A polytopeP is called asubpolytope of a polytopeQ if vertP ⊆ vertQ. The purpose of this paper is to construct examples of 3-dimensional polytopes which are not subpolytopes of stack polytopes. Previously no such examples were known.

The idea of a net will be familiar to anyone who has made a model of a three-dimensional convex polytope (3-polytope) out of a flat sheet of card or similar material. To begin with, one cuts out a polygon, and then the model is formed by folding this and joining its edges in an appropriate manner. For example, Fig. 1 illustrates two nets for a regular tetrahedron. The folds are indicated by broken lines and lettering (either of the edges or of the vertices) indicates how the edges are to be joined.

Subpolytopes of stack polytopes

Convex polytopes with convex nets

Higherdimensional Analogues of Csaszar's Torus

background

/pdf/geometric-representations-of-graphs-with-low-polygonal-2kwb3qk6se.pdf

Geometric Representations of Graphs with Low Polygonal Complexity

For a planar graph H and a positive integer n we study the maximal number f = f(n, H ), such that there exists a planar graph on n vertices containing f subgraphs isomorphic to H  We determine f(n, H ) precisely if H is either a complete bipartite (planar) graph or a maximal planar graph without triangles that are not faces, and estimate f(n, H ) for every maximal planar graph H 

On The Number of Subgraphs of Prescribed Type of Planar Graphs With A Given Number of Vertices

Let P be a (convex) d -polytope in the Euclidean space E d and p a point of E d not contained in P or in a supporting hyperplane of a facet of P (we use the terminology of Grunbaum [2]). The part of the boundary of P which is “visible” from p , i.e. the union of those facets whose supporting hyperplanes separate p and P , form a (d – l)-ball, whose boundary is a ( d – 2)-sphere S (all balls, spheres and manifolds to be considered are piecewise-linear). The boundary-complex ℬ( P ) induces a subdivision of S , which we call a (sharp) shadow-boundary of P . is combinatorially isomorphic to the boundary complex of a poly tope which is a central projection of P from the point p on a hyperplane.

Subpolytopes of stack polytopes

Citations

Convex polytopes with convex nets

Higherdimensional Analogues of Csaszar's Torus

Geometric Representations of Graphs with Low Polygonal Complexity

On The Number of Subgraphs of Prescribed Type of Planar Graphs With A Given Number of Vertices

Shadow-boundaries and cuts of convex polytopes

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