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

Ito, T.; Kaminski, M.; Paulusma, D.; Thilikos, D.M.

Abstract

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.

Citation

Ito, T., Kaminski, M., Paulusma, D., & Thilikos, D. (2011). Parameterizing cut sets in a graph by the number of their components. Theoretical Computer Science, 412(45), 6340-6350. https://doi.org/10.1016/j.tcs.2011.07.005

• Journal Article Type: Article
• Publication Date: Oct 1, 2011
• Deposit Date: Dec 6, 2011
• Journal: Theoretical Computer Science
• Print ISSN: 0304-3975
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 412
• Issue: 45
• Pages: 6340-6350
• DOI: https://doi.org/10.1016/j.tcs.2011.07.005
• Keywords: Cut set, 2K2-partition, Graph contractibility.
• Public URL: https://durham-repository.worktribe.com/output/1502622