  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).
