Algorithms for Finding a Maximum Non-k-linked Graph

A graph with at least 2k vertices is said to be k-linked if for any ordered k-tuples (s1, . . . , sk) and (t1, . . . , tk) of 2k distinct vertices, there exist pairwise vertex-disjoint paths P1, . . . , Pk such that Pi connects si and ti for i = 1, . . . , k. For a given graph G, we consider the problem of finding a maximum induced subgraph of G that is not k-linked. This problem is a common generalization of computing the vertex-connectivity and testing the k-linkedness of G, and it is closely related to the concept of H-linkedness. In this paper, we give the first polynomial-time algorithm for the case of k = 2, whereas a similar problem that finds a maximum induced subgraph without 2-vertex-disjoint paths connecting fixed terminal pairs is NP-hard. For the case of general k, we give an (8k−2)-additive approximation algorithm. We also investigate the computational complexities of the edge-disjoint case and the directed case.