## Abstract

The Induced Minor Containment problem takes as input two graphs G and H, and asks whether G has H as an induced minor. We show that this problem is fixed parameter tractable in | ^{VH}| if G belongs to any nontrivial minor-closed graph class and H is a planar graph. For a fixed graph H, the H-Contractibility problem is to decide whether a graph can be contracted to H. The computational complexity classification of this problem is still open. So far, H has a dominating vertex in all cases known to be solvable in polynomial time, whereas H does not have such a vertex in all cases known to be NP-complete. Here, we present a class of graphs H with a dominating vertex for which H-Contractibility is NP-complete. We also present a new class of graphs H for which H-Contractibility can be solved in polynomial time. Finally, we study the (H,v)-Contractibility problem, where v is a vertex of H. The input of this problem is a graph G and an integer k, and the question is whether G is H-contractible such that the "bag" of G corresponding to v contains at least k vertices. We show that this problem is NP-complete whenever H is connected and v is not a dominating vertex of H.

Original language | English (US) |
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Pages (from-to) | 799-809 |

Number of pages | 11 |

Journal | Discrete Applied Mathematics |

Volume | 160 |

Issue number | 6 |

DOIs | |

State | Published - Apr 2012 |

## Keywords

- Graph contraction
- Graph induced minor
- Graph minor

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics