Hereditary properties of finite ultrametric spaces

Abstract

A characterization of finite homogeneous ultrametric spaces and finite ultrametric spaces generated by unrooted labeled trees has been made in terms of representing trees. Finite ultrametric spaces having perfect strictly n-ary trees have been characterized in terms of special graphs connected with the space. A detailed survey of some special classes of finite ultrametric spaces, which were considered in the last ten years, has been given and their hereditary properties have been studied. 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?