Upward star-shaped polyhedral graphs

Abstract

We introduce a new wider class of polyhedra called upward (star-shaped) polyhedra, and present a graph-theoretic characterization. Our proof includes a drawing algorithm which constructs an upward polyhedron with n vertices in O(n 1.5) time. Moreover, we can test whether a given plane graph is an upward polyhedral graph in linear time. Our result is the first graph-theoretic characterization of non-convex polyhedra, which solves an open problem posed by Grünbaum [6], and a generalization of the Steinitz' theorem [9]. © 2009 Springer-Verlag Berlin Heidelberg.