### Abstract

The well-known theorem of Erdo{double acute accent}s and Pósa says that a graph G has either k vertex-disjoint cycles or a vertex set X of order at most f(k) such that G\X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we generalize Erdo{double acute accent}s-Pósa's result to cycles that are required to go through a set S of vertices. Given an integer k and a vertex subset S (possibly unbounded number of vertices) in a given graph G, we prove that either G has k vertex-disjoint cycles, each of which contains at least one vertex of S, or G has a vertex set X of order at most f(k)=40k^{2}log_{2}k such that G\X has no cycle that intersects S.

Original language | English |
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Pages (from-to) | 378-381 |

Number of pages | 4 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 101 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2011 Sep 1 |

Externally published | Yes |

### Keywords

- Disjoint cycles
- Erdo{double acute accent}s-Pósa property
- Feedback vertex sets

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*101*(5), 378-381. https://doi.org/10.1016/j.jctb.2011.03.004