Determining triangulations and quadrangulations by boundary distances

We show that if all internal vertices of a disc triangulation have degree at least 6, then the full structure can be deter-mined from the pairwise graph distances between boundary vertices. A similar result holds for disc quadrangulations with all internal vertices having degree at least 4. This confirms a conjecture of Itai Benjamini. Both degree bounds are best possible, and correspond to local non-positive curvature. How-ever, we show that a natural conjecture for a "mixed" version of the two results is not true.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).