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：给定平面上的对象集合$O$，其对应的交集图定义如下：每个对象对应一个顶点，当两个对象相交时，其对应顶点之间连一条边。本文研究恰好包含$k$个转折点的单位线段与折线情形。在识别问题中，我们已知一个图，需要判断该图是否能够表示为特定几何对象的交集图。先前的研究表明，多种识别问题均为$\exists\mathbb{R}$-完全的，而单位线段与折线是剩余少数自然情形。我们证明这两类对象的识别问题均属于$\exists\mathbb{R}$-完全类。