Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, Gabriel graph, or modified Gabriel graph of a set of points in the plane. The characterizations give rise to linear-time algorithms for determining whether a tree has such a drawing; if such a drawing exists one can be constructed in linear time in the real RAM model. The characterization of Gabriel graphs settles several conjectures of Matula and Sokal [17].

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doi.org/10.1007/BF02086609
Algorithmica
School of Computer Science

Bose, P, Lenhart, W. (W.), & Liotta, G. (G.). (1996). Characterizing Proximity Trees. Algorithmica, 16(1), 83–110. doi:10.1007/BF02086609