# Graphs with large total angular resolution

## TL;DR

This paper investigates the maximum total angular resolution in straight-line graph drawings, establishing bounds and computational complexity results, which enhance understanding of graph readability and drawing constraints.

## Contribution

It proves a tight bound on the number of edges for graphs with high total angular resolution and shows that deciding this property is NP-hard.

## Key findings

- Bound of 2n-6 edges for graphs with >60° total angular resolution
- NP-hardness of deciding total angular resolution ≥ 60°
- Tightness of the established edge bound

## Abstract

The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph. We prove that, up to a finite number of well specified exceptions of constant size, the number of edges of a graph with $n$ vertices and a total angular resolution greater than $60^{\circ}$ is bounded by $2n-6$. This bound is tight. In addition, we show that deciding whether a graph has total angular resolution at least $60^{\circ}$ is NP-hard.