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Parameterizing cut sets in a graph by the number of their components.

Ito, T. and Kaminski, M. and Paulusma, Daniel and Thilikos, D.M. (2011) 'Parameterizing cut sets in a graph by the number of their components.', Theoretical computer science., 412 (45). pp. 6340-6350.


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.

Item Type:Article
Keywords:Cut set, 2K2-partition, Graph contractibility.
Full text:Full text not available from this repository.
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Record Created:07 Dec 2011 14:51
Last Modified:03 Apr 2013 16:19

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