Proximity Drawings: Three Dimensions Are Better than Two ⋆ (Extended Abstract) Paolo Penna 1 and Paola Vocca 2 1 Dipartimento di Scienze dell’Informazione, Universit`a di Roma “La Sapienza”, Via Salaria 113, I-00198 Rome, Italy. penna@mat.uniroma2.it, 2 Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Rome, Italy. vocca@axp.mat.uniroma2.it 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, gener- ally 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 co- ordinates 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. 1 Introduction. Three–dimensional drawings of graphs have received increasing attention re- cently due to the availability of low–cost workstations and of applications that require three–dimensional representations of graphs [6, 13, 18, 22, 20]. Even though there are several theoretical results [1, 7, 8, 9], there is still the need for a better theoretical understanding of three-dimensional capabilities. In this paper we tackle the problem of drawing proximity drawings in the three–dimensional space. Proximity drawings have been deeply investigated in the two–dimensional space because of their interesting graphical features (see, e.g. [17, 2, 5, 10, 11, 4, 16]). Nevertheless, only preliminary results are available for three–dimensional proximity drawings [15]. ⋆ Work partially supported by the Italian Project MURST-“Algorithms and Data Structure” S.H. Whitesides (Ed.): GD’98, LNCS 1547, pp. 275–287, 1998. c  Springer-Verlag Berlin Heidelberg 1998