# Proximity Drawings: Three Dimensions Are Better than Two

## Abstract

We consider weak Gabriel drawings of unbounded degree trees in the three-dimensional space. We assume a minimum distance between any two vertices. Under the same assumption, there exists an exponential area lower bound for general graphs. Moreover, all previously known algorithms to construct (weak) proximity drawings of trees, generally produce exponential area layouts, even when we restrict ourselves to binary trees. In this paper we describe a *linear-time polynomial-volume* algorithm that constructs a *strictly-upward weak Gabriel drawing* of any *rooted tree* with *O*(log *n*)-bit requirement. As a special case we describe a Gabriel drawing algorithm for binary trees which produces integer coordinates and *n* ^{3}-*area* representations. Finally, we show that an infinite class of graphs requiring exponential area, admits *linear-volume Gabriel drawings.* The latter result can also be extended to *β*-drawings, for any 1 < *β* < 2, and relative neighborhood drawings.

## Keywords

Binary Tree Proximity Region Adjacent Vertex Outerplanar Graph Antipodal Point## References

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