Tree-based representation of discrete totally bounded ultrametric spaces

Establish the equivalence for every discrete nonempty totally bounded ultrametric space (X, d) between the following two conditions: (i) Construct a labeled tree T = T(l) whose induced ultrametric space (V(T), d_l) is isometric to (X, d); and (ii) Show that for every open ball B in (X, d) with positive diameter, at least one part of the diametrical graph of the metric subspace (B, d|_{B×B}) is a single-point set.

Background

The paper develops a general framework for representing totally bounded ultrametric spaces via labeled trees, extending the Gurvich–Vyalyi representation of finite ultrametric spaces. A key theme is linking metric properties of ultrametric spaces with combinatorial structures on trees and graphs.

In Section 10, the authors investigate when a labeled tree T = T(l) induces an ultrametric space (V(T), d_l) that coincides (up to isometry) with a given ultrametric space. For discrete totally bounded ultrametric spaces, they propose a precise characterization in terms of the diametrical graph of each ball, asking for a singleton part in that graph.

The conjecture encapsulates this characterization as an equivalence. The authors note partial progress: it has been proven for finite ultrametric spaces and one direction is known for certain locally finite settings, but the full equivalence remains unresolved in the general discrete totally bounded case.