The Order Dimension of Planar Maps Revisited

Abstract

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.