A Tipping Point for the Planarity of Small and Medium Sized Graphs

Abstract

This paper presents an empirical study of the relationship between the density of small-medium sized random graphs and their planarity. It is well known that, when the number of vertices tends to infinite, there is a sharp transition between planarity and non-planarity for edge density d = 0.5. However, this asymptotic property does not clarify what happens for graphs of reduced size. We show that an unexpectedly sharp transition is also exhibited by small and medium sized graphs. Also, we show that the same "tipping point" behavior can be observed for some restrictions or relaxations of planarity (we considered outerplanarity and near-planarity, respectively).

CoSEP: A compound spring embedder layout algorithm with support for ports

“…We tested our algorithm CoSEP in comparison to the CoSE algorithm with a test suite, the set named RND/BU4P in reference 25 of randomly generated biconnected, undirected, and 4-planar graphs, which has been widely used in graph drawing studies. 26,27 Among the 531 graphs available, we have used a set of 500 graphs which are considered to be small to medium-sized ( 10 − 500 nodes) 28,29 for interactive visualization purposes. When testing a particular graph, five executions are performed and the average values for the test criteria are taken.…”