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Schnyder characterized planar graphs in terms of order dimension. Brightwell and Trotter proved that the dimension of the vertex-edge-face poset PM of a planar map M is at most four. In this paper we investigate cases where dim(PM) ≤ 3 and also where dim(QM) ≤ 3; here QM denotes the vertex-face poset of M. We show: If M contains a K4-subdivision, then dim(PM) = dim(QM) = 4. ...
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