  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.
