A new upper bound for angular resolution

TL;DR

This paper improves the upper bound on the angular resolution of planar straight-line graph drawings, showing that for any small epsilon, certain graphs have angular resolution bounded by a tighter expression involving logarithms.

Contribution

It introduces a new construction demonstrating a sharper upper bound on angular resolution for planar graphs with maximum degree d.

Findings

Established a new upper bound involving (log d)^epsilon / d^{3/2}

Constructed specific planar graphs meeting this bound

Refined previous bounds by Garg and Tamassia

Abstract

The angular resolution of a planar straight-line drawing of a graph is the smallest angle formed by two edges incident to the same vertex. Garg and Tamassia (ESA '94) constructed a family of planar graphs with maximum degree $d$ that have angular resolution $O((\log d)^{\frac{1}{2}}/d^{\frac{3}{2}})$ in any planar straight-line drawing. This upper bound has been the best known upper bound on angular resolution for a long time. In this paper, we improve this upper bound. For an arbitrarily small positive constant $\varepsilon$, we construct a family of planar graphs with maximum degree $d$ that have angular resolution $O((\log d)^\varepsilon/d^{\frac{3}{2}})$ in any planar straight-line drawing.