## 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.

Original language | English |
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Pages (from-to) | 6340-6350 |

Number of pages | 11 |

Journal | Theoretical Computer Science |

Volume | 412 |

Issue number | 45 |

DOIs | |

Publication status | Published - 2011 Oct 21 |

## Keywords

- 2K2-partition
- Cut set
- Graph contractibility

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)