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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 ν be a function from E to R + . We prove two theorems on the extensibility of ν. Every function ν extends to a metric on F iff G is a forest. The function ν extends to an ultrametric on F if and only if for all non-trivial cycles p in G, mult(p)>1, where mult(p) depends on the values of ν on paths.
MSC:
54E35Metric spaces, metrizability
68R10Graph theory in connection with computer science (including graph drawing)
05C05Trees
68Q25Analysis of algorithms and problem complexity
54C20Extension of maps on topological spaces