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

Takehiro Ito, Marcin Kamiński, Daniël Paulusma, Dimitrios M. Thilikos

Research output: Chapter in Book/Report/Conference proceedingConference contribution


For a connected graph G=(V,E), a subset U⊂V is called a k-cut if U disconnects G, and the subgraph induced by U contains exactly k (≥1) components. More specifically, a k-cut U is called a (k,ℓ)-cut if V \U induces a subgraph with exactly ℓ (≥2) components. We study two decision problems, called k-Cut and (k,ℓ)-Cut, which determine whether a graph G has a k-cut or (k,ℓ)-cut, respectively. By pinpointing a close relationship to graph contractibility problems we first show that (k,ℓ)-Cut is in P for k=1 and any fixed constant ℓ≥2, while the problem is NP-complete for any fixed pair k,ℓ≥2. We then prove that k-Cut is in P for k=1, and is NP-complete for any fixed k≥2. On the other hand, we present an FPT algorithm that solves (k,ℓ)-Cut on apex-minor-free graphs when parameterized by k+ℓ. By modifying this algorithm we can also show that k-Cut is in FPT (with parameter k) and Disconnected Cut is solvable in polynomial time for apex-minor-free graphs. The latter problem asks if a graph has a k-cut for some k≥2.

Original languageEnglish (US)
Title of host publicationAlgorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings
Number of pages11
StatePublished - 2009
Event20th International Symposium on Algorithms and Computation, ISAAC 2009 - Honolulu, HI, United States
Duration: Dec 16 2009Dec 18 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5878 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference20th International Symposium on Algorithms and Computation, ISAAC 2009
Country/TerritoryUnited States
CityHonolulu, HI

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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