Optimal right angles crossing graphs

Abstract

A graph is an optimal right angle crossing graph (also called an optimal RAC graph for short) if it has n vertices and 4n–10 edges and admits a straight-line drawing in the plane such that each edge is crossed at most once and edges cross only at a right angle. This implies that the drawing is 3T- or TTX-framed, that is, the outer face is a triangle that is adjacent to three triangles or to two triangles and a crossing. An optimal pseudo-RAC graph is the topological version of an optimal RAC graph, where the restrictions to straight-line edges and right angle crossings are dropped.

We show that every 3T-framed optimal pseudo-RAC graph is an optimal RAC graph, that is, 3T-framed optimal pseudo-RAC embeddings can be stretched and orthogonalized. This is not true for TTX-framed embeddings. There are n-vertex 3T- and TTX-framed optimal RAC graphs for every $n\ge 9$, and eleven optimal RAC and fourteen optimal pseudo-RAC graphs with at most eight vertices. Optimal pseudo-RAC graphs can be recognized in $O\left(n^3\right)$ time, where the recognition algorithm demonstrates that every optimal pseudo-RAC graph has at most three 1-planar embeddings, in which edges are crossed at most once.