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.

翻译：我们显示,如果磁盘三角体具有所有内部脊椎度至少6度,那么完全三角体可以从边界顶端之间的对相图形距离中确定。至少4度的四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四度四次四度四度四度四度四度四次四度四度四度四次四度四度四次四度四次四度四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次四次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次三次