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 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.KeywordsLinear OrderingDirected PathDirected Acyclic GraphComplete Bipartite GraphMinimum IntegerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.