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ISAAC
2009
Springer

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

9 years 10 months ago
Parameterizing Cut Sets in a Graph by the Number of Their Components
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 ape...
Takehiro Ito, Marcin Kaminski, Daniël Paulusm
Added 25 Jul 2010
Updated 25 Jul 2010
Type Conference
Year 2009
Where ISAAC
Authors Takehiro Ito, Marcin Kaminski, Daniël Paulusma, Dimitrios M. Thilikos
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