  Schnyder characterized planar graphs in terms of order dimension. Brightwell
and Trotter proved that the dimension of the vertex-edge-face poset $\Pvef{M}$
of a planar map $M$ is at most four. In this paper we investigate cases where
$\dim(\Pvef{M}) \leq 3$ and also where $\dim(\Qvf{M}) \leq 3$; here $\Qvf{M}$
denotes the vertex-face poset of $M$. We show: - If $M$ contains a
$K_4$-subdivision, then $\dim(\Pvef{M}) = \dim(\Qvf{M}) = 4$. - If $M$ or the
dual $M^*$ contains a $K_{2,3}$-subdivision, then $\dim(\Pvef{M}) = 4$. Hence,
a map $M$ with $\dim(\Pvef{M}) \leq 3$ must be outerplanar and have an
outerplanar dual. We concentrate on the simplest class of such maps and prove
that within this class $\dim(\Pvef{M}) \leq 3$ 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 $\dim(\Qvf{M}) \leq 3$. There are, however, outerplanar maps
with $\dim(\Qvf{M}) = 4$. We construct the first such example.
