An analogue of a theorem of Steinitz for ball polyhedra in $\mathbb{R}^3$

TL;DR

This paper extends Steinitz's theorem from convex polyhedra to ball polyhedra in three-dimensional space, characterizing their edge-graphs through combinatorial properties.

Contribution

It provides a new analogue of Steinitz's theorem specifically for intersections of finitely many unit balls in r3, broadening the understanding of polyhedral graph characterizations.

Findings

Characterization of edge-graphs of ball polyhedra

Extension of Steinitz's theorem to non-convex intersections

Conditions for graphs to correspond to ball polyhedra

Abstract

Steinitz's theorem states that a graph $G$ is the edge-graph of a $3$-dimensional convex polyhedron if and only if, $G$ is simple, plane and $3$-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in $\mathbb{R}^3$.