1-Bend RAC Drawings of NIC-Planar Graphs in Quadratic Area

A drawing of a graph is called 1-planar if every edge is crossed at most once. A 1-planar drawing is called independent-crossing planar (IC-planar) if no two pairs of crossing edges share a vertex. A 1-planar drawing is called near-independent-crossing planar (NIC-planar) if any two pairs of crossing edges share at most one vertex. The 1-planar, NIC-planar, and IC-planar graphs are the graphs that admit a 1-planar, NIC-planar, and IC-planar drawing, respectively. The NIC-planar graphs are a subset of the 1-planar graphs and a superset of the IC-planar graphs, which are important beyond-planar graph classes. We constructively show that every n-vertex NIC-plane graph admits a NIC-planar drawing with only right-angle crossings (RAC) and at most one bend per edge on a grid of size O(n) × O(n). Our construction takes linear time. We also give an overview of the relationships between several classes of 1-planar and RAC graphs.