Directed VR-representable graphs have unbounded dimension
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 ℝ3 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 Ri and Rj 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 Ri and Rj, 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<iv for all 1 ≤i ≤d. In this note we show that the dimension of the class of directed VR-representable graphs is unbounded.
KeywordsLinear Ordering Directed Path Directed Acyclic Graph Complete Bipartite Graph Minimum Integer
- [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