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

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.

Figures (3)

• Figure 1: A line realization of the graph $H$ constructed from a realizable rank-3 matroid on 5 elements with rank-2 flats $\mathcal{S}=\{\{1,2\}, \{2,3\}, \{3,4\}, \{1,4\}, \{1,3,5\}, \{2,4,5\}\}$.
• Figure 2: A pseudoline arrangement and an isomorphic line arrangement.
• Figure 3: A Krupp, and a great-pseudocircle arrangement realizing two copies of the pseudoline arrangement of Figure \ref{['fig:lines']}.

Theorems & Definitions (10)

• theorem 1
• theorem 2
• lemma 1
• proof
• proof : Proof of Theorem \ref{['thm:lineIntersection']}
• lemma 2
• lemma 3
• proof
• theorem 3: Kang and Müller KM14, Felsner and Scheucher FS18
• proof : Proof of Theorem \ref{['thm:circle']}