Recognition of Unit Segment and Polyline Graphs is ∃ℝ-Complete

Given a set of objects O in the plane, the corresponding intersection graph is defined as follows. A vertex is created for each object 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 the intersection graph of certain geometric objects. In previous work it was shown that various recognition problems are ∃ℝ-complete, leaving unit segments and polylines as few remaining natural cases. We show that recognition for both families of objects is ∃ℝ-complete.