Characterizing Graphs of Zonohedra

TL;DR

This paper characterizes the graphs of zonohedra, a special class of convex polyhedra, and provides a linear-time algorithm to recognize such graphs, along with bounds on their structural properties.

Contribution

It offers a new characterization of zonohedron graphs and introduces an efficient recognition algorithm with proven structural bounds.

Findings

Recognition algorithm runs in linear time.

Number of zones and faces per zone is O(√n).

Provides a structural characterization of zonohedron graphs.

Abstract

A classic theorem by Steinitz states that a graph G is realizable by a convex polyhedron if and only if G is 3-connected planar. Zonohedra are an important subclass of convex polyhedra having the property that the faces of a zonohedron are parallelograms and are in parallel pairs. In this paper we give characterization of graphs of zonohedra. We also give a linear time algorithm to recognize such a graph. In our quest for finding the algorithm, we prove that in a zonohedron P both the number of zones and the number of faces in each zone is O(square root{n}), where n is the number of vertices of P.