Unfolding orthogonal polyhedra with linear refinement

Abstract

An unfolding of a polyhedron is a single connected planar piece without overlap resulting from cutting and flattening the surface of the polyhedron. Even for orthogonal polyhedra, it is known that edge-unfolding, i.e., cuts are performed only along the edges of a polyhedron, is not sufficient to guarantee a successful unfolding in general. However, if additional cuts parallel to polyhedron edges are allowed, it has been shown that every orthogonal polyhedron of genus zero admits a grid-unfolding with quadratic refinement. Using a new unfolding technique developed in this paper, we improve upon the previous result by showing that linear refinement suffices. Our approach not only requires fewer cuts but is also much simpler.