Strictly-Convex Drawings of $3$-Connected Planar Graphs

TL;DR

This paper introduces a simple, implementable method for creating strictly-convex straight-line drawings of 3-connected planar graphs within a polynomial-sized integer grid, improving upon previous complex techniques.

Contribution

A new straightforward technique for strictly-convex planar graph drawings that achieves polynomial grid size, simplifying previous complex methods.

Findings

Achieves strictly-convex drawings on a grid of size 2(n-1) x (5n^3-4n^2)

Simplifies the process compared to previous perturbation-based methods

Provides an explicit construction method for such drawings.

Abstract

Strictly-convex straight-line drawings of $3$-connected planar graphs in small area form a classical research topic in Graph Drawing. Currently, the best-known area bound for such drawings is $O(n^2) \times O(n^2)$, as shown by B\'{a}r\'{a}ny and Rote by means of a sophisticated technique based on perturbing (non-strictly) convex drawings. Unfortunately, the hidden constants in such area bound are in the $10^4$ order. We present a new and easy-to-implement technique that yields strictly-convex straight-line planar drawings of $3$-connected planar graphs on an integer grid of size $2(n-1) \times (5n^3-4n^2)$.