# Drawing Planar Graphs with Few Geometric Primitives

## TL;DR

This paper introduces methods to draw planar graphs using a minimal number of geometric primitives like line segments and arcs, optimizing visual complexity on polynomial and quasi-polynomial grids.

## Contribution

It presents new algorithms for drawing trees, planar 3-trees, and maximal planar graphs with fewer primitives than previous bounds, improving visual simplicity.

## Key findings

- Trees can be drawn with 3n/4 segments on a polynomial grid.
- Planar 3-trees can be drawn with (8n-17)/3 segments.
- Maximal planar graphs can be drawn with (5n-11)/3 arcs.

## Abstract

We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let $n$ denote the number of vertices of a graph. We show that trees can be drawn with $3n/4$ straight-line segments on a polynomial grid, and with $n/2$ straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with $(8n-17)/3$ segments on an $O(n)\times O(n^2)$ grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with $3n/2$ edges on an $O(n)\times O(n^2)$ grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only $(5n - 11)/3$ arcs. This is significantly smaller than the lower bound of $2n$ for line segments for a nontrivial graph class.