Exact and approximate results on the least size of a graph with a given degree set

The degree set of a finite simple graph G is the set of distinct degrees of vertices of G. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set D is 1+maxD. Tripathi & Vijay considered the analogous problem concerning the least size of graphs with degree set D. We expand on their results, and determine the least size of graphs with degree set D when (i) minD divides d for each d∈D; (ii) minD=2; (iii) D={m,m+1,…,n}. In addition, given any D, we produce a graph G whose size is within minD of the optimal size, giving a (1+2d1+1)-approximation, where d1=maxD.