# Directed VR-representable graphs have unbounded dimension

## Abstract

Visibility representations of graphs map vertices to sets in Euclidean space and express edges as visibility relations between these sets. A three-dimensional visibility representation that has been studied is one in which each vertex of the graph maps to a closed rectangle in ℝ^{3} and edges are expressed by vertical visibility between rectangles. The rectangles representing vertices are disjoint, contained in planes perpendicular to the *z*-axis, and have sides parallel to the *x* or *y* axes. Two rectangles *R*_{i} and *R*_{j} are considered visible provided that there exists a closed cylinder *C* of non-zero length and radius such that the ends of *C* are contained in *R*_{i} and *R*_{j}, the axis of *C* is parallel to the *z*-axis, and *C* does not intersect any other rectangle. A graph that can be represented in this way is called *VR-representable*.

A VR-representation of a graph can be directed by directing all edges towards the positive *z* direction. A directed acyclic graph *G* has *dimension d* if *d* is the minimum integer such that the vertices of *G* can be ordered by *d* linear orderings, <_{1},..., <_{d}, and for vertices *u* and *v* there is a directed path from *u* to *v* if and only if *u*<_{i}*v* for all 1 ≤*i* ≤*d*. In this note we show that the dimension of the class of directed VR-representable graphs is unbounded.

## Keywords

Linear Ordering Directed Path Directed Acyclic Graph Complete Bipartite Graph Minimum Integer## References

- [BEF94][BEF
^{+}94] Prosenjit Bose, Hazel Everett, Sandor Fekete, Anna Lubiw, Henk Meijer, Kathleen Romanik, Tom Shermer, and Sue Whitesides. On a Visibility Representation for Graphs in Three Dimensions. In David Avis and Prosenjit Bose, editors,*Snapshots in Computational and Discrete Geometry, Volume III*. McGill University, July 1994. Technical Report SOCS-94.50.Google Scholar - [BETT93]Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Algorithms for Automatic Graph Drawing: An Annotated Bibliography. Technical report, Department of Computer Science, Brown University, 1993.Google Scholar
- [BT88]Giuseppe Di Battista and Roberto Tamassia. Algorithms for Plane Representations of Acyclic Digraphs.
*Theoretical Computer Science*, 61:175–198, 1988.Google Scholar - [RU88]Ivan Rival and Jorge Urrutia. Representing Orders by Translating Convex Figures in the Plane.
*Order 4*, pages 319–339, 1988.Google Scholar - [RU92]Ivan Rival and Jorge Urrutia. Representing Orders by Moving Figures in Space.
*Discrete Mathematics*, 109:255–263, 1992.Google Scholar - [Tro92]William T. Trotter.
*Combinatorics and Partially Ordered Sets: Dimension Theory*. Johns Hopkins University Press, Baltimore, MD, 1992.Google Scholar - [TT86]Roberto Tamassia and Ioannis G. Tollis. A Unified Approach to Visibility Representations of Planar Graphs.
*Discrete Computational Geometry*, 1:321–341, 1986.Google Scholar - [Wis85]Stephen K. Wismath. Characterizing Bar Line-of-Sight Graphs. In
*Proceedings of the First Annual Symposium on Computational Geometry*, pages 147-152, Baltimore, MD, June 5–7 1985. ACM Press.Google Scholar