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 D is 1 + max(D). We look at the analogous problem concerning the least size of a graph with a given degree set D. We determine the least size for the sets D when (i) vertical bar D vertical bar < 3; (ii) D = {1, 2,..., n}; and (iii) every element in D is at least vertical bar D vertical bar. In addition, we give sharp upper and lower bounds in all cases. (c) 2006 Elsevier B.V. All rights reserved.