Recognition of Unit Segment and Polyline Graphs is $\exists\mathbb{R}$-Complete

Abstract

Given a set of objects $O$ in the plane, the corresponding intersection graph is defined as follows. Each object defines a vertex and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit segments and polylines with exactly $k$ bends. In the recognition problem, we are given a graph and want to decide whether the graph can be represented as an intersection graph of certain geometric objects. In previous work it was shown that various recognition problems are $\exists\mathbb{R}$-complete, leaving unit segments and polylines among the few remaining natural cases where the recognition complexity remained open. We show that recognition for both families of objects is $\exists\mathbb{R}$-complete.