The Minimum Size of a Graph with Given Tree Connectivity

For a graph G = (V, E) and a set S ⊆ V of at least two vertices, an S-tree is a such subgraph T of G that is a tree with S ⊆ V (T). Two S-trees T1 and T2 are said to be internally disjoint if E(T1) ∩ E(T2) = ∅ and V (T1) ∩ V (T2) = S, and edge-disjoint if E(T1) ∩ E(T2) = ∅. The generalized local connectivity κG(S) (generalized local edge-connectivity λG(S), respectively) is the maximum number of internally disjoint (edge-disjoint, respectively) S-trees in G. For an integer k with 2 ≤ k ≤ n, the generalized k-connectivity (generalized k-edge-connectivity, respectively) is defined as κk(G) = min{κG (S) | S ⊆ V (G), |S| = k} (λk(G) = min{λG(S) | S ⊆ V (G), |S| = k}, respectively)