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

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 ${\mathbb{R}}^3$.

9 pages