### 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 |
---|---|

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 |

### Fingerprint

### 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

**Fixed-parameter tractability for subset feedback set problems with parity constraints.** / Kakimura, Naonori; Kawarabayashi, Ken ichi.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 576, no. 1, pp. 61-76. https://doi.org/10.1016/j.tcs.2015.02.004

}

TY - JOUR

T1 - Fixed-parameter tractability for subset feedback set problems with parity constraints

AU - Kakimura, Naonori

AU - Kawarabayashi, Ken ichi

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Fixed-parameter algorithm

KW - Graph minor theory

KW - Parity constraints

KW - Subset feedback set problem

UR - http://www.scopus.com/inward/record.url?scp=84930246544&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930246544&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2015.02.004

DO - 10.1016/j.tcs.2015.02.004

M3 - Article

AN - SCOPUS:84930246544

VL - 576

SP - 61

EP - 76

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 1

ER -