# New results on a visibility representation of graphs in 3D

## Abstract

This paper considers a 3-dimensional visibility representation of cliques *K*_{n}. In this representation, the objects representing the vertices are 2-dimensional and lie parallel to the *x, y*-plane, and two vertices of the graph are adjacent if and only if their corresponding objects see each other by a line of sight parallel to the *z*-axis that intersects the interiors of the objects. In particular, we represent vertices by unit discs and by discs of arbitrary radii (possibly different for different vertices); we also represent vertices by axis-aligned unit squares, by axis-aligned squares of arbitrary size (possibly different for different vertices), and by axis-aligned rectangles.

a significant improvement (from 102 to 55) of the best known upper bound for the size of cliques representable by rectangles or squares of arbitrary size;

a sharp bound for the representation of cliques by unit squares (

*K*_{7}can be represented but*K*_{n}for n>7 cannot);a representation of

*K*_{n}by unit discs.

### References

- 1.H. Alt, M. Godau and S. Whitesides. Universal 3-dimensional visibility representations for graphs. See elsewhere in these proceedings.Google Scholar
- 2.P. Bose, H. Everett, S. Fekete, A.Lubiw, H. Meijer, K. Romanik, T. Shermer and S. Whitesides. On a visibility representation for graphs in three dimensions.
*Proc. Graph Drawing '93*, Paris (Sèvres), 1993, pp. 38–39.Google Scholar - 3.P. Bose, H. Everett, S. Fekete, A. Lubiw, H. Meijer, K. Romanik, T. Shermer and S. Whitesides. On a visibility representation for graphs in three dimensions. Snapshots of Computational and Discrete Geometry, v. 3,
*eds*. D. Avis and P. Bose, McGill University School of Computer Science Technical Report SOCS-94.50, July 1994, pp. 2–25.Google Scholar - 4.R. Cohen, P. Eades, T. Lin and F. Ruskey. Three-dimensional graph drawing.
*Proc. Graph Drawing '94*, Princeton N J, 1994, Lecture Notes in Computer Science LNCS #894, Springer-Verlag, 1995, pp. 1–11.Google Scholar - 5.F. R. K. Chung. On unimodal subsequences.
*J. Combinatorial Theory*, Series A, v. 29, 1980, pp. 267–279.Google Scholar - 6.A. Dean and J. Hutchison. Rectangle visibility representations of bipartite graphs.
*Proc. Graph Drawing '94*, Princeton NJ, 1994. Lecture Notes in Computer Science LNCS #894, Springer-Verlag, 1995, pp. 159–166.Google Scholar - 7.J. M. Hammersley. A few seedlings of research.
*Proc. 6*^{th}Berkeley Symp. Math. Stat. Prob., U. of California Press, 1972, pp. 345–394.Google Scholar - 8.H. Koike. An application of three-dimensional visualization to object-oriented programming.
*Proc. of Advanced Visual Interfaces AVI '92*, Rome, May 1992, v. 36 of World Scientific Series in Computer Science, 1992, pp. 180–192.Google Scholar - 9.E. Kranakis, D. Krizanc and J. Urrutia. On the number of directions in visibility representations of graphs.
*Proc. Graph Drawing '94*, Princeton NJ, 1994, Lecture Notes in Computer Science LNCS #894, Springer-Verlag, 1995, pp. 167–176.Google Scholar - 10.J. Mackinley, G. Robertson and S. Card. Cone trees: animated 3d visualizations of hierarchical information.
*Proc. of the SIGCHI Conf. on Human Factors in Computing*, 1991, pp. 189–194.Google Scholar - 11.K. Romanik. Directed VR-representable graphs have unbounded dimension.
*Proc. Graph Drawing '94*, Princeton, NJ, 1994, Lecture Notes in Computer Science LNCS #894, Springer-Verlag, 1995, pp. 177–181.Google Scholar - 12.R. Tamassia and I. Tollis. A unified approach to visibility representations of planar graphs.
*Discrete Comput. Geom.*v. 1, 1986, pp. 321–341.CrossRefGoogle Scholar - 13.S. Wismath. Characterizing bar line-of-sight graphs.
*Proc. ACM Symp. on Computational Geometry*, 1985, pp. 147–152.Google Scholar