Intersection graphs of segments and $\exists\mathbb{R}$

Abstract

A graph $G$ with vertex set $\{v_1,v_2,\ldots,v_n\}$ is an intersection graph of segments if there are segments $s_1,\ldots,s_n$ in the plane such that $s_i$ and $s_j$ have a common point if and only if $\{v_i,v_j\}$ is an edge of~$G$. In this expository paper, we consider the algorithmic problem of testing whether a given abstract graph is an intersection graph of segments. It turned out that this problem is complete for an interesting recently introduced class of computational problems, denoted by $\exists\mathbb{R}$. This class consists of problems that can be reduced, in polynomial time, to solvability of a system of polynomial inequalities in several variables over the reals. We discuss some subtleties in the definition of $\exists\mathbb{R}$, and we provide a complete and streamlined account of a proof of the $\exists\mathbb{R}$-completeness of the recognition problem for segment intersection graphs. Along the way, we establish $\exists\mathbb{R}$-completeness of several other problems. We also present a decision algorithm, due to Muchnik, for the first-order theory of the reals.