Note on the hardness of generalized connectivity

Abstract

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 T ₁,T ₂,…,T _ℓ in G such that V(T _i )∩V(T _j )=S for every pair i,j of distinct integers with 1≤i,j≤ℓ. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by κ _k (G), of G is defined by κ _k (G)=min{κ(S)}, where the minimum is taken over all k-subsets S of V(G). Thus κ ₂(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 determining the generalized connectivity of a graph. At first, we obtain that for two fixed positive integers k ₁ and k ₂, given a graph G and a k ₁-subset S of V(G), the problem of deciding whether G contains k ₂ internally disjoint trees connecting S can be solved by a polynomial-time algorithm. Then, we show that when k ₁ is a fixed integer of at least 4, but k ₂ is not a fixed integer, the problem turns out to be NP-complete. On the other hand, when k ₂ is a fixed integer of at least 2, but k ₁ is not a fixed integer, we show that the problem also becomes NP-complete.