The Complexity of Drawing Graphs on Few Lines and Few Planes

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 minimum number of lines in $\mathbb {R} ^d$ that together can cover all edges of a drawing of G. For $d=2$, G must be planar. We investigate the complexity of computing these 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.

• Let $\rho ^2_3(G)$ be the minimum number of planes in $\mathbb {R} ^3$ needed to cover a straight-line drawing of a graph G. 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 $\mathrm {P}=\mathrm {NP}$.

K. Fleszar was supported by Conicyt PCI PII 20150140 and Millennium Nucleus Information and Coordination in Networks RC130003.

F. Lipp was supported by Cusanuswerk.

O. Verbitsky was supported by DFG grant VE 652/1-2.