不连通割与2K ~ 2-剖分的计算复杂性
The Computational Complexity of Disconnected Cut and 2K2-Partition
摘要：
对于连通图G=（V，E），如果U不连通图G，且由U导出的子图也不连通，则称V的子集U为不连通割.我们证明了测试图是否有不连通割的问题是NP-完全的。这个问题多项式等价于以下问题：测试一个图是否有2K 2-划分，测试一个图是否允许顶点满射同态到自反4-圈，测试一个图是否有一个至多由两个biclique组成的生成子图。因此，作为直接的结果，这三个决策问题也是NP完全的。这就解决了在这四种情况下经常出现的一个悬而未决的问题。
英文摘要：
For a connected graph G=(V,E), a subset U of V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is NP-complete. This problem is polynomially equivalent to the following problems: testing if a graph has a 2K2-partition, testing if a graph allows a vertex-surjective homomorphism to the reflexive 4-cycle and testing if a graph has a spanning subgraph that consists of at most two bicliques. Hence, as an immediate consequence, these three decision problems are NP-complete as well. This settles an open problem frequently posed in each of the four settings.
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