Improved Outerplanarity Bounds for Planar Graphs

TL;DR

This paper improves bounds on the outerplanarity of planar graphs by relating it to parameters like fence-girth and diameter, providing tight bounds and linear-time algorithms for optimal embeddings.

Contribution

The authors establish new upper bounds on outerplanarity based on fence-girth and diameter, and show these bounds are tight with linear-time embedding algorithms.

Findings

Outerplanarity bounds of n/(2g)+2g+O(1) for planar graphs

Outerplanarity at most (1/2)diam(G)+O(√n)

Linear-time algorithms for optimal outerplanarity embeddings

Abstract

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.