On the Order Dimension of Outerplanar Maps

Schnyder characterized planar graphs in terms of order dimension. Brightwell and Trotter proved that the dimension of the vertex-edgeface 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. • IfM or the dualM∗ contains aK2,3-subdivision, then dim(PM ) = 4. Hence, a map M with dim(PM ) ≤ 3 must be outerplanar and have an outerplanar dual. We concentrate on the simplest class of such maps and prove that within this class dim(PM ) ≤ 3 is equivalent to the existence of a certain oriented coloring of edges. This condition is easily checked and can be turned into a linear time algorithm returning a 3-realizer. Additionally, we prove that if M is 2-connected and M and M∗ are outerplanar, then dim(QM ) ≤ 3. There are, however, outerplanar maps with dim(QM ) = 4. We construct the first such example.