### Abstract

The well-known Erdocombining double acute accents-Pósa theorem says that for any integer k and any graph G, either G contains k vertex-disjoint cycles or a vertex set X of order at most c-k log k (for some constant c) such that G-X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science. In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdocombining double acute accents-Pósa theorem still holds for this subset version. In this paper, we extend both Thomassen's result and Reed's result in this way. In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G-X has no S-cycle, or conclude that such a vertex set does not exist in O(n ^{2}m) time (independently obtained in [7]). 2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(n ^{2}m) time. We also extend the above results to those with the parity constraints as follows; 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 even S-cycle, 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 odd S-cycle, or concludes that such a vertex set does not exist. 3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint even S-cycles, or concludes that such k disjoint cycles do not exist. 4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist.

Original language | English |
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Title of host publication | Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 |

Pages | 1726-1736 |

Number of pages | 11 |

Publication status | Published - 2012 |

Externally published | Yes |

Event | 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan Duration: 2012 Jan 17 → 2012 Jan 19 |

### Other

Other | 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 |
---|---|

Country | Japan |

City | Kyoto |

Period | 12/1/17 → 12/1/19 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012*(pp. 1726-1736)

**Erdocombining double acute accents-Pósa property and its algorithmic applications - Parity constraints, subset feedback set, and subset packing.** / Kakimura, Naonori; Kawarabayashi, Ken Ichi; Kobayashi, Yusuke.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012.*pp. 1726-1736, 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, 12/1/17.

}

TY - GEN

T1 - Erdocombining double acute accents-Pósa property and its algorithmic applications - Parity constraints, subset feedback set, and subset packing

AU - Kakimura, Naonori

AU - Kawarabayashi, Ken Ichi

AU - Kobayashi, Yusuke

PY - 2012

Y1 - 2012

N2 - The well-known Erdocombining double acute accents-Pósa theorem says that for any integer k and any graph G, either G contains k vertex-disjoint cycles or a vertex set X of order at most c-k log k (for some constant c) such that G-X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science. In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdocombining double acute accents-Pósa theorem still holds for this subset version. In this paper, we extend both Thomassen's result and Reed's result in this way. In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G-X has no S-cycle, or conclude that such a vertex set does not exist in O(n 2m) time (independently obtained in [7]). 2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(n 2m) time. We also extend the above results to those with the parity constraints as follows; 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 even S-cycle, 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 odd S-cycle, or concludes that such a vertex set does not exist. 3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint even S-cycles, or concludes that such k disjoint cycles do not exist. 4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist.

AB - The well-known Erdocombining double acute accents-Pósa theorem says that for any integer k and any graph G, either G contains k vertex-disjoint cycles or a vertex set X of order at most c-k log k (for some constant c) such that G-X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science. In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdocombining double acute accents-Pósa theorem still holds for this subset version. In this paper, we extend both Thomassen's result and Reed's result in this way. In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G-X has no S-cycle, or conclude that such a vertex set does not exist in O(n 2m) time (independently obtained in [7]). 2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(n 2m) time. We also extend the above results to those with the parity constraints as follows; 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 even S-cycle, 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 odd S-cycle, or concludes that such a vertex set does not exist. 3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint even S-cycles, or concludes that such k disjoint cycles do not exist. 4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist.

UR - http://www.scopus.com/inward/record.url?scp=84860116441&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860116441&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84860116441

SN - 9781611972108

SP - 1726

EP - 1736

BT - Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012

ER -