Planar graphs and poset dimension
- Walter Schnyder
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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.
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- Planar graphs and poset dimension
Volume 5, Issue 4 , pp 323-343
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Primary 06A10
- secondary 05C10, 05C75
- Planar graph
- poset dimension
- straight line embedding
- Industry Sectors
- Walter Schnyder (1)
- Author Affiliations
- 1. Department of Mathematics, Louisiana State University, 70803-4918, Baton Rouge, LA, USA