# Graphs of zonohedra

A paper appeared yesterday on arXiv that I would find very interesting if I believed it to be true: “Characterizing graphs of zonohedra” by Adnan and Hasan. For the purposes of this paper a zonohedron is a centrally symmetric convex polyhedron in which each face is a parallelogram. My understanding is that the graphs of such things (that is, their vertices and edges) are the planar duals of simple line arrangements in the oriented projective plane, while the characterization they claim is equivalent to saying that they are the duals of simple pseudoline arrangements. But the two are not the same, and in fact it's known to be NP-hard to distinguish pseudoline arrangements from line arrangements [Shor,

If a graph is the dual of a pseudoline arrangement but not of a line arrangement, it is still possible to represent it as the skeleton of a polyhedron with parallelogram faces in which some of the dihedral angles are convex and others are flat (exactly π); see my paper on simplicial arrangements for a construction that works equally well for simple arrangements. So I suppose the new paper's characterization could be made to be true by suitably broadening the definition of a convex polyhedron...

*The Victor Klee Festschrift*, 1991], making a characterization of the type they claim unlikely.If a graph is the dual of a pseudoline arrangement but not of a line arrangement, it is still possible to represent it as the skeleton of a polyhedron with parallelogram faces in which some of the dihedral angles are convex and others are flat (exactly π); see my paper on simplicial arrangements for a construction that works equally well for simple arrangements. So I suppose the new paper's characterization could be made to be true by suitably broadening the definition of a convex polyhedron...