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 et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189–194] asserts that the least order of a graph with a given degree set DD is 1+max(D)1+max(D). We look at the analogous problem concerning the least size of a graph with a given degree set DD. We determine the least size for the sets DD when (i) |D|⩽3|D|⩽3; (ii) D={1,2,…,n}D={1,2,…,n}; and (iii) every element in DD is at least |D||D|. In addition, we give sharp upper and lower bounds in all cases.