The vertex connectivity k (G) of a graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph, that is, a graph with only one vertex. The edge connectivity (G) of a nontrivial graph G is the minimum number of edges whose removal results in a disconnected graph. It is known that the vertex connectivity k (G) of a graph G, the edge connectivity (G), and the minimum vertex degree (G) are related by the Whitney inequality k (G) (G) (G). Previously, the authors solved the problem of finding the minimum number of vertices and edges in graphs with given k (G), (G), and (G). The present paper continues this study and essentially uses its results. Formulas for the minimum number of vertices and edges in graphs with given k (G) and (G) are presented. Graphs with the minimum number of edges among all n -vertex graphs with given k (G) and (G) are considered. A family of such graphs with n/2 edges is described. A special case of this family for k= is the Harary graphs H₊, ₍, that is, n -vertex k -connected graphs with the minimum number of edges. It is known that the number of edges in such graphs equals k n/2. The result is also related to the problem of determining the minimum number of edges in an n -vertex graph with given (G), which was solved by Fulkerson and Shapley.
Abrosimov et al. (Mon,) studied this question.