# Planar graphs and poset dimension

- Received:
- Accepted:

DOI: 10.1007/BF00353652

- Cite this article as:
- Schnyder, W. Order (1989) 5: 323. doi:10.1007/BF00353652

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## Abstract

We view the incidence relation of a graph *G=(V. E)* as an order relation on its vertices and edges, i.e. *a*<_{G}*b* if and only of *a* is a vertex and *b* is an edge incident on *a*. This leads to the definition of the *order-dimension* of *G* as the minimum number of total orders on *V ∪ E* whose intersection is <_{G}. Our main result is the characterization of planar graphs as the graphs whose order-dimension does not exceed three. Strong versions of several known properties of planar graphs are implied by this characterization. These properties include: each planar graph has arboricity at most three and each planar graph has a plane embedding whose edges are straight line segments. A nice feature of this embedding is that the coordinates of the vertices have a purely combinatorial meaning.