### Abstract

In this paper, we show the Erdos-Pósa property for edge-disjoint packing of S-closed walks with parity constraints in 4-edge-connected graphs. More precisely, we prove that for any 4-edge-connected graph G and any vertex subset S, either G has k edge-disjoint elementary closed odd walks, each of which has at least one vertex of S, or G has an edge set F with |F| ≥ f(k) such that G - F has no such walks. The 4-edge-connectivity is the best possible in the sense that 3-edge-connected graphs do not satisfy the statement. Since the proof is constructive, we can design a fixed-parameter algorithm for finding k edge-disjoint walks satisfying the conditions in a 4-edge-connected graph for a parameter k. In addition, this gives a simple fixed-parameter algorithm for the parity edge-disjoint walks problem with k terminal pairs.

Original language | English |
---|---|

Pages (from-to) | 766-782 |

Number of pages | 17 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2017 |

### Fingerprint

### Keywords

- Cycle packing
- Erdos-Pósa property
- Fixed-parameter algorithm

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*31*(2), 766-782. https://doi.org/10.1137/15M1022239

**Packing edge-disjoint odd Eulerian subgraphs through prescribed vertices in 4-edge-connected graphs.** / Kakimura, Naonori; Kawarabayashi, Ken Ichi; Kobayashi, Yusuke.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 31, no. 2, pp. 766-782. https://doi.org/10.1137/15M1022239

}

TY - JOUR

T1 - Packing edge-disjoint odd Eulerian subgraphs through prescribed vertices in 4-edge-connected graphs

AU - Kakimura, Naonori

AU - Kawarabayashi, Ken Ichi

AU - Kobayashi, Yusuke

PY - 2017

Y1 - 2017

N2 - In this paper, we show the Erdos-Pósa property for edge-disjoint packing of S-closed walks with parity constraints in 4-edge-connected graphs. More precisely, we prove that for any 4-edge-connected graph G and any vertex subset S, either G has k edge-disjoint elementary closed odd walks, each of which has at least one vertex of S, or G has an edge set F with |F| ≥ f(k) such that G - F has no such walks. The 4-edge-connectivity is the best possible in the sense that 3-edge-connected graphs do not satisfy the statement. Since the proof is constructive, we can design a fixed-parameter algorithm for finding k edge-disjoint walks satisfying the conditions in a 4-edge-connected graph for a parameter k. In addition, this gives a simple fixed-parameter algorithm for the parity edge-disjoint walks problem with k terminal pairs.

AB - In this paper, we show the Erdos-Pósa property for edge-disjoint packing of S-closed walks with parity constraints in 4-edge-connected graphs. More precisely, we prove that for any 4-edge-connected graph G and any vertex subset S, either G has k edge-disjoint elementary closed odd walks, each of which has at least one vertex of S, or G has an edge set F with |F| ≥ f(k) such that G - F has no such walks. The 4-edge-connectivity is the best possible in the sense that 3-edge-connected graphs do not satisfy the statement. Since the proof is constructive, we can design a fixed-parameter algorithm for finding k edge-disjoint walks satisfying the conditions in a 4-edge-connected graph for a parameter k. In addition, this gives a simple fixed-parameter algorithm for the parity edge-disjoint walks problem with k terminal pairs.

KW - Cycle packing

KW - Erdos-Pósa property

KW - Fixed-parameter algorithm

UR - http://www.scopus.com/inward/record.url?scp=85021888664&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021888664&partnerID=8YFLogxK

U2 - 10.1137/15M1022239

DO - 10.1137/15M1022239

M3 - Article

VL - 31

SP - 766

EP - 782

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 2

ER -