  In the area of beyond-planar graphs, i.e. graphs that can be drawn with some
local restrictions on the edge crossings, the recognition problem is prominent
next to the density question for the different graph classes. For 1-planar
graphs, the recognition problem has been settled, namely it is NP-complete for
the general case, while optimal 1-planar graphs, i.e. those with maximum
density, can be recognized in linear time. For 2-planar graphs, the picture is
less complete. As expected, the recognition problem has been found to be
NP-complete in general. In this paper, we consider the recognition of simple
optimal 2-planar graphs. We exploit a combinatorial characterization of such
graphs and present a linear time algorithm for recognition and embedding.
