Polytopes with low excess degree

We study the existence and structure of d-polytopes for which the number f₁ of edges is small compared to the number f₀ of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as 2f₁-df₀. We show that the excess degree of a d-polytope cannot lie in the range $[d+3,2d-7]$, complementing the known result that values in the range $[1,d-3]$ are impossible. In particular, many pairs (f₀,f₁) are not realised by any polytope. For d-polytopes with excess degree $d-2$, strong structural results are known; we establish comparable results for excess degrees d, $d+2$, and $2d-6$. Frequently, in polytopes with low excess degree, say at most $2d-6$, the nonsimple vertices all have the same degree and they form either a face or a missing face. We show that excess degree $d+1$ is possible only for $d=3,5$, or $7$, complementing the known result that an excess degree $d-1$ is possible only for $d=3$ or $5$.