Steinitz theorems for orthogonal polyhedra

  • Authors:

  • David Eppstein;Elena Mumford

  • Affiliations:

  • Univ. of California, Irvine, Irvine, CA, USA;Technische Universiteit Eindhoven, Eindhoven, Netherlands

  • Venue:
  • Proceedings of the twenty-sixth annual symposium on Computational geometry
  • Year:
  • 2010

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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 characterize the graphs of simple orthogonal polyhedra: they are exactly the 3-regular bipartite planar graphs in which the removal of any two vertices produces at most two connected components. We also characterize two subclasses of these polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, and xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs