The Complexity of Drawing Graphs on Few Lines and Few Planes

TL;DR

The paper investigates the computational complexity of covering graph drawings with a small number of lines or planes, focusing on $\rho^1_2(G)$, $\rho^1_3(G)$, and $\rho^2_3(G)$. It establishes that deciding $\rho^1_2(G)\le k$ and $\rho^1_3(G)\le k$ is $\exists\mathbb{R}$-complete, but these problems are fixed-parameter tractable in $k$ via kernelization to $O(k^4)$ and Renegar's algorithm, enabling an $k^{O(k^2)}+O(n+m)$-time decision procedure and a combinatorial FPT description. In contrast, deciding $\rho^2_3(G)\le k$ is NP-hard for fixed $k\ge2$, ruling out FPT algorithms under standard assumptions. Additionally, there exist graphs whose $\rho^1_2$-optimal drawings require irrational coordinates, illustrating subtle realizability issues, and the paper extends the complexity landscape to weak affine cover numbers $\pi^l_3(G)$, proving NP-hardness and approximation barriers. Collectively, the results delineate sharp contrasts between line-based and plane-based cover problems and highlight rich connections to existential geometry, kernelization, and planarity-related SAT variants.

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 $ρ^1_d(G)$ 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 $ρ^2_3(G)$ 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 $ρ^1_d(G)\le k$ for a given graph $G$ and integer $k$ is ${\exists\mathbb{R}}$-complete. - Since $\mathrm{NP}\subseteq{\exists\mathbb{R}}$, deciding $ρ^1_d(G)\le k$ 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 $ρ^1_2(G)$ and $ρ^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $ρ^1_2$ or $ρ^1_3$ sometimes require irrational coordinates. - We prove that deciding whether $ρ^2_3(G)\le k$ is NP-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $\mathrm{P}=\mathrm{NP}$.

Figures (5)

• Figure 1: Two drawings of the same planar 3-tree in 2D and a drawing of $K_5$ in 3D. In the 3D drawing (c), the four white vertices lie in a common (gray) plane, whereas the black vertex lies above it.
• Figure 2: A simple line arrangement of four lines (depicted in black) with the corresponding arrangement graph and the modification used in the proof of Theorem \ref{['thm:rho12-hard']}.
• Figure 3: The Perles configuration and a supergraph of it.
• Figure 8: The intersection line gadget and how it is depicted in Fig. \ref{['fig:rho-np-graph-construction']}.
• Figure 11: Blocking gadget that occupies a whole plane. Again, the clauses are depicted by the boxes and the dashed line is the intersection of the $k$ planes.

Theorems & Definitions (15)

• Theorem 1
• Lemma 2
• Corollary 3
• Theorem 4
• Theorem 5: Renegar Renegar92aRenegar92bRenegar92c
• Corollary 6
• Theorem 7
• Theorem 8
• Definition 9
• Definition 10