative proofs.

Findings

Every simple polygon contains a degree 3 tree covering prescribed vertices.

Established tight bounds on the minimal number of degree 3 vertices.

Reproved a known result on binary trees in disjoint line segments.

Abstract

We prove that every simple polygon contains a degree 3 tree encompassing a prescribed set of vertices. We give tight bounds on the minimal number of degree 3 vertices. We apply this result to reprove a result from Bose et al. that every set of disjoint line segments in the plane admits a binary tree.