ISAAC 2009: Algorithms and Computation pp 605-615

# Parameterizing Cut Sets in a Graph by the Number of Their Components

• Takehiro Ito
• Marcin Kamiński
• Daniël Paulusma
• Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5878)

## Abstract

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.

## Keywords

Planar Graph Connected Graph Line Graph Chordal Graph Span Subgraph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Takehiro Ito
• 1
• Marcin Kamiński
• 2
• Daniël Paulusma
• 3
• Dimitrios M. Thilikos
• 4
1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
2. 2.Computer Science DepartmentUniversité Libre de BruxellesBrusselsBelgium
3. 3.Department of Computer ScienceUniversity of Durham, Science LaboratoriesDurhamEngland
4. 4.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece