Improved Algorithms for Grid-Unfolding Orthogonal Polyhedra

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. For 1-layer orthogonal polyhedra of genus [Formula: see text], we show a grid-unfolding algorithm using only [Formula: see text] additional cuts, affirmatively answering an open problem raised in a recent literature. Our approach not only requires fewer cuts but yields much simpler algorithms.