Simultaneous Geometric Embedding (SGE) asks whether, for a given collection of graphs on the same vertex set V, there is an embedding of V in the plane that admits a crossing-free drawing with straightline edges for each of the given graphs. It is known that SGE is $\exists\mathbb{R}$-complete, that is, the problem is polynomially equivalent to deciding whether a system of polynomial equations and inequalities with integer coefficients has a real solution. We prove that SGE remains $\exists\mathbb{R}$-complete for edge-disjoint input graphs, that is, for collections of graphs without so-called public edges. As an intermediate result, we prove that it is $\exists\mathbb{R}$-complete to decide whether a directional walk without repeating edges is realizable. Here, a directional walk consists of a sequence of not-necessarily distinct vertices (a walk) and a function prescribing for each inner position whether the walk shall turn left or shall turn right. A directional walk is realizable, if there is an embedding of its vertices in the plane such that the embedded walk turns according to the given directions. Previously it was known that realization is $\exists\mathbb{R}$-complete to decide for directional walks repeating each edge at most 336 times. This answers two questions posed by Schaefer ["On the Complexity of Some Geometric Problems With Fixed Parameters", JGAA 2021].

翻译：同时几何嵌入（SGE）问题询问：对于同一顶点集V上给定的一组图，是否存在V在平面上的嵌入，使得每个给定图都能用直线边实现无交叉绘制。已知SGE问题是$\exists\mathbb{R}$-完全的，即该问题在多项式时间上等价于判定一个具有整数系数的多项式方程与不等式系统是否存在实数解。我们证明，对于无公共边输入图（即无所谓公共边的图集合），SGE问题仍保持$\exists\mathbb{R}$-完全性。作为中间结果，我们证明判定无重复边的有向行走是否可实现也是$\exists\mathbb{R}$-完全的。这里，有向行走包含一个顶点序列（顶点可重复）和一个函数，该函数为每个中间位置指定行走应左转还是右转。若有向行走可实现，则存在其顶点在平面上的嵌入，使得嵌入后的行走按给定方向转弯。此前已知对于每条边最多重复336次的有向行走，其可实现性判定是$\exists\mathbb{R}$-完全的。这回答了Schaefer提出的两个问题[《关于某些固定参数几何问题的复杂性》，JGAA 2021]。