Subdominant pseudoultrametric on graphs

TL;DR

This paper characterizes when a weighted graph's edge weights can be extended to a pseudoultrametric on vertices, providing conditions for uniqueness and identifying graph structures where the subdominant pseudoultrametric is an ultrametric.

Contribution

It establishes necessary and sufficient conditions for extending weights to pseudoultrametrics and characterizes graphs with unique minimal pseudoultrametrics.

Findings

Conditions for extending weights to pseudoultrametrics

Graph structure characterization for unique minimal pseudoultrametrics

Criteria for subdominant pseudoultrametric to be an ultrametric

Abstract

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.