Improved Outerplanarity Bounds for Planar Graphs

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 n6+ (1), and not difficult to show that the outerplanarity can never be bigger. We give here improved bounds of the form n2g+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 n8+O (1). We also show that the outerplanarity of a planar graph G is at most 12diam (G) +O (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.