19 Polytope Skeletons and Paths
نویسنده
چکیده
The k-dimensional skeleton of a d-polytope P is the set of all faces of the polytope of dimension at most k. The 1-skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 19.1 we briefly describe the situation for 3polytopes. In Section 19.2 we consider general properties of polytopal graphs— subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 19.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and the Hirsch conjecture. The short Section 19.4 is devoted to polytopal digraphs. Section 19.5 is devoted to skeletons of polytopes, connectivity, collapsibility and shellability, empty faces and polytopes with “few vertices,” and the reconstruction of polytopes from their lowdimensional skeletons; we also consider what can be said about the collections of all k-faces of a d-polytope, first for k = d−1 and then when k is fixed and d is large compared to k. Section 19.6 describes some results on counting high dimensional polytopes and spheres.
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