WADS 2017: Algorithms and Data Structures pp 265-276

# The Complexity of Drawing Graphs on Few Lines and Few Planes

• Steven Chaplick
• Krzysztof Fleszar
• Fabian Lipp
• Alexander Ravsky
• Oleg Verbitsky
• Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

## 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 $$\mathrm {P}=\mathrm {NP}$$.

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