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Quasi-isometries between graphs with variable edge lengths

James Davies, Meike Hatzel, Robert Hickingbotham

TL;DR

The paper addresses whether multiplicative distortion in quasi-isometries between graphs with variable edge lengths can be removed by adjusting edge weights. It constructs a counterexample: for every $C$, there exist graphs $G$ and $H$ and a $(2,1)$-quasi-isometry $\varphi: V(G)\to V(H)$ such that no edge weighting $w$ on $H$ yields a $(1,C)$-quasi-isometry $\varphi: G\to (H,w)$. The proof uses a high-girth, high-chromatic graph $H$, a vertex-splitting construction to obtain $G$, and a case analysis based on light/heavy edges and geodesic paths to derive contradictions, thereby showing multiplicative distortion cannot be dispensed with in general. This refutes the NSS2025treewidth conjecture and highlights fundamental limits in coarse graph theory regarding edge-weight tuning for quasi-isometries.

Abstract

This paper investigates quasi-isometries between graphs with variable edge lengths. A quasi-isometry is a mapping between metric spaces that approximately preserves distances, allowing for a bounded amount of additive and multiplicative distortion. Recently, Nguyen, Scott, and Seymour conjectured that, by appropriately adjusting the edge lengths of the target graph along with modifying the additive distortion constant, the multiplicative distortion factor could be eliminated. We disprove this conjecture.

Quasi-isometries between graphs with variable edge lengths

TL;DR

The paper addresses whether multiplicative distortion in quasi-isometries between graphs with variable edge lengths can be removed by adjusting edge weights. It constructs a counterexample: for every , there exist graphs and and a -quasi-isometry such that no edge weighting on yields a -quasi-isometry . The proof uses a high-girth, high-chromatic graph , a vertex-splitting construction to obtain , and a case analysis based on light/heavy edges and geodesic paths to derive contradictions, thereby showing multiplicative distortion cannot be dispensed with in general. This refutes the NSS2025treewidth conjecture and highlights fundamental limits in coarse graph theory regarding edge-weight tuning for quasi-isometries.

Abstract

This paper investigates quasi-isometries between graphs with variable edge lengths. A quasi-isometry is a mapping between metric spaces that approximately preserves distances, allowing for a bounded amount of additive and multiplicative distortion. Recently, Nguyen, Scott, and Seymour conjectured that, by appropriately adjusting the edge lengths of the target graph along with modifying the additive distortion constant, the multiplicative distortion factor could be eliminated. We disprove this conjecture.

Paper Structure

This paper contains 5 sections, 6 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof
  4. Conclusion
  5. Acknowledgement:

Key Result

Theorem 1

For all $L,C\in \mathbb{N}$, there exists $C'\in \mathbb{N}$ such that if a graph $G$ is $(L, C)$-quasi-isometric to a tree, then $G$ is $(1,C')$-quasi-isometric to a tree.

Figures (1)

  • Figure 1: Starting with $H$, we orientate the edges to obtain $\stackon[0.2ex]{H}{ }$ then we split the vertices to obtain $G$.

Theorems & Definitions (12)

  • Theorem 1: CDNRV2012Kerr2023
  • Theorem 2: NSS2025treewidth
  • Conjecture 3: NSS2025treewidth
  • Theorem 4
  • Theorem 5: burr1980subtrees
  • proof : Proof of \ref{['MainTheorem']}
  • Claim 1
  • Claim 2
  • Claim 3
  • Corollary 6
  • ...and 2 more