Abstract
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.
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Felsner, S. Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes. Order 18, 19–37 (2001). https://doi.org/10.1023/A:1010604726900
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DOI: https://doi.org/10.1023/A:1010604726900