We prove the following variant of Levi's Enlargement Lemma: for an arbitrary arrangement $\mathcal{A}$ of $x$-monotone pseudosegments in the plane and a pair of points $a,b$ with distinct $x$-coordinates and not on the same pseudosegment, there exists a simple $x$-monotone curve with endpoints $a,b$ that intersects every curve of $\mathcal{A}$ at most once. As a consequence, every simple monotone drawing of a graph can be extended to a simple monotone drawing of a complete graph. We also show that extending an arrangement of cylindrically monotone pseudosegments is not always possible; in fact, the corresponding decision problem is NP-hard.

翻译：我们证明了Levi扩张引理的以下变体：对于平面上任意$x$-单调伪线段排列$\mathcal{A}$以及一对$x$坐标不同且不在同一伪线段上的点$a,b$，存在一条以$a,b$为端点的简单$x$-单调曲线，该曲线与$\mathcal{A}$中每条曲线最多相交一次。由此可得，任意图的简单单调画法均可扩展为完全图的简单单调画法。我们还证明了柱面单调伪线段排列的扩展并非总是可行；事实上，相应的判定问题是NP困难的。