Constructing certain families of 3-polytopal graphs

Let (Formula presented.) and (Formula presented.) be a 3-polytopal graph such that for every (Formula presented.), (Formula presented.) has at least one vertex of degree (Formula presented.). We find the minimal vertex count for (Formula presented.). We then describe an algorithm to construct the graphs (Formula presented.). A dual statement may be formulated for faces of 3-polytopes. The ideas behind the algorithm generalise readily to solve related problems. Moreover, given a 3-polytope (Formula presented.) comprising a vertex of degree (Formula presented.) for all (Formula presented.), (Formula presented.) fixed, we define an algorithm to output for (Formula presented.) a 3-polytope (Formula presented.) comprising a vertex of degree (Formula presented.), for all (Formula presented.), and such that the initial (Formula presented.) is a subgraph of (Formula presented.). The vertex count of (Formula presented.) is asymptotically optimal, in the sense that it matches the aforementioned minimal vertex count up to order of magnitude, as (Formula presented.) gets large. In fact, we only lose a small quantity on the coefficient of the second highest term, and this quantity may be taken as small as we please, with the tradeoff of first constructing an accordingly large auxiliary graph.