The size of graphs with given edge inclusive connectivity

In [Discrete Math. 46 (1983) 191 - 198], the concept of inclusive edge connectivity was introduced and discussed. Given a vertex upsilon is an element of V(G), the inclusive edge connectivity of upsilon, denoted by lambda(i)(upsilon, G), is the minimum number of edges whose deletion results in a subgraph of G in which upsilon is a cut-vertex. Define lambda(i)(G) = min{lambda(i)(upsilon, G) : upsilon is an element of V(G), and d(G)(upsilon)greater than or equal to 2} to be the inclusive edge connectivity of G. Extremal problems on lambda(i)(G) are studied in this paper.