Right Angle Crossing Graphs and 1-planarity

Abstract

A Right Angle Crossing Graph (also called RAC graph for short) is a graph that has a straight-line drawing where any two crossing edges are orthogonal to each other. A 1-planar graph is a graph that has a drawing where every edge is crossed at most once. We study the relationship between RAC graphs and 1-planar graphs in the extremal case that the RAC graphs have as many edges as possible. It is known that a maximally dense RAC graph with n∈>∈3 vertices has 4n - 10 edges. We show that every maximally dense RAC graph is 1-planar. Also, we show that for every integer i such that i∈≥∈0, there exists a 1-planar graph with n∈=∈8∈+∈4i vertices and 4n - 10 edges that is not a RAC graph. © 2012 Springer-Verlag Berlin Heidelberg.