## Abstract

The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to boundedsize subproblems. One of the main open problems in this direction is whether k-Path admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size k^{O(1)}? We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and K_{3,t}-minor-free graphs. Moreover, we show that k-Path even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path has a separation that can safely be reduced after communication with the oracle.

Original language | English |
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Title of host publication | 12th International Symposium on Parameterized and Exact Computation, IPEC 2017 |

Editors | D. Lokshtanov, N. Nishimura |

Place of Publication | Dagstuhl |

Publisher | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |

Number of pages | 13 |

ISBN (Electronic) | 978-3-95977-051-4 |

DOIs | |

Publication status | Published - 1 Feb 2018 |

Event | 12th International Symposium on Parameterized and Exact Computation (IPEC 2017) - Vienna, Austria Duration: 6 Sep 2017 → 8 Sep 2017 Conference number: 12 https://algo2017.ac.tuwien.ac.at/ipec |

### Publication series

Name | Leibniz International Proceedings in Informatics (LIPIcs) |
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Volume | 89 |

### Conference

Conference | 12th International Symposium on Parameterized and Exact Computation (IPEC 2017) |
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Abbreviated title | IPEC 2017 |

Country | Austria |

City | Vienna |

Period | 6/09/17 → 8/09/17 |

Internet address |

## Keywords

- Excluded topological minor
- K-path
- Long path
- Modulator
- Turing kernel