### Abstract

The subset feedback set problem, which is a generalization of the well-known feedback vertex set problem, is that we are given an undirected graph G with a vertex subset S and a positive integer k, and the goal is to find a vertex set X of size at most k such that G-. X has no S-cycle, where an S-cycle is a cycle having at least one vertex of S. It was recently shown that this problem is fixed parameter tractable, where k is the parameter. In this paper, we further generalize this problem to one with the parity constraints, and show the fixed parameter tractability:. 1.For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k such that G-X has no S-cycle of even length, or concludes that such a vertex set does not exist.2.For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k such that G-X has no S-cycle of odd length, or concludes that such a vertex set does not exist.

Original language | English |
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Pages (from-to) | 61-76 |

Number of pages | 16 |

Journal | Theoretical Computer Science |

Volume | 576 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

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### Keywords

- Fixed-parameter algorithm
- Graph minor theory
- Parity constraints
- Subset feedback set problem

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*576*(1), 61-76. https://doi.org/10.1016/j.tcs.2015.02.004