Hardness results on generalized connectivity

Let G be a nontrivial connected graph of order n and let k be an integer with 2 ≤ k ≤ n. For a set S of k vertices of G, let κ(S) denote the maximum number l of edge-disjoint trees T1, T2, . . . , Tl in G such that V (Ti)∩V (Tj) = S for every pair i, j of distinct integers with 1 ≤ i, j ≤ l. A collection {T1, T2, . . . , Tl} of trees in G with this property is called an internally disjoint set of trees connecting S. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by κk(G), of G is defined by κk(G) =min{κ(S)}, where the minimum is taken over all k-subsets S of V (G). Thus κ2(G) = κ(G), where κ(G) is the connectivity of G, for which there are polynomial-time algorithms to solve it. This paper mainly focus on the complexity of the generalized connectivity. At first, we obtain that for two fixed positive integers k1 and k2, given a graph G and a k1-subset S of V (G), the problem of deciding whether G contains k2 internally disjoint trees connecting S can be solved by a polynomial-time algorithm. Then, we show that when k1 is a fixed integer of at least 4, but k2 is not a fixed integer, the problem turns out to be NP-complete. On the other hand, when k2 is a fixed integer of at least 2, but k1 is not a fixed integer, we show that the problem also becomes NP-complete. Finally we give some open problems.