## Prescribed ultrametrics.(English)Zbl 0786.54031

Summary: Let $$G=(S,E)$$ be a subgraph of $$K_ n= (S,F)$$, the complete graph on $$n$$ vertices. Let $$\nu$$ be a function from $$E$$ to $$R^ +$$. 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:

 54E35 Metric spaces, metrizability 68R10 Graph theory (including graph drawing) in computer science 05C05 Trees 68Q25 Analysis of algorithms and problem complexity 54C20 Extension of maps

### Keywords:

metric; forest; ultrametric
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### References:

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