## Abstract

In this paper we consider the following question: how many vertices of the discrete torus must be deleted so that no topologically nontrivial cycles remain? We look at two different edge structures for the discrete torus. For (ℤ _{m} ^{d} ) _{1}, where two vertices in ℤ _{m} are connected if their ℓ _{1} distance is 1, we show a nontrivial upper bound of d ^{log} _{2}(3/2)m ^{d-1} d ^{0.6}m ^{d-1} on the number of vertices that must be deleted. For (ℤ _{m} ^{d} ) _{∞}, where two vertices are connected if their ℓ _{∞} distance is 1, Saks et al. (Combinatorica 24(3):525-530, 2004) already gave a nearly tight lower bound of d(m-1) ^{d-1} using arguments involving linear algebra. We give a more elementary proof which improves the bound to m ^{d} -(m-1) ^{d} , which is precisely tight.

Original language | American English |
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Pages (from-to) | 446-454 |

Number of pages | 9 |

Journal | Algorithmica |

Volume | 50 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2008 |

Externally published | Yes |

## Keywords

- Discrete torus
- Foam
- Multicut
- Tiling