The Excess Degree of a Polytope

Source abstract

We define the excess degree $\xi(P)$ of a $d$-polytope $P$ as $2f_1-df_0$, where $f_0$ and $f_1$ denote 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 $d\ne5$, 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 $(f_0,f_1)$, for which there exists a 5-polytope with $f_0$ vertices and $f_1$ edges, are determined.