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

Ito, T. and Kaminski, M. and Paulusma, Daniel and Thilikos, D. M. (2009) 'Parameterizing cut sets in a graph by the number of their components.', in Algorithms and computation : 20th International Symposium, ISAAC 2009, 16-18 December 2009, Honolulu, Hawaii, USA ; proceedings. Berlin: Springer, pp. 605-615. Lecture notes in computer science. (5878).


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.

Item Type:Book chapter
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Date of first online publication:2009
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