Let (G, w) be a weighted graph. The necessary and sufficient conditions under which a weight w : E(G) → R + can be extended to a pseudoultrametric on V (G) are found. A criterion of the uniqueness of this extension is also obtained. It is proved that G is complete k-partite with k ≥ 2 if and only if, for every pseudoultrametrizable weight w, there exists the smallest pseudoultrametric agreed with w. We characterize the structure of graphs for which the subdominant pseudoultrametric is an ultrametric for every strictly positive pseudoultrametrizable weight.
We give an elementary proof of the classical Menger result according to which any metric space X that consists of more than four points is isometrically imbedded into R if every three-point subspace of X is isometrically imbedded into R. A series of corollaries of this theorem is obtained. We establish new criteria for finite metric spaces to be isometrically imbedded into R.
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