## Abstract

Let C_{H} be the class of graphs containing some fixed graph H as a minor. We define c^{v} _{H}(G) (resp. c^{e} _{H}(G)) as the minimun number of vertices (resp. edges) whose removal from G produces a graph without any subgraph isomorphic to a graph in C_{H}. Also p^{v} _{H}(G) (resp. p^{e} _{H}(G)) is the the maximum number of vertex- (resp. edge-) disjoint subgraphs of G that are isomorphic to some graph in C_{H}. We denote by θ_{r} the graph with two vertices and r parallel edges between them. When H = θ_{r}, the parameters c^{v/e} _{H} and p^{v/e} _{H} are NP-complete to compute (for sufficiently large r). In this paper we prove a series of combinatorial and algorithmic lemmata that imply that if p^{v/e} _{θr} (G) ≤ k, then c^{v/e} _{θr} (G) = O(k log k). This means that for every r, the class C_{θr} has the vertex/edge Erdős-Pósa property. Using the combinatorial ideas from our proofs we introduce a unified approach for the design of an O(log OPT)-approximation algorithm for c^{v} _{θr}, p^{v} _{θr}, c^{e} _{θr} and p^{e} _{θr} that runs in O(n ・ log(n) ・ m) steps.

Original language | English (US) |
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Title of host publication | Approximation and Online Algorithms - 13th International Workshop, WAOA 2015, Revised Selected Papers |

Editors | Martin Skutella, Laura Sanità |

Publisher | Springer Verlag |

Pages | 122-132 |

Number of pages | 11 |

ISBN (Print) | 9783319286839 |

DOIs | |

State | Published - 2015 |

Event | 13th International Workshop on Approximation and Online Algorithms, WAOA 2015 - Patras, Greece Duration: Sep 17 2015 → Sep 18 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9499 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 13th International Workshop on Approximation and Online Algorithms, WAOA 2015 |
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Country/Territory | Greece |

City | Patras |

Period | 9/17/15 → 9/18/15 |

## Keywords

- Covering
- Erdős-Pósa properties
- Graph packing
- Minors

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science

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