Axis-Parallel Right Angle Crossing Graphs

TL;DR

This paper introduces and studies axis-parallel RAC (apRAC) graphs, a subclass of RAC graphs with crossings restricted to axis-parallel segments, exploring their properties, bounds, and drawing algorithms.

Contribution

It establishes inclusion relationships, bounds on edge density, and provides a linear-time drawing algorithm for 2-bend apRAC graphs, advancing understanding of this graph class.

Findings

apRAC graphs have specific inclusion relationships with RAC graphs

Maximum degree 8 graphs are 2-bend apRAC with a linear drawing algorithm

Some results improve the understanding of general RAC graph properties

Abstract

A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in which each crossing occurs at a right angle. Originally motivated by psychological studies on readability of graph layouts, RAC graphs form one of the most prominent graph classes in beyond planarity. In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or apRAC, for short), that restricts the crossings to pairs of axis-parallel edge-segments. apRAC drawings combine the readability of planar drawings with the clarity of (non-planar) orthogonal drawings. We consider these graphs both with and without bends. Our contribution is as follows: (i) We study inclusion relationships between apRAC and traditional RAC graphs. (ii) We establish bounds on the edge density of apRAC graphs. (iii) We show that every graph with maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some of our results on apRAC graphs also improve the state of the art for general RAC graphs. We conclude our work with a list of open questions and a discussion of a natural generalization of the apRAC model.