Journal of Graph Algorithms and Applications

Special Issue on Parameterized and Approximation Algorithms in Graph Drawing

The Complexity of Drawing Graphs on Few Lines and Few Planes

Steven Chaplick, Krzysztof Fleszar, Fabian Lipp, Alexander Ravsky, Oleg Verbitsky, and Alexander Wolff

Vol. 27, no. 6, pp. 459-488, 2023. Regular paper.

Abstract 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 $d=2$, the graph $G$ 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 $NP \subseteq \exists\mathbb{R}$, 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 $\exists\mathbb{R} \subseteq PSPACE$, 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 $P=NP$.