Steinitz Theorems for Orthogonal Polyhedra

TL;DR

This paper characterizes the graphs of simple orthogonal polyhedra, including corner and xyz types, providing graph-theoretic criteria and efficient algorithms for their construction.

Contribution

It introduces new graph-theoretic characterizations of simple orthogonal polyhedra and develops algorithms to construct these polyhedra from their graphs.

Findings

Graphs of xyz polyhedra are bipartite cubic polyhedral graphs.

Every bipartite cubic polyhedral graph with a 4-connected dual is a corner polyhedron.

Efficient algorithms are provided for constructing orthogonal polyhedra from their graphs.

Abstract

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.