The Order Dimension of Planar Maps Revisited

Abstract

Abstract. Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the Brightwell-Trotter Theorem about planar maps. It states that the order dimension of the incidence poset P M of vertices, edges and faces of a planar map M has dimension at most 4. The original proof generalizes the machinery of Schnyder-paths and Schnyder-regions. In this note we use a simple result about the order dimension of grid intersection graphs to show a slightly stronger result: dim(split(P M )) 4.This may be the rst result in the area that is obtained without using the tools introduced by Schnyder.Mathematics Subject Classications (2010) 05C10, 06A07, 68R10

Grid Intersection Graphs and Order Dimension

“…It has previously been observed that idim(G) ≤ 4 when G is a GIG [6]. As already shown in [13] this can be strengthened to dim(G) ≤ 4.…”

Tree-width and dimension

“…Actually, this was obtained as an easy corollary to the following theorems of Brightwell and Trotter [8,7], published in 1997 and 1993, respectively (a new and quite elegant proof of this result has just been obtained by Felsner [13]).…”

“…Returning to the general subject of the dimension of posets with planar cover graphs, Felsner, Li and Trotter [14] proved the following result in 2010: Actually, this was obtained as an easy corollary to the following theorems of Brightwell and Trotter [8,7], published in 1997 and 1993, respectively (a new and quite elegant proof of this result has just been obtained by Felsner [13]).…”

Intersection Graphs of Rays and Grounded Segments

“…They introduced several classes of intersection graphs that are the topic of this paper. Geometric intersection graphs are now ubiquitous in discrete and computational geometry, and deep connections to other fields such as complexity theory [20,25,26] and order dimension theory [8,9,11] have been established.…”