Maximizing Maximal Angles for Plane Straight-Line Graphs

TL;DR

This paper investigates the maximum possible incident angle at each vertex in plane straight-line graphs on a point set, providing tight bounds for various graph classes such as triangulations, spanning trees, and paths.

Contribution

It establishes tight bounds on the maximum incident angle for different classes of plane graphs on point sets, advancing understanding of geometric graph properties.

Findings

Tight bounds for maximum incident angles in triangulations.

Tight bounds for maximum incident angles in spanning trees.

Tight bounds for maximum incident angles in paths.

Abstract

Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $\phi$-open if each vertex has an incident angle of size at least $\phi$. In this paper we study the following type of question: What is the maximum angle $\phi$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $\phi$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.