Hereditary properties of finite ultrametric spaces

TL;DR

This paper characterizes finite ultrametric spaces using representing trees, explores their hereditary properties, and surveys recent special classes, providing insights into their structural and subspace properties.

Contribution

It offers new characterizations of finite ultrametric spaces via trees and graphs, and investigates hereditary properties of specific ultrametric classes.

Findings

Finite homogeneous ultrametric spaces characterized by representing trees.

Finite ultrametric spaces generated by unrooted labeled trees characterized.

Survey of recent classes of finite ultrametric spaces and their hereditary properties.

Abstract

A characterization of finite homogeneous ultrametric spaces and finite ultrametric spaces generated by unrooted labeled trees is found in terms of representing trees. A characterization of finite ultrametric spaces having perfect strictly $n$-ary trees is found in terms of special graphs connected with the space. Further, we give a detailed survey of some special classes of finite ultrametric spaces, which were considered in the last ten years, and study their hereditary properties. More precisely, we are interested in the following question. Let $X$ be an arbitrary finite ultrametric space from some given class. Does every subspace of $X$ also belong to this class?