  An edge-unfolding of a polyhedron is produced by cutting along edges and
flattening the faces to a *net*, a connected planar piece with no overlaps. A
*grid unfolding* allows additional cuts along grid edges induced by coordinate
planes passing through every vertex. A vertex-unfolding permits faces in the
net to be connected at single vertices, not necessarily along edges. We show
that any orthogonal polyhedron of genus zero has a grid vertex-unfolding.
(There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of
"gridding" of the faces is necessary.) For any orthogonal polyhedron P with n
vertices, we describe an algorithm that vertex-unfolds P in O(n^2) time.
Enroute to explaining this algorithm, we present a simpler vertex-unfolding
algorithm that requires a 3 x 1 refinement of the vertex grid.

Computational Geometry;Discrete Mathematics