A Quantitative Steinitz Theorem for Plane Triangulations

TL;DR

This paper presents a new proof of Steinitz's theorem for plane triangulations, providing a bound on the grid size for embedding such triangulations as projections of convex polyhedral surfaces, with implications for geometric graph theory.

Contribution

It introduces a novel proof of Steinitz's theorem for plane triangulations and establishes a new bound on the grid size needed for embedding these triangulations as convex polyhedral surfaces.

Findings

Vertices can be embedded in a polynomial-sized integer grid.

The embedding is the vertical projection of a convex polyhedral surface.

The grid size depends on the shedding diameter of the triangulation.

Abstract

We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane triangulation $G$ with $n$ vertices can be embedded in $\mathbb{R}^2$ in such a way that it is the vertical projection of a convex polyhedral surface. We show that the vertices of this surface may be placed in a $4n^3 \times 8n^5 \times \zeta(n)$ integer grid, where $\zeta(n) \leq (500 n^8)^{\tau(G)}$ and $\tau(G)$ denotes the shedding diameter of $G$, a quantity defined in the paper.