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A graph with at least 2k vertices is said to be k-linked if for any two 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 vertex-connectivity and testing 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 to find 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 complexity of the edge-disjoint case and the directed case. [ABSTRACT FROM AUTHOR]

Endo proved that every toroidal graph has a book embedding with at most seven pages. In this paper, we prove that every toroidal bipartite graph has a book embedding with at most five pages. In order to do so, we prove that every bipartite torus quadrangulation with Q n vertices admits two disjoint noncontractible simple closed curves cutting the torus into two annuli so that each of the two annuli contains a spanning connected subgraph of with exactly edges Q n satisfying a certain condition. [ABSTRACT FROM AUTHOR]