The excess degree of a polytope

© 2018 Society for Industrial and Applied Mathematics. We define the excess degree \xi (P) of a d-polytope P as 2f1 - df0, where f0and f1denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple. It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d - 2, and the value d - 1 only occurs when d = 3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., \xi (P) < d) have a very particular structure: provided d
ot = 5, either there is a unique nonsimple vertex, or every nonsimple vertex has degree d + 1. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Second, we characterize completely the decomposable d-polytopes with 2d + 1 vertices (up to combinatorial equivalence). Third, all pairs (f0, f1), for which there exists a 5-polytope with f0vertices and f1 edges, are determined.