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 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 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.
Supported by IRIS National Network of Centres of Excellence, NSERC, and DIMACS. DIMACS is an NSF Science and Technology Center, funded under contract STC-88-09648, and also receives support from the New Jersey Commission on Science and Technology.
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© 1995 Springer-Verlag Berlin Heidelberg
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Romanik, K. (1995). Directed VR-representable graphs have unbounded dimension. In: Tamassia, R., Tollis, I.G. (eds) Graph Drawing. GD 1994. Lecture Notes in Computer Science, vol 894. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58950-3_369
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DOI: https://doi.org/10.1007/3-540-58950-3_369
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