Parameterizing cut sets in a graph by the number of their components
Parameterizing cut sets in a graph by the number of their components
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DOI：
10.1016/j.tcs.2011.07.005
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发表时间：
2009-12
期刊：
影响因子：
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通讯作者：
Takehiro Ito;M. Kaminski;D. Paulusma;D. Thilikos
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文献类型：
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作者：
Takehiro Ito;M. Kaminski;D. Paulusma;D. Thilikos
对于连通图G=（V，E），如果U不连通G且由U导出的子图G[U]也不连通，则子集U ∈ V是不连通割.一个割U是k-割，如果G[U]恰好包含k（≥1）个分支。更具体地说，一个k-割U是一个（k，n）-割，如果V <$U诱导一个子图恰好有n（≥2）个分支。不连通割问题是检验一个图是否有不连通割，并且已知它是NP完全的。问题k-Cut和（k，n）-Cut分别用来检验一个图是否有k-割或（k，n）-割。通过指出与图的可压缩性问题的密切关系，证明了（k，n）-割在P中，其中k=1，且任意固定常数n ≥2，而对于任意固定对k，n ≥2，（k，n）-割是NP完全的.证明了当k=1时，k-割在P中，当k≥2时，k-割是NP-完全的.另一方面，对于任意固定整数g≥0，我们给出了一个FPT算法，当参数为k+ ∞时，该算法能求解Euler亏格图的（k，∞）-Cut。通过修改这个算法，我们还可以证明，当用k参数化时，对于这个图类，k-割是在FPT中。最后，我们证明了不连通割是多项式时间可解的少数封闭类的图，不包括一些顶点图。
For a connected graph G=(V,E), a subset U⊆V is a disconnected cut if U disconnects G and the subgraph G[U] induced by U is disconnected as well. A cut U is a k-cut if G[U] contains exactly k(≥1) components. More specifically, a k-cut U is a (k,ℓ)-cut if V∖U induces a subgraph with exactly ℓ(≥2) components. The Disconnected Cut problem is to test whether a graph has a disconnected cut and is known to be NP-complete. The problems k-Cut and (k,ℓ)-Cut are to test whether a graph has a k-cut or (k,ℓ)-cut, respectively. By pinpointing a close relationship to graph contractibility problems we show that (k,ℓ)-Cut is in P for k=1 and any fixed constant ℓ≥2, while it is NP-complete for any fixed pair k,ℓ≥2. We then prove that k-Cut is in P for k=1 and NP-complete for any fixed k≥2. On the other hand, for every fixed integer g≥0, we present an FPT algorithm that solves (k,ℓ)-Cut on graphs of Euler genus at most g when parameterized by k+ℓ. By modifying this algorithm we can also show that k-Cut is in FPT for this graph class when parameterized by k. Finally, we show that Disconnected Cut is solvable in polynomial time for minor-closed classes of graphs excluding some apex graph.