Compact drawings of 1-planar graphs with right-angle crossings and few bends

Abstract

We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two pairs of crossing edges share two vertices.We study the relations of these beyond-planar graph classes (beyond-planar graphs is a collective term for the primary attempts to generalize the planar graphs) to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends. We present four drawing algorithms that preserve the given embeddings. First, we show that every n-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size O(n)×O(n). Then, we show that every n-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size O(n3)×O(n3). Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line.