Determining triangulations and quadrangulations by boundary distances

Haslegrave, John (2023) Determining triangulations and quadrangulations by boundary distances. Journal of Combinatorial Theory, Series B, 163. pp. 233-255. ISSN 0095-8956

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Abstract

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.