It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the smallest number of lines in $\mathbb{R}^d$ whose union contains a crossing-free straight-line drawing of $G$. For $d=2$, $G$ must be planar. Similarly, let $\rho^2_3(G)$ denote the smallest number of planes in $\mathbb{R}^3$ whose union contains a crossing-free straight-line drawing of $G$. We investigate the complexity of computing these three parameters and obtain the following hardness and algorithmic results. - For $d\in\{2,3\}$, we prove that deciding whether $\rho^1_d(G)\le k$ for a given graph $G$ and integer $k$ is ${\exists\mathbb{R}}$-complete. - Since $\mathrm{NP}\subseteq{\exists\mathbb{R}}$, deciding $\rho^1_d(G)\le k$ is NP-hard for $d\in\{2,3\}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. - Since ${\exists\mathbb{R}}\subseteq\mathrm{PSPACE}$, both $\rho^1_2(G)$ and $\rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $\rho^1_2$ or $\rho^1_3$ sometimes require irrational coordinates. - We prove that deciding whether $\rho^2_3(G)\le k$ is NP-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $\mathrm{P}=\mathrm{NP}$.

翻译：众所周知，任何图在$\mathbb{R}^3$中都存在无交叉直线型绘制，且任何平面图在$\mathbb{R}^2$中也存在同样的绘制。对于图$G$和$d \in \{2,3\}$，设$\rho^1_d(G)$表示$\mathbb{R}^d$中包含$G$的无交叉直线型绘制所需的最少直线数量。对于$d=2$，$G$必须为平面图。类似地，设$\rho^2_3(G)$表示$\mathbb{R}^3$中包含$G$的无交叉直线型绘制所需的最少平面数量。我们研究了计算这三个参数的复杂度，并得到了以下困难性结果和算法结果： - 对于$d\in\{2,3\}$，我们证明：对于给定图$G$和整数$k$，判定是否$\rho^1_d(G)\le k$是${\exists\mathbb{R}}$-完备的。 - 由于$\mathrm{NP}\subseteq{\exists\mathbb{R}}$，对于$d\in\{2,3\}$，判定$\rho^1_d(G)\le k$是NP难的。从正面看，我们证明该问题关于参数$k$是固定参数可处理的。 - 由于${\exists\mathbb{R}}\subseteq\mathrm{PSPACE}$，$\rho^1_2(G)$和$\rho^1_3(G)$均可在多项式空间内计算。从负面看，我们证明关于$\rho^1_2$或$\rho^1_3$的最优绘制有时需要无理数坐标。 - 我们证明：对于任意固定$k \ge 2$，判定是否$\rho^2_3(G)\le k$是NP难的。因此，除非$\mathrm{P}=\mathrm{NP}$，该问题关于参数$k$不是固定参数可处理的。