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.

翻译：本文研究平面图的外平面性，即在我们最初可自由选择的平面嵌入中，必须移除外层面顶点多少次才能使图变为空。众所周知，存在外平面性为 $\tfrac{n}{6}+\Theta(1)$ 的 $n$ 顶点图，且不难证明外平面性永远不会大于该值。本文给出形式为 $\tfrac{n}{2g}+2g+O(1)$ 的改进界，其中 $g$ 为围栏周长，即两侧均有顶点的最短环的长度。该参数 $g$ 至少为图的连通度，且通常更大；例如，我们的结果意味着平面二分图具有 $\tfrac{n}{8}+O(1)$ 的外平面性。我们还证明平面图 $G$ 的外平面性至多为 $\tfrac{1}{2}$diam$(G)+O(\sqrt{n})$，其中 diam$(G)$ 为图的直径。我们得到的所有界在低阶项范围内都是紧的，且可在线性时间内找到达到外平面性界的平面嵌入。