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.