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

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 $\mathrm{\ell }(v)$ in ${\mathbb{R}}^3$ so that $vw$ is an edge in $E$ if and only if $\mathrm{\ell }(v)$ and $\mathrm{\ell }(w)$ 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 $\mathrm{\exists }\mathbb{R}$-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.