Determining triangulations and quadrangulations by boundary distances

TL;DR

This paper proves that certain boundary distances uniquely determine triangulations and quadrangulations with specific degree conditions, confirming a conjecture and highlighting the role of non-positive curvature.

Contribution

It establishes that boundary distances uniquely determine these maps under degree constraints, confirming a conjecture of Itai Benjamini and showing the bounds are optimal.

Findings

Boundary distances determine triangulations with internal degree ≥6

Boundary distances determine quadrangulations with internal degree ≥4

Degree bounds are optimal and relate to non-positive curvature

Abstract

We show that if a disc triangulation has all internal vertex degrees at least 6, then the full triangulation may be determined from the pairwise graph distance between boundary vertices. A similar result holds for quadrangulations with all internal degrees at least 4. This confirms a conjecture of Itai Benjamini. Both degree bounds are best possible, and correspond to local non-positive curvature. However, we show that a natural conjecture for a "mixed" version of the two results is not true.