### Abstract

The main question discussed in this paper is how well a finite metric space of size n can be embedded into a graph with certain topological restrictions. The existing constructions of graph spanners imply that any n-point metric space can be represented by a (weighted) graph with n vertices and n^{1+0(1/r)} edges, with distances distorted by at most r. We show that this tradeoff between the number of edges and the distortion cannot be improved, and that it holds in a much more general setting. The main technical lemma claims that the metric space induced by an unweighted graph H of girth g cannot be embedded in a graph G (even if it is weighted) of smaller Euler characteristic, with distortion less than g/4 - 3/2. In the special case when | V(G)\ = | V(H)| and G has strictly less edges than H, an improved bound of g/3 - 1 is shown. In addition, we discuss the case χ(G) < χ(H) - 1, as well as some interesting higher-dimensional analogues. The proofs employ basic techniques of algebraic topology.

Original language | English (US) |
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Pages (from-to) | 79-94 |

Number of pages | 16 |

Journal | Discrete and Computational Geometry |

Volume | 19 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1998 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Discrete and Computational Geometry*,

*19*(1), 79-94. https://doi.org/10.1007/PL00009336