Intersection Graphs of Rays and Grounded Segments

TL;DR

This paper explores the relationships between various classes of intersection graphs of segments and rays in the plane, resolving open problems and characterizing the complexity of recognition problems for these classes.

Contribution

It establishes equality and separation results among classes of segment and ray intersection graphs, solving open problems and confirming conjectures in the field.

Findings

Intersection graphs of grounded segments and downward rays are the same class.

Not all intersection graphs of rays are representable as downward rays.

Recognition problems for outer segment, grounded segment, and ray graphs are complete for the existential theory of the reals.

Abstract

We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that: (1) intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class, (2) not every intersection graph of rays is an intersection graph of downward rays, and (3) not every intersection graph of rays is an outer segment graph. The first result answers an open problem posed by Cabello and Jej\v{c}i\v{c}. The third result confirms a conjecture by Cabello. We thereby completely elucidate the remaining open questions on the containment relations between these classes of segment graphs. We further characterize the complexity of the recognition problems for the classes of outer segment, grounded segment, and ray intersection graphs. We prove that these recognition problems are complete for the existential theory of the reals. This holds even if a 1-string realization is given as additional input.