, Volume 5, Issue 4, pp 323-343

Planar graphs and poset dimension

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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.

Communicated by W. T. Trotter