Convex drawings of Planar Graphs and the Order Dimension of 3-Polytopes
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Freie Universitat Berlin, Fachbereich Mathematik und Informatik,
Takustr. 9, 14195 Berlin, Germany
We define an analogue of Schnyder's tree decompositions for 3-connected planar graphs. Based on this structure we obtain: Let G be a 3-connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3-connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3. The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.