Abstract
Let G=(V,E) be an undirected graph with a node set V and an arc set E. G has k pairwise disjoint subsets T 1,T 2,...,T k of nodes, called resource sets, where |T i | is even for each i. The partition problem with k resource sets asks to find a partition V 1 and V 2 of the node set V such that the graphs induced by V 1 and V 2 are both connected and |V 1 ∩ T i |=|V 2 ∩ T i |=|T i |/2 holds for each i=1,2,...,k. The problem of testing whether such a bisection exists is known to be NP-hard even in the case of k=1. On the other hand, it is known that that if G is (k+1)-connected for k=1,2, then a bisection exists for any given resource sets, and it has been conjectured that for k≥ 3, a (k+1)-connected graph admits a bisection. In this paper, we show that for k=3, the conjecture does not hold, while if G is 4-connected and has K 4 as its subgraph, then a bisection exists and it can be found in O(|V|3 log |V|) time. Moreover, we show that for an arc-version of the problem, the (k+1)-edge-connectivity suffices for k=1,2,3.
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Ishii, T., Iwata, K., Nagamochi, H. (2005). Bisecting a Four-Connected Graph with Three Resource Sets. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_19
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DOI: https://doi.org/10.1007/11602613_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30935-2
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