On spaces extremal for the Gomory-Hu inequality

TL;DR

This paper characterizes when the Gomory-Hu inequality for finite ultrametric spaces becomes an equality, providing necessary and sufficient conditions, counting non-isometric spaces, and exploring mappings from ultrametric to semimetric spaces.

Contribution

It introduces new conditions for equality in the Gomory-Hu inequality, counts non-isometric ultrametric spaces satisfying this, and shows how finite semimetric spaces relate to ultrametric spaces via specific mappings.

Findings

Necessary and sufficient conditions for equality in Gomory-Hu inequality.

Count of non-isometric ultrametric spaces with given spectrum.

Representation of finite semimetric spaces via ultrametric spaces and specific mappings.

Abstract

Let $(X,d)$ be a finite ultrametric space. In 1961 E.C. Gomory and T.C. Hu proved the inequality $|Sp(X)|\leqslant |X|$ where $Sp(X)=\{d(x,y)\colon x,y \in X\}$. Using weighted Hamiltonian cycles and weighted Hamiltonian paths we give new necessary and sufficient conditions under which the Gomory-Hu inequality becomes an equality. We find the number of non-isometric $(X,d)$ satisfying the equality $|Sp(X)|=|X|$ for given $Sp(X)$. Moreover it is shown that every finite semimetric space $Z$ is an image under a composition of mappings $f\colon X\to Y$ and $g\colon Y\to Z$ such that $X$ and $Y$ are finite ultrametric space, $X$ satisfies the above equality, $f$ is an $\varepsilon$-isometry with an arbitrary $\varepsilon>0$, and $g$ is a ball-preserving map.