# A 2-Approximation for the Height of Maximal Outerplanar Graph Drawings

## TL;DR

This paper improves the approximation factor for drawing maximal outerplanar graphs with minimal height from 4 to 2 by introducing the umbrella depth parameter and an efficient algorithm.

## Contribution

It introduces the umbrella depth parameter and provides a 2-approximation algorithm for the height of maximal outerplanar graph drawings.

## Key findings

- The algorithm achieves height at most twice the umbrella depth.
- Umbrella depth is a lower bound for the height of any poly-line drawing.
- The approximation factor is improved from 4 to 2.

## Abstract

In this paper, we study planar drawings of maximal outerplanar graphs with the objective of achieving small height. A recent paper gave an algorithm for such drawings that is within a factor of 4 of the optimum height. In this paper, we substantially improve the approximation factor to become 2. The main ingredient is to define a new parameter of outerplanar graphs (the so-called umbrella depth, obtained by recursively splitting the graph into graphs called umbrellas). We argue that the height of any poly-line drawing must be at least the umbrella depth, and then devise an algorithm that achieves height at most twice the umbrella depth.