The Order Dimension of Planar Maps Revisited

• Published in: SIAM journal on discrete mathematics Vol. 28; no. 3; pp. 1093 - 1101
• Main Author: Felsner, Stefan
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
• Subjects: Codes; Graphs; Incidence; Machinery; Mathematical analysis; Proving; Set theory; Theorems
• ISSN: 0895-4801; 1095-7146
• DOI: 10.1137/130945284

Schnyder characterized planar graphs in terms of order dimension. The structures used for the proof have found many applications. Researchers also found several extensions of the seminal result. A particularly far-reaching extension is the Brightwell--Trotter theorem about planar maps. It states that the order dimension of the incidence poset $\mathbf{P}_{\mathbf{M}}$ of vertices, edges, and faces of a planar map $\mathbf{M}$ has dimension at most 4. The original proof generalizes the machinery of Schnyder paths and Schnyder regions. In this short paper we use a simple result about the order dimension of grid intersection graphs to show a slightly stronger result: $\dim(\mathsf{split}(\mathbf{P}_{\mathbf{M}})) \leq 4$. Here, $\mathsf{split}(P)$ refers to a particular order of height two associated with $P$. The Brightwell--Trotter theorem follows because $\dim(\mathsf{split}(P)) \geq \dim(P)$ holds for every $P$. This may be the first result in the area that is obtained without using the tools introduced by Schnyder. [PUBLICATION ABSTRACT]