## Abstract

In his PhD thesis [1], Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G, denoted Cn(G), deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies two conditions: each path between distinct essential vertices (vertices of degree not equal to 2) is of length at least n+. 1 edges, and each path from a vertex to itself which is not nullhomotopic is of length at least n+. 1 edges. Using Forman's discrete Morse theory for CW-complexes, we show the first condition can be relaxed to require only that each path between distinct essential vertices is of length at least n- 1.

Original language | English (US) |
---|---|

Pages (from-to) | 136-145 |

Number of pages | 10 |

Journal | Topology and its Applications |

Volume | 178 |

DOIs | |

State | Published - Sep 30 2014 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2014 Elsevier B.V.

## Keywords

- Configuration space
- Discrete Morse theory
- Graph braid group