Convex and linear orientations of polytopal graphs (Q1580764)

Suppose \(G\) is the graph formed by the vertices and edges of some \(d\)-polytope. Then a graph is \(d\)-polytopal if it is isomorphic to \(G\) and a digraph is \(d\)-polytopal if it is isomorphic to a digraph that results when \(G\) is oriented by means of some affine function on the \(d\)-polytope. It is proven that for each fixed \(d\)-polytope and any acyclic orientation of the graph, there exist both concave and convex functions that induce the orientation. A characterization is given of the orientation induced by an affine function acting on a member of each combinatorial class of 3-polytopes.