T. Ito
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 |
You might also like
Matching cuts in graphs of high girth and H-free graphs
(2023)
Conference Proceeding
Solving problems on generalized convex graphs via mim-width
(2023)
Journal Article
On the price of independence for vertex cover, feedback vertex set and odd cycle transversal
(2023)
Journal Article
Computing Subset Vertex Covers in H-Free Graphs
(2023)
Conference Proceeding
Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius
(2023)
Conference Proceeding
Downloadable Citations
About Durham Research Online (DRO)
Administrator e-mail: dro.admin@durham.ac.uk
This application uses the following open-source libraries:
SheetJS Community Edition
Apache License Version 2.0 (http://www.apache.org/licenses/)
PDF.js
Apache License Version 2.0 (http://www.apache.org/licenses/)
Font Awesome
SIL OFL 1.1 (http://scripts.sil.org/OFL)
MIT License (http://opensource.org/licenses/mit-license.html)
CC BY 3.0 ( http://creativecommons.org/licenses/by/3.0/)
Powered by Worktribe © 2024
Advanced Search