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 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,…, <1 v for all 1 ≤ i ≤ d. In this note we show that the dimension of the class of directed VR-representable graphs is unbounded.