Algorithms for finding a maximum non-kappa;-linked graph

Abstract

A graph with at least 2k vertices is said to be k-linked if for any ordered k-tuples (s1, ⋯, sk ) and (t1,⋯, t k ) of &k distinct vertices, there exist pairwise vertex-disjoint paths P 1, ⋯, P k such that P i connects s i and t i 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. © 2011 Springer-Verlag Berlin Heidelberg.