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Zbl 0786.54031
Higgins, J.; Campbell, D.
Prescribed ultrametrics. (English)
[J] RAIRO, Inform. Théor. Appl. 27, No.1, 1-5 (1993). ISSN 0988-3754

Summary: Let $G=(S,E)$ be a subgraph of $K\sb n= (S,F)$, the complete graph on $n$ vertices. Let $\nu$ be a function from $E$ to $R\sp +$. We prove two theorems on the extensibility of $\nu$. Every function $\nu$ extends to a metric on $F$ iff $G$ is a forest. The function $\nu$ extends to an ultrametric on $F$ if and only if for all non-trivial cycles $p$ in $G$, $\text{mult}(p)>1$, where $\text{mult}(p)$ depends on the values of $\nu$ on paths.

MSC 2000:: *54E35 Metric spaces, metrizability
68R10 Graph theory in connection with computer science
05C05 Trees
68Q25 Analysis of algorithms and problem complexity
54C20 Extension of maps on topological spaces

Keywords: metric; forest; ultrametric

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