Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends

TL;DR

This paper introduces algorithms for drawing beyond-planar graphs with right-angle crossings, achieving compact grid representations with minimal bends, thus advancing graph visualization techniques.

Contribution

It presents new algorithms for embedding-preserving RAC drawings of NIC-planar, 1-planar, and IC-planar graphs with optimized bends and grid sizes.

Findings

NIC-planar graphs admit RAC drawings with one bend per edge on O(n) x O(n) grid.

1-planar graphs admit RAC drawings with two bends per edge on O(n^3) x O(n^3) grid.

Existing algorithms for 1-planar and IC-planar graphs are extended to preserve embeddings and improve drawing quality.

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 $O(n) \times O(n)$. 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 $O(n^3) \times O(n^3)$. 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.