NIC-Planar Graphs

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 show that optimal NICplanar graphs with 3.6(n − 2) edges have a unique embedding and can be recognized in linear time, whereas the recognition problem of NICplanar graphs is NP-complete. In addition, we show that there are NICplanar graphs that do not admit right angle crossing drawings, which distinguishes NIC-planar from IC-planar graphs.