Intersection graphs of segments and ∃R

A graph G with vertex set {v1, v2,..., vn} is an intersection graph of segments if there are segments s1,..., sn in the plane such that si and sj have a common point if and only if {vi, vj} 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 ∃R. This class consists of problems that can be reduced, in polynomial time, to solv-ability of a system of polynomial inequalities in several variables over the reals. We discuss some subtleties in the definition of ∃R, and we provide a complete and streamlined account of a proof of the ∃R-completeness of the recognition problem for segment intersection graphs. Along the way, we establish ∃R-completeness of several other problems. We also present a decision algorithm, due to Muchnik, for the first-order theory of the reals.