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 $\exists\mathbb{R}$-complete, leaving unit segments and polylines as few remaining natural cases. We show that recognition for both families of objects is $\exists\mathbb{R}$-complete.

翻译：给定平面上一组对象 O，其对应的交图定义如下：为每个对象创建一个顶点，且当两个对应对象相交时，对应的顶点之间连一条边。本文研究单位线段和恰好具有 k 个转折点的折线的情况。在识别问题中，给定一个图，需要判定该图是否可以表示为某些几何对象的交图。已有研究表明，多种识别问题是 $\exists\mathbb{R}$-完全的，这使得单位线段和折线成为少数尚未解决的自然情形。我们证明，对于这两类对象，其识别问题均为 $\exists\mathbb{R}$-完全的。