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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
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References:

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