2016/10/27 by Brandenburg, Franz J. · 1 citation
#05C85 #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.1610.08884
We prove that triangulated IC-planar and NIC-planar graphs can be recognized in cubic time. A graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. A drawing is IC-planar if, in addition, each vertex is incident to at most one crossing edge and NIC-planar if two pairs of crossing edges share at most one vertex. In a triangulated drawing each face is a triangle. In consequence, planar-maximal and maximal IC-planar and NIC-planar graphs can be recognized in O(n5) time and maximum and optimal ones in O(n3) time. In contrast, recognizing 3-connected IC-planar and NIC-planar graphs is NP-complete, even if the graphs are given with a rotation system which describes the cyclic ordering of the edges at each vertex. Our results complement similar ones for 1-planar graphs.