### Abstract

Define a minimal detour subgraph of the n-dimensional cube to be a spanning subgraph G of Q_{n} having the property that for vertices x, y of Q_{n}, distances are related by d_{G}(x, y) ≤ d_{Qn}(x, y) + 2. Let f(n) be the minimum number of edges of such a subgraph of Q_{n}. After preliminary work on distances in subgraphs of product graphs, we show that f(n)/|E(Q_{n}| < c/√n. The subgraphs we construct to establish this bound have the property that the longest distances are the same as in Q_{n}, and thus the diameter does not increase. We establish a lower bound for f(n), show that vertices of high degree must be distributed throughout a minimal detour subgraph of Q_{n}, and end with conjectures and questions.

Original language | English |
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Pages (from-to) | 119-128 |

Number of pages | 10 |

Journal | Journal of Graph Theory |

Volume | 23 |

Issue number | 2 |

DOIs | |

Publication status | Published - Oct 1996 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*23*(2), 119-128. https://doi.org/10.1002/(SICI)1097-0118(199610)23:2<119::AID-JGT3>3.0.CO;2-W