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Forbidden Four Cycle, Star Graphs and Isometric Embeddings

Oleksiy Dovgoshey, Olga Rovenska

TL;DR

This work characterizes when an infinite ultrametric space admits an isometric embedding into a space generated by labeled star graphs (a $oldsymbol{US}$-space). The authors reduce the problem to four-point subspace configurations, showing that embedding is possible iff every four-point diametrical graph is one of $K_{1,1,1}$, $K_{1,1,2}$, or $K_{1,3}$ (equivalently, no four-point subspace is weakly similar to $(X_4,d_4)$ or $(Y_4, ho_4)$). The proof combines a metric adjustment to produce a Cauchy sequence, an embedding into a star-graph ultrametric via completion, and a lifting argument to recover the original space, yielding a practical criterion for embedding. The results link four-point invaraints to US-embeddability and inspire conjectures about embeddings into special ultrametric spaces like $(oldsymbol{R}^+, d^+)$, with equidistant spaces characterized by a distinct four-point graph type.

Abstract

We prove the necessary and sufficient conditions under which ultrametric spaces of arbitrary infinite cardinality admit isometric embeddings into ultrametric spaces generated by labeled star graphs.

Forbidden Four Cycle, Star Graphs and Isometric Embeddings

TL;DR

This work characterizes when an infinite ultrametric space admits an isometric embedding into a space generated by labeled star graphs (a -space). The authors reduce the problem to four-point subspace configurations, showing that embedding is possible iff every four-point diametrical graph is one of , , or (equivalently, no four-point subspace is weakly similar to or ). The proof combines a metric adjustment to produce a Cauchy sequence, an embedding into a star-graph ultrametric via completion, and a lifting argument to recover the original space, yielding a practical criterion for embedding. The results link four-point invaraints to US-embeddability and inspire conjectures about embeddings into special ultrametric spaces like , with equidistant spaces characterized by a distinct four-point graph type.

Abstract

We prove the necessary and sufficient conditions under which ultrametric spaces of arbitrary infinite cardinality admit isometric embeddings into ultrametric spaces generated by labeled star graphs.

Paper Structure

This paper contains 6 sections, 25 theorems, 101 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries. Metric spaces
  3. Preliminaries. Graphs
  4. Preliminaries. Star graphs and US-spaces
  5. Lemmas and Main result
  6. Conclusion. Expected results

Key Result

Proposition 2.2

Let $(X,d)$ be an ultrametric space with $|X|\geq 2$. If $x_1$ and $x_2$ are two distinct points of $X$ such that for all distinct points $y_1,y_2 \in X$, then the mapping $\Phi : X \to X$ defined, for every $x\in X$, as is a self-isometry of the space $(X,d)$.

Figures (6)

  • Figure 1: The four-point ultrametric spaces $(X_4,d_4)$ and $(Y_4,\rho_4)$.
  • Figure 2: The diametrical graphs of $(X_4,d_4)$ and $(Y_4,\rho_4)$.
  • Figure 3: The diametrical graph $G_{X,\rho}$.
  • Figure 4: $G_{A,d_A}$ is isomorphic to one of the graphs $K_{1,1,1,1}$, $K_{1,1,2}$ or $K_{1,3}$.
  • Figure 5: The four-point ultrametric spaces $(W_4,\delta_4)$ and its diametrical graph $G_{W_4,\delta_4}$.
  • ...and 1 more figures

Theorems & Definitions (55)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 45 more