  In this paper, we study the outerplanarity of planar graphs, i.e., the number
of times that we must (in a planar embedding that we can initially freely
choose) remove the outerface vertices until the graph is empty. It is
well-known that there are $n$-vertex graphs with outerplanarity
$\tfrac{n}{6}+\Theta(1)$, and not difficult to show that the outerplanarity can
never be bigger. We give here improved bounds of the form
$\tfrac{n}{2g}+2g+O(1)$, where $g$ is the fence-girth, i.e., the length of the
shortest cycle with vertices on both sides. This parameter $g$ is at least the
connectivity of the graph, and often bigger; for example, our results imply
that planar bipartite graphs have outerplanarity $\tfrac{n}{8}+O(1)$. We also
show that the outerplanarity of a planar graph $G$ is at most
$\tfrac{1}{2}$diam$(G)+O(\sqrt{n})$, where diam$(G)$ is the diameter of the
graph. All our bounds are tight up to smaller-order terms, and a planar
embedding that achieves the outerplanarity bound can be found in linear time.
