Algorithms for finding a maximum non-k-linked graph

Authors:
Yusuke Kobayashi;Yuichi Yoshida
Affiliations:
Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo;School of Informatics, Kyoto University, and Preferred Infrastructure, Inc.
Venue:
ESA'11 Proceedings of the 19th European conference on Algorithms
Year:
2011

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Abstract

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.