Voronoi drawings of trees

We study the problem of characterizing sets of points whose Voronoi diagrams are trees and if so, what are the combinatorial properties of these trees. The second part of the problem can be naturally turned into the following graph drawing question: Given a tree T, can one represent T so that the resulting drawing is a Voronoi diagram of some set of points? We investigate the problem both in the Euclidean and in the Manhattan metric. The major contributions of this paper are as follows. • We characterize those trees that can be drawn as Voronoi diagrams in the Euclidean metric. • We characterize those sets of points whose Voronoi diagrams are trees in the Manhattan metric. • We show that the maximum vertex degree of any tree that can be drawn as a Manhattan Voronoi diagram is at most five and prove that this bound is tight. • We characterize those binary trees that can be drawn as Manhattan Voronoi diagrams.

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