# Drawing planar graphs with few segments on a polynomial grid

## TL;DR

This paper investigates how to draw planar graphs using a minimal number of segments, achieving improved area bounds for trees and providing new segment bounds for other subclasses.

## Contribution

It presents new methods for drawing planar graphs with fewer segments and polynomial area, improving previous bounds for trees and extending results to other subclasses.

## Key findings

- Trees can be drawn with 3n/4-1 segments and quadratic area.
- 3-connected planar graphs have drawings with 8n/3 segments.
- Biconnected outerplanar graphs can be drawn with 3n/2 segments.

## Abstract

The visual complexity of a graph drawing can be measured by the number of geometric objects used for the representation of its elements. In this paper, we study planar graph drawings where edges are represented by few segments. In such a drawing, one segment may represent multiple edges forming a path. Drawings of planar graphs with few segments were intensively studied in the past years. However, the area requirements were only considered for limited subclasses of planar graphs. In this paper, we show that trees have drawings with $3n/4-1$ segments and $n^2$ area, improving the previous result of $O(n^{3.58})$. We also show that 3-connected planar graphs and biconnected outerplanar graphs have a drawing with $8n/3-O(1)$ and $3n/2-O(1)$ segments, respectively, and $O(n^3)$ area.