The Complexity of Intersection Graphs of Lines in Space and Circle Orders

TL;DR

This paper proves that recognizing intersection graphs of lines in space and circle orders is computationally complex, specifically $orall ext{R}$-complete, linking these problems to solving polynomial systems over the reals.

Contribution

It establishes the $orall ext{R}$-completeness of recognition problems for both intersection graphs of lines in space and circle orders, resolving an open problem.

Findings

Recognition problems are $orall ext{R}$-complete.

Both problems are polynomial-time equivalent.

Addresses an open problem by Brightwell and Luczak.

Abstract

We consider the complexity of the recognition problem for two families of combinatorial structures. A graph $G=(V,E)$ is said to be an intersection graph of lines in space if every $v\in V$ can be mapped to a straight line $\ell (v)$ in $\mathbb{R}^3$ so that $vw$ is an edge in $E$ if and only if $\ell(v)$ and $\ell(w)$ intersect. A partially ordered set $(X,\prec)$ is said to be a circle order, or a 2-space-time order, if every $x\in X$ can be mapped to a closed circular disk $C(x)$ so that $y\prec x$ if and only if $C(y)$ is contained in $C(x)$. We prove that the recognition problems for intersection graphs of lines and circle orders are both $\exists\mathbb{R}$-complete, hence polynomial-time equivalent to deciding whether a system of polynomial equalities and inequalities has a solution over the reals. The second result addresses an open problem posed by Brightwell and Luczak.