We show that if all internal vertices of a disc triangulation have degree at least 6, then the full structure can be determined 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. However, we show that a natural conjecture for a “mixed” version of the two results is not true.

中文翻译：

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通过边界距离确定三角剖分和四边形剖分

我们证明，如果圆盘三角剖分的所有内部顶点的度数至少为 6，则可以根据边界顶点之间的成对图距离确定完整结构。类似的结果也适用于所有内部顶点的度数至少为 4 的圆盘四边形。这证实了 Itai Benjamini 的猜想。两个度数界限都是最好的，并且对应于局部非正曲率。然而，我们表明，对两个结果的“混合”版本的自然猜想是不正确的。