### Abstract

We attempt to shed new light on the notion of ‘tree-like’ metric spaces by focusing on an approach that does not use the four-point condition. Our key question is: Given metric space M on n points, when does a fully labelled positive-weighted tree T exist on the same n vertices that precisely realises M using its shortest path metric? We prove that if a spanning tree representation, T, of M exists, then it is isomorphic to the unique minimum spanning tree in the weighted complete graph associated with M, and we introduce a fourth-point condition that is necessary and sufficient to ensure the existence of T whenever each distance in M is unique. In other words, a finite median graph, in which each geodesic distance is distinct, is simply a tree. Provided that the tie-breaking assumption holds, the fourth-point condition serves as a criterion for measuring the goodness-of-fit of the minimum spanning tree to M, i.e., the spanning tree-likeness of M. It is also possible to evaluate the spanning path-likeness of M. These quantities can be measured in O(n^{4}) and O(n^{3}) time, respectively.

Original language | English |
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Pages (from-to) | 51-57 |

Number of pages | 7 |

Journal | Discrete Applied Mathematics |

Volume | 226 |

DOIs | |

Publication status | Published - 2017 Jul 31 |

Externally published | Yes |

### Keywords

- Median graph
- Minimum spanning tree
- Tree-like metric space

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*226*, 51-57. https://doi.org/10.1016/j.dam.2017.04.001