  A graph is NIC-planar if it admits a drawing in the plane with at most one
crossing per edge and such that two pairs of crossing edges share at most one
common end vertex. NIC-planarity generalizes IC-planarity, which allows a
vertex to be incident to at most one crossing edge, and specializes
1-planarity, which only requires at most one crossing per edge.
  We characterize embeddings of maximal NIC-planar graphs in terms of
generalized planar dual graphs. The characterization is used to derive tight
bounds on the density of maximal NIC-planar graphs which ranges between
3.2(n-2) and 3.6(n-2). Further, we prove that optimal NIC-planar graphs with
3.6(n-2) edges have a unique embedding and can be recognized in linear time,
whereas the general recognition problem of NIC-planar graphs is NP-complete. In
addition, we show that there are NIC-planar graphs that do not admit right
angle crossing drawings, which distinguishes NIC-planar from IC-planar graphs.
