# Internally-Convex Drawings of Outerplanar Graphs in Small Area

## TL;DR

This paper presents improved algorithms for drawing outerplanar graphs with convex internal faces in significantly smaller area, advancing visualization techniques for such graphs.

## Contribution

It introduces an algorithm for embedding-preserving convex drawings of outerplanar graphs in O(n^{1.5}) area and explores strictly-convex drawings with optimized area for graphs with a path dual.

## Key findings

- Convex drawings in O(n^{1.5}) area for outerplanar graphs
- Strictly-convex drawings with area Θ(nk^2) for graphs with a path dual
- Improved area bounds over previous O(n^2) results

## Abstract

A well-known result by Kant [Algorithmica, 1996] implies that n-vertex outerplane graphs admit embedding-preserving planar straight-line grid drawings where the internal faces are convex polygons in $O(n^2)$ area. In this paper, we present an algorithm to compute such drawings in $O(n^{1.5})$ area. We also consider outerplanar drawings in which the internal faces are required to be strictly-convex polygons. In this setting, we consider outerplanar graphs whose weak dual is a path and give a drawing algorithm that achieves $\Theta(nk^2)$ area, where $k$ is the maximum size of an internal facial cycle.